Trigonometry Total concept Shortcut tricks Smart method

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[00:00:00]Hi, good morning everyone. So today very important topic discussion. So here the most important is a trigonometry. The trigonometry is nothing but the trigonometry. It is a uses of all competitive exam or IIT or 10th class or any competitive exam here uses of trigonometry topic. This is the most most important topic. So in uses of trigonometry topic but all competitive exam. So this is a uses of all students is lengthy method. So here now I'm using your shortcut method. How to use the shortcut method in trigonometry in IIT.

[00:00:35]So look here the trigonometry definition. First definition of the trigonometry trigonometry tree is mean tree is mean three tree is mean three.

[00:00:45]Go is mean. Gono is mean angle angle. Mry is mean.

[00:00:54]Mry is nothing but measures.

[00:00:58]measures. So the trigonometry is nothing but tree is mean tree go is mean angle metry is mean measure. The definition of the trigonometry the definition of the trigonometry is a the measures of three angles and three sides known as is a trigonometry. So the measurements of three angles the measurements of the measurements measurements of measurements of angles angles and so here sides three sides the measurements of angles and sides known as is a trigonometry. The definition of the trigonometry the measurements of three angles and three sides it is called trigonometry. Right? So what is the trigonometry? The most important you are using trigonometry in trigonometry you're using the right triangle. Right triangle. So right triangle right triangle. So right triangle is nothing but what is the definition of the right triangle? The trigonometry here uses of more than times in right triangle triangle. So what is the definition of the right triangle triangle? The any angle is exactly 90°. The any angle is exactly 90° that is called as a right triangle. The any angle one angle is a 90° that is called is a right triangle.

[00:02:23]So now I'm draw the right triangle here.

[00:02:25]So this is a right triangle. So here 90° here 90°. So this is exactly right angle here. Now I'm taking theta here. Here now I'm taking theta. Theta is nothing but the theta is nothing but acute angle. Theta is nothing but acute angle or any angle. Not only acute angle, it's any angle. But here, so acute angle is a definition more than of 0° more than of more than of 0°. More than of 0° less than less than 90°. So more than of 0° more than of 0° less than 90°. The more than of 0° less than 90° less than 90° it is called as a acute angle. Right?

[00:03:10]The acute angle definition finished the right triangle is mean the any angle is a 90° that is called is a right triangle. So here theta so here remaining angle is a 90 - theta 90 minus theta. So 90 minus theta here 90 minus theta. So what is a look here suppose the sum of the three angles is 180° here 90° 90° plus here theta plus 90 minus theta 90 minus theta here both are three angles in a right triangle so this is a 90° this is a theta this is a remaining is a 90 minus theta sum of the three angles in a triangle 180° 180° so here 90 90 plus plus theta minus theta cancel remaining 90 is equal to 180°. So 90 + 90 180 is equal to 180 both are same. So both are equal. So here the most important key point here very important in every exam competitive exam you are asking this type of question here. So, sum of the some of the some of the in a right triangle in a right triangle in a right triangle in a right triangle sum of the other two sides other two sides other two sides in a right triangle sum of the other two sides is how much so here for example alpha plus beta now I'm taking here for example here triangle right triangle again right triangle so here 90° °. So remaining here theta here alpha. So alpha plus theta. Alpha plus theta is equal how many degrees?

[00:04:52]Remaining 90°. So 90°. Alpha is also acute angle. Theta is also acute angle.

[00:05:00]Most most very important. So in a right triangle sum of the other two sides is a 90°. Each one angle is a acute angle.

[00:05:08]Each one angle is a acute angle. The question here exam purpose competitive exam purpose you're asking here sometimes here in a right triangle in a right triangle other two sides other two sides is dash angles other two sides is dash angle that is a acute angle so why because this is acute angle this is acute angle suppose look here so now I'm draw the again now I'm draw the here right angle triangle so again right triangle triangle so suppose here right triangle So this is a right angle. So this is a right angle. It is a 90° it is a 90° here. Now I'm taking suppose here 30° here 60° here 30° 60° 90 30 + 60 90 90 + 90 180.

[00:05:58]So here satisfied. So each one angle this is also this is also acute angle.

[00:06:04]Acute angle. This is also acute angle.

[00:06:06]This is also acute angle. So acute angle. So this is also acute angle. So 60° is acute angle. 30° is acute angle.

[00:06:15]So here what is the important key point here? The sum of the the sum of the in a right triangle sum of the other two sides is a or sum of the other two angles is 90° or sometimes question asking in a right triangle in a right triangle other two angles is dash acute angle. This is a 60 is acute angle 30 is also acute angle. It is very very very important. or sometimes other draw the other right angle here. So this is a 90° 90° now I'm taking here 45 here also 45 both are here 45 45 is a acute angle 45 is also acute angle in the specially your key point is what the in a right triangle other two angles is a acute angles the sum of the other two angles is also 90° it is very very important concept right in exam purpose all competitive exam purpose you're asking the more than times this type of question Right. So listen. So remaining the right triangle. So right triangle right triangle. So right triangle right triangle. So here right triangle.

[00:07:25]So right triangle rather than I'm taking here for example this is a right triangle.

[00:07:30]So this is a right triangle. So here this is exactly 90°. 90° 90° remaining here. Now I'm taking here theta theta here. Take the theta here possible also here possible but don't take here why because already here 90° so here 90° so here theta so 90° exactly opposite very very very important the 90° opposite side the 90° opposite side is always hypotenus hypotenus so very very important hypotenus the theta theta exactly opposite is a opposite side opposite side opposite side. Very very important. The remaining is a opposite side opposite side. Opposite side. So here 90° exactly opposite hypotenus. Every time very important 90° opposite side is always take the hypotenus the exactly theta here. Now I'm taking theta here.

[00:08:32]Same theta exactly opposite is opposite side. The remaining side is it is nothing but adjacent adjacent side.

[00:08:41]Adjacent side it is very important adjacent side. So okay. So sometimes opposite side is nothing but perpendicular. So sometimes opposite side is nothing but perpendicular. So P is mean nothing but perpendicular. So here this is a base. This is a base.

[00:08:57]Very very important case. So first here perpendicular opposite side it is a base. Sometimes here take the theta. Tha exactly opposite. This is opposite. This is a adjacent side. But here hypotenus is no change. In a right triangle the largest side in a right triangle the largest side is a hypotenus. Very very important. So exact 90° is hypotenus.

[00:09:21]Right? So here base perpendicular and hypotenus. Right? Now I'm taking here this is a name. Now I'm taking for example this name is a a this name is a a. So now I'm taking this name is A.

[00:09:34]This is a A. This is a B. This is a C.

[00:09:37]So angle B. Angle B is equal to how many degrees? 90°. So angle B is nothing but right. Right angle. Right angle. So here right angle. The angle B is it is nothing but right angle. So okay. Right.

[00:09:55]This is remaining here the here the uses of the trigonometry ratios. the trigonometry ratios here. How to get the trigonometry ratios? The trigonometry ratios is a the here the trigonometry ratios here three trigonometry ratios.

[00:10:12]The primary primary three trigonometry ratios is a sin theta sin theta sin theta theta is nothing but angle. So remaining sin theta and remaining cos theta remaining tan theta most most very important sin theta cos theta tan theta.

[00:10:31]So sin theta, cos theta, tan theta. This is a primary three trigonometry ratios.

[00:10:38]The primary three trigonometry ratios is a sin theta, cos theta, tan theta. It is most most important. But actually total how many ratios in trigonometry? Six ratios. But here primary ratios is a three. What is that? Sin theta, cos theta, tan theta. Right? So sin theta.

[00:10:59]Sin theta is nothing but use the formulas here very very important sin theta formulas here so now I'm here taking the sin theta formula first number one so sin theta sin theta formula opposite side opposite side OP short name here opposite side opposite side okay opposite side by hypotenus opposite side by hypotenus opposite side by hypotenus is a sin theta formula second one remaining cos theta formula cos theta formula adjacent side adjacent side short name here adjacent side by hypotenus now I'm taking here third one is a tan theta so tan theta is nothing but opposite side by so adjacent side it is a three trigonometry ratios here primary three trigonometry ratios s theta is opposite side by hypotenus the cos theta is the adjacent side by hypotenus the tan theta is the opposite by adjacent side right this is a three trigonometry ratios Here remaining sin theta reciprocal. So here sin theta reciprocal is a sin theta reciprocal is a cosec theta. Sin theta reciprocal is a cosec theta. So cosec theta cosec theta cosec theta is can be written as hypotenus by short name cosec theta is nothing but opposite opposite side hypotenus by opposite opposite. So cosec theta is can be written as hypotenus by opposite side. So remaining cos theta the cos theta the cos theta. So your cos theta reciprocal is a se sec se theta se theta is can be written as reverse hypotenus by adjacent side adjacent side. This is a second theta. Second theta remaining tan theta reciprocal is a c theta. So tan theta reciprocal is a c theta. Reciprocal is nothing but the multiple the any number we get the one that is a reciprocal values opposite by adjacent. This is a reciprocal can be written as adjacent side by opposite side. This is a co theta. So the total trigonometric six ratios here six ratios. What is that sin theta? Sin theta is mean opposite side by hypotenus. Cos theta is mean adjacent side by hypotenus. Tan theta or tangent is a name actually tangent. Tangent is nothing but tan theta in the short form.

[00:13:20]So tan theta is equal to opposite side by adjacent. It is a cosecant not cosecant actually name is a cosecant but short name is cosec theta cosec theta is reciprocal sign. So can be written as hypotenus by opposite side. The cos theta reciprocal second theta. Second theta is nothing but hypotenus by adjacent side. C theta is equal to adjacent side by opposite side. This is a six trigonometry ratios. This is most most important. This is a you're using your formulas. But here so remaining is a very very important the shortcut. So lot of students here confused say more than of students here confused in trigonometry formulas. But now I'm taking here the shortcut here. So now I'm taking this is a right triangle.

[00:14:06]This is a right triangle. So it is a right triangle. Right? So this is a 90°.

[00:14:11]90° exactly opposite is a hypotenus.

[00:14:14]short name hypotenus hypotenus so this is a theta theta is nothing but exactly theta opposite is a opposite side opposite side short name now I'm taking so remaining this is a adjacent side adjacent side so adjacent side adjacent side so here 90° exactly opposite hypotenus theta exactly opposite opposite side this is remaining is a adjacent side so but now I'm taking here so very important here so this is opposite is nothing but now I'm taking perpendicular P is mean perpendicular so adjacent side is nothing but what base is nothing but B so sometimes adjacent sometimes base is the same name so adjacent is nothing but base opposite side is nothing but perpendicular perpendicular is P so remaining hypotenus is no change it is as it is right hypotenus right so it is a short form now I'm taking for example so here first so First now I'm taking so first now I'm taking here sin theta so first I'm taking sin theta sin theta comma cos theta comma pan theta it is three primary trigonometry ratios so now I'm taking sin theta what is the sin theta is a short name now I'm taking your short name so here short name now I'm taking your short name so pbh so first now I'm write the pbh So PBH PBH PBH PB PBP is a PBP. Okay. PBP is a shortcut. So PBP okay.

[00:15:59]So hi high B is mean papi high it is a shortcut it is a easy it is easy how so p by h p by h p by h is mean hypotenus p by h is mean opposite by hypotenus it is a what sin theta sin theta sin theta this is a sin theta so remaining base adjacent by opposite so adjacent by look here so here PBH so adjacent side adjacent side is what so adjacent side is a so base by base by hypotenus so base by hypotenus is nothing but cos theta now I'm taking cos theta here so here look here so pp P.

[00:16:58]Okay. P D Papu Ble Papu Ble Papu P. So here short name P D P here highi high H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D So Papu Ble Papu II D.

[00:17:15]So here perpendicular by H, perpendicular by H opposite by hypotenus, it is a sin theta. So it is a sin theta. It is a sin theta. Remaining B. B is mean base by H is mean hypotenus. Base is mean adjacent side.

[00:17:30]By hypotenus is a what? Here cos theta.

[00:17:33]Cos theta. Remaining t opposite opposite b adjacent t it is a tan theta. It is a tan theta. It is a directly now I'm taking here. So simple shortcut here again remaining. So here now I'm here.

[00:17:48]So pu ble papu. So h b so papu ble papu highi high bul this is a short name. So here now I'm taking here so it is easy. So here first numerator now I'm taking numerator is mean sin theta sin theta. So here remaining sin theta reciprocal cosec theta cosec theta. So cosec theta cosec theta sin theta formula perpendicular by height or perpendicular is nothing but look here the perpendicular is nothing but perpendicular is mean opposite side. So perpendicular is mean opposite side it is a short name. So base base is mean adjacent side adjacent side adjacent side remaining P is mean opposite side.

[00:18:44]So opposite side opposite side it is a B is mean adjacent side adjacent side it is a it is easy look here. So sin theta is mean so it is H is mean hypotenus hypotenus hypotenus. So look here sin theta is mean opposite side by hypotenus P by H. So cosec theta is reverse okay hypotenus by opposite short name so it is a cos theta it is a cos theta it is a cos theta here so now I'm taking here cos theta so cos theta cos theta is a numerator denominator is a cos reciprocal same second theta how cos theta is mean base by height or adjacent by hypotenus second theta is mean hypotenus by adjacent side So remaining it is a tan tan tan is a numerator. So opposite side. So adjacent side is what here? So adjacent side. So adjacent side. So here tan reciprocal cut. So this is a very very important. This is very very important concept. This is very very important. Easy easy method here. Okay. The formulas here learn the easy method here. What is that? Sin theta is a opposite side by hypotenus.

[00:19:58]So cosec theta is a hypotenus by opposite. cos theta formula adjacent side by hypotenus or second theta formula hypotenus by adjacent side. So remaining ta formula opposite side by adjacent side or carta formula adjacent by opposite this is easy learning method so this is easy method so most most important lot of students here confuse okay so don't confuse here this is a short method shortcut here so okay clear so sa formula cost thea formula it is finished so here now I'm taking some problems okay so competitive exam some problems is given so more than times you are asking the question okay How to ask in the questions. Look here.

[00:20:39]So now I'm draw the right triangle. So here take the right triangle. So this is a right triangle. This is a right triangle. So this is a 90° 90° exactly opposite hypotenus. Theta exactly opposite opposite side. So opposite side. So this is a adjacent side.

[00:20:56]Adjacent side. Adjacent side. Look here.

[00:20:59]So here now I'm taking here. This is a uh three. This is a four. This is a five. So Pythagorean triplets 3 4 5 3 4 5 You ask the some questions here. So remaining the sin theta first sin theta formula sin theta is how much remaining cos theta is how much remaining tan theta is how much dash. So here sin theta sin theta formula is what? Sin theta formula actually opposite side by hypotenus opposite is what here? Three three by hypotenus is how much here?

[00:21:35]Five. So sin theta formula cos theta cos theta formula is what? Adjacent side by hypotenus. So cos theta is equal to what? Adjacent is what here? Four by hypotenus is what here? Five. Remaining tan theta. Tan theta formula opposite side by adjacent side. This is a tan theta formula. So tan theta opposite is what here? Three by adjacent is how much here? Four. So this is it sin reciprocal is what sin reciprocal is what cosec cosec theta. So sin theta reciprocal is cosec theta. Sin theta is reciprocal theta is mean reciprocal is mean reverse 3x 5. So 3x 5. So reverse is what 5x3 it is a cosec. So cos reciprocal is what?

[00:22:20]Second theta. Second theta. Cos reciprocal is a second theta. So 4x 5 4x 5 is nothing but reverse 5x4. So remaining here. So remaining here. So 5x4. This is a 5x4. This is a 5x3.

[00:22:35]Remaining tan theta reciprocal is what c theta. So tan theta reciprocal c theta it is a reverse is how much? So 4x3 it is a sometimes you are asking the question so sin theta how much cos theta how much tan theta how much or cosec theta second theta tan theta this type of question you are asking in exam purpose all competitive exam so right so now I'm taking here the remaining concept so here some problems so actually this type of take the this method or right angle here rather here also this is a right angle also this is a right angle it is a 90° 90° exactly opposite hypotenus. So suppose you take the theta theta exactly opposite opposite side it is a very very important basic concept is very important remaining this is a adjacent side adjacent side so this is a 90° exactly opposite hypotenus always 90° exactly opposite hypotenus theta exactly opposite opposite side adjacent s suppose now I'm take taking the theta here so now I'm taking theta here 90° hypotenus so here theta here not taking here take the theta that and theta exactly opposite opposite side. Okay.

[00:23:46]The theta opposite side is opposite 90° exactly opposite hypot is nothing but so adjacent side. Adjacent side. So this is very very important basic concept. First of all first of all learn the basic concept. It is finished. So remaining remaining some problems here. Observe here some problems. Okay. Right. So some problems here looking uh some problem now take the some problems. So here right. So here now I'm taking the some problems.

[00:24:20]So here now I'm taking some problems.

[00:24:24]So how to ask the in exam purpose? Look here. So here the question is this question is given the question is look here. This question is given look here.

[00:24:37]The question is uh so here the question is question is now I'm taking here question so now I'm taking the question some question so here question is so here I questions very very important more than times repeated questions this is a all competitive exam how to use the short method or smart method don't use a long method here. Okay. What is for now I'm taking first second question. So question number two. What is the question here? If if if pi sin theta if 5 sin theta is equal to 3. If 5 sin theta is equal to 3 then then c theta is equal to how much? Here c theta is equal to how much? Question mark. Option A is given. Option A is given. Option A is how much you given?

[00:25:30]Option A. So here 3x4 option B is given.

[00:25:35]So 4x 5 so option C is given so 3x 5 option D is given 4x3 this is the options is given so how to use this question so look here the first of all now I'm taking the here so here right triangle here use the right triangle so first now I'm taking the question sin theta 5 sin theta is equal nothing but three so here five is a multiple it becomes two division the sin theta is nothing but 3x 5 sin theta is nothing but 3x 5 but 3x 5 the pythagorian triplet here 3a 5 and now I'm draw the right triangle right triangle so look here so here so sin theta formula opposite side by opposite side by hypotenus other we learn so 90° 90° exactly opposite hypotenus theta exactly opposite opposite side it is a adjacent side look here opp Opposite side is how much? Three. Three is a three. So remaining hypotenus is how much? Here five. Already three five. The remaining pythagorian triplet very very important.

[00:26:43]Pythagorean triplets is most most important. So here three five. Remaining pythagoran triplet is a four. Remaining pythagoran triplet is a four. So how to use the pythagoran formula in a in a right triangle? In a right triangle any two sides is given. Find the third side that time use the Pythagorean formula or Pythagoras theorem. Pythagoras theorem it is a what? Hypotenus short name.

[00:27:06]Hypotenus square is equal to opposite side whole square plus adjacent side whole square. This is a Pythagoras formula. This is a Pythagoras formula.

[00:27:15]Hypotenus square is equal to opposite square plus adjacent square. What is the definition of the Pythagoras theorem?

[00:27:21]The hypotenus square is equal to sum of the other two side squares. It is called as a Pythagoras theorem. So look here hypotenus hypotenus is how much here? Pi five square is equal opposite.

[00:27:36]So look here opposite is how much?

[00:27:38]Three. Opposite is a three. Suppose this is a don't know. For example, this is a don't know. Four is don't know. So how to find adjacent is mean don't know. Now I'm taking this is x x. So here 5 square is mean 25. So 3 square is mean 9 + x².

[00:27:52]So 25 + 9 is becomes to - 9 x². So here x² is equal to 25 - 9 is a 16. So root is applying on both side it is we will get x= 4. This is a four. So 3 five is a 3 five is given that I'm directly write the four. This is a pythog. So okay right. So now I'm taking here. So actually ask the question is what look here. So 3 pi 4. This is a four. This is a four. What is c theta? You're asking c theta. Ca formula adjacent side by adjacent side by opposite side. This is a C theta formula. The C theta C theta is equal to adjacent adjacent side is what? Here adjacent is a four by opposite side is how much? Three. The answer is 4x3. 4x3. 4x3 is a which option? Option D is a correct answer.

[00:28:44]This is a the concept method. Very very important. This type of question you are given in more than times or competitive exam or IIT also. Right? So this is clear. So remaining now I'm taking other.

[00:28:57]So here look here the remaining is here.

[00:29:00]So remaining here also. Now I'm taking for example question. This is very very important question. So here so remaining. So four tan a is equal to three. Then value of sign. This is very important. So now I'm taking fifth question. Fifth question. What is the fifth question here? 4 tan a 4 tan a = 3. 4 tan a= 3 the value of the value of sin a plus the it is a question is very very important so it is very very important question so here sin a + cos a sin a by sin a + cos a by sin a minus cos a you're asking the question so right so here using the options here also option a option a is a - 7 option b is also 7 option c is also 2x one. So option B is also 1 by two. This is a options is given in this question, right? It is a what is the method here?

[00:29:58]Tan very very important. So here so tan tan a is equal to 3 4 is a multiple becomes to division. So tan theta formula can be written as sin a tan theta actually tan theta formula tan theta formula sin theta by sin theta. So tan theta formulas very important here.

[00:30:20]So sometimes now I'm using the formulas here also. So here tan theta formula tan theta formula sin theta by sin theta by cos theta or c theta formula cos theta by sin theta. So remaining so this is most important. So tan theta is a sin theta by cos theta c theta is cos theta by sin theta. Right? So here look here.

[00:30:45]So 4 tan a is equal to 3. So tan a = 3x 4 3x 4 tan a can be written as sin a by cos a is equal to 3x4. So look here very very important here. So remaining now I'm taking so what is the question sign here denominator is what here denominator observe the denominator observe the denominator denominator is what cos divided by the cos in the question sin by cos sin by cos so actually sin by sin a by cos a plus remaining cos is mean cos a by cos a cos a by cos a by sin a by cos a minus this is a minus - cos a by cos a look here very important cos cos cancel here one here also one so sin by cos sin by cos is how much sin a by cos a sin a by cos a value how much 3x4 now here uh sin a by cos okay sin a by cos a here so look here directly here sin a by cos a sin a by cos a is mean 3x4 plus it is a Remaining sin a by cos a sin a by cos a mean 3x 4 - 1 there is nothing 1 4 1 4 so remaining 4 1's are 4 3 1 are 3 + 4 1 are 4 divided by so remaining there is nothing 1 4 1 are 4 so 3 - 4 4 cancel 3 + 4 7 7 by 4 - 3 - 7 answer - 7 option A is a correct answer this is a very smart method this is a Smart method. Don't use the long method. Your smart method directly is tan a is a given that time sin a by cos a 3x4 is a divided by cos sin a by cos a plus cos a by cos a cos a by cos a 1. So sin a by cos a minus so this is a 3x4 this is a 3x4 minus 1 + 1.

[00:32:49]So answer - 7. So sometimes for example here is given c is given that time cos theta by sin theta. So we will get the answer. This is very important method.

[00:33:00]So it is a remaining.

[00:33:02]So remaining question. So here remaining question. So it is finished. So remaining question. So remaining questions here.

[00:33:13]The remaining questions.

[00:33:21]H look here if ninth question for example ninth question. Now I'm taking ninth question. Ninth question is what?

[00:33:27]If cos theta is equal to <unk>3x 4 <unk>3x 4 then what is sin square theta + cos square theta sin square theta + cos² sorry sin square theta + tan square theta so here sin square theta sin square theta + tan square theta here asking question question mark so options is given there here also options is how much a = 7 by 12 option b = 1 option c= 1x2 2 option D is equal to 3x4 3x4 cos theta so here very important so cos theta cos theta is nothing but so cos theta is nothing but so here cos theta formula cos theta formula adjacent side by hypotenus so here adjacent is how much <unk>3 hypotenus is how much one so adjacent is a adjacent side is a <unk>3 hypotenus is a fourth but opposite is directly one directly one.

[00:34:29]Use the pythagoras formula. All right.

[00:34:31]So sin square theta plus tan square theta. Sin theta is nothing but opposite side by hypotenus. So opposite is how much? Here opposite is a one. 1 by hypotenus is how much? Four 1x 4. Next to tan theta tan theta is a opposite side by adjacent side. So opposite is what? Opposite is what? Opposite is a one by adjacent is what? Root three. So this is a values but here substituting in this given question s is mean how much here 1x4 1x 4² plus so tan is mean 1x roo<unk>3 whole square 1x roo<unk>3 whole square so 1 square is mean 1 16 + 1 square root cancel so remaining three so remaining three so here remaining three suppose here now I'm taking for example right triangle triangle so listen so here right triangle triangle So suppose here right triangle roo<unk>3x2 sorry. So cos theta is equal to= <unk>3x2 here roo<unk>3x2. So that time now I'm taking right triangle triangle. So here 90° exactly hypotenus theta exactly opposite side. This is a adjacent side. So here cos theta cos theta is mean adjacent. It is a root three. So hypotenus hypotenus is a two.

[00:35:48]So it is directly one. So sin theta sin theta formula. So opposite side by hypotenus opposite is how much? 1x <unk>3 1x roo<unk>2 sorry 1x2 1x2 why because so here so here sin theta formula sin theta formula opposite side by hypotenus opposite is how much one by hypotenus is how much two 1 by 2 so remaining is what tan theta now I'm fine tan theta formula opposite side by adjacent side opposite is how much here one by adjacent is how much? <unk>3. So this is a tan theta.

[00:36:29]Sin square theta plus cos square theta.

[00:36:31]So sin square theta is nothing but look here sin theta is mean 1x2 1x2 whole square 1x2 whole square plus tan square tan square theta is mean 1x <unk>3 whole² right so 1 by so this is a 4 + 1 by so 3 4 3's are 12 12 okay remaining cross multiplication 3 + 4 so 7 by 12 7 by 12 so answer is a option A is a correct answer 7 by 12 so this is a very important So here 7 by 12 it is a very important 7x 12 option a 7 x 12 it is a correct answer 7 by 12 it is a correct answer this use the pythagoras formula here right triangle triangle so very important so remaining so remaining here so remaining question now I'm taking remaining question uh suppose here remaining questions here now I'm taking remaining questions here so here this type of question is given.

[00:37:32]So second theta 1 - sin theta. So here question is suppose second theta second theta 1 minus 1 - sin theta 1 - sin theta 2 theta plus tan theta value how much we are asking the question options is given options is given option A zero option B 1 option C uh two option D option D is given 4 3 right so this this question is given second theta 1 - sin theta second theta + tan data this time here using the question here so here don't use the lengthy method okay the short method smart method so here trigonometry box is very very important so trigonometry box is very important the first important key point here trigonometry so now now example for example first one sin theta Yeah. Sin 45° cos 45°. So it is always 1 by <unk>2 1x <unk>2. Next to tan 45 value C 45 value is always 1. Tan 45 C 45 1. So remaining second 45 second 45 cosecond 45. So is always <unk>2 <unk>2.

[00:38:57]This is a roo<unk>2. This is very very important. So this is a very very important. So sin 45 cos 45 value 1 by <unk>2 tan 45 c 45 value 1. So second 45 cosec 45 is also roo<unk>2. So here now I'm taking theta is equal to 45. Now I'm taking smart method. You are using the theta is equal to 45. It is a shortcut method. So it is a shortcut method. So shortcut method. Shortcut method is very very important. So here theta is equal to now I'm taking 45. Look here. Huh. So your second second theta second 45 second 45 into 1 - sin 45 theta plus now I'm taking 45. So remaining second 45 plus tan 45 it is a now I'm taking the data plus now I'm substituting here 45 second 45 value how much <unk>2 second 45 value is how much <unk>2 remaining 1 minus sin 45 1 by <unk>2 remaining second 45 <unk>2 plus tan 45 1 right so <unk>2 there so <unk>2 1 are <unk>2 - 1 by <unk>2 so remaining <unk>2 + 1. So <unk>2<unk>2 get cancel. So remaining this is a a minus b into a + b <unk>2 square minus 1 square. So square root cancel 2 - 1 is nothing but one answer is a one. Option B is a correct answer.

[00:40:22]This is a short method but here learn the most important here box but now I'm easy method here. Okay. So it is a easy method here box. So this is answer it is finished. So remaining so remaining concept so remaining concept here right. So here remaining here concept uh so it is clear. So trigonometry box value is it is finished. So remaining so remaining here.

[00:40:56]So very very important remaining.

[00:41:00]All right.

[00:41:03]So here very important concept now I'm discussion here remaining concepts.

[00:41:08]So here very important questions here in every competitive exam here more than times repeated questions here. So it is very very very important concept.

[00:41:18]So every competitive exam lot of students here use the lengthy method or in exam purpose four marks question six marks question types here a step by step don't use the step step by step. Why?

[00:41:29]Because here are all competitive exam here use the smart way and shortcut method. So how to smart way or how to shortcut method here listen. So now I'm taking for example. So here the formulas. So here the very very important four formulas it is it is very important four formulas. What is the four formulas here? So under root of So under root of under root of 1 + sin theta 1 + sin theta by 1 - sin theta first case number one number one here now I'm writing here look here so listen so here first number one number one number one under root of 1 + sin theta by 1 - sin theta first case so second one this type of questions you are asking in more than times. So remaining 1 - sin theta by 1 + sin theta. So this type of question you're asking dash. So third one. So very very important. Third one. So third one. So third one here.

[00:42:33]Third one you are using under root of 1 + cos theta by 1 - cos theta dash. So fourth one. So under root of 1 - cos theta by 1 + cos theta. So here look here. So now I'm taking here very important. So here sign is given observe very important very very important this type of question here more than times you're asking in any exam any competitive exam it is a use the short method shortcut method smart way so you are given the sign it is a given the sin theta that I'm divided by the cos theta suppose your cos theta is given that I'm divided by the sin theta it is very important so both 1 by sin so very very important trigonometry formulas so sin theta 1x sin theta so here. So 1x sin theta 1x sin theta 1x sin theta is nothing but cosecant theta. So 1x cos theta is nothing but second theta.

[00:43:34]So 1x tan theta is nothing but hot theta most most important here very very important questions you are very very important models here formulas here 1x sin theta cosec theta 1x cos theta sec theta 1x tan theta theta so here look here now I'm divided by the sin theta sin theta here now I'm divided by cos so 1x cos 1 by cos 1x cos so here sin theta is given divided by the um now I'm taking divided by cos 1x cos 1 by cos is mean what? Second theta. Second theta plus second theta. Sin theta by cos theta. It is a tan theta. Tan theta. It is a tan theta. Look here. So remaining for example sin here given divided by the cos 1x cos 1x cos is a second theta - sin theta by cos theta. It is a tan theta. It is a smart method. It is finished. Remaining 1 + cos here given the cos that divided by the sin sin theta 1 by sin 1 by sin is what cosec theta plus cos theta by sin theta c theta c theta so remaining so here 1 - cos cos is given divided by the s 1 by sin cosec theta 1 by sin is a cosca minus cos theta by sin theta is nothing but co theta this is most most important here So what is the asking the question you are asking here? What is the question here? Look here 1 + sin theta by 1 - sin theta question asking 1 by divided by cos theta 1 by cos 1 by cos 1x cos is mean what? Second theta 1x cos theta is mean second theta plus sin theta by cos theta tan theta. The answer is so second theta plus tan theta.

[00:45:22]Second theta plus tan theta is which option? Option A is a correct answer.

[00:45:25]This is a smart method but is a more than times this question or this question or this question or this question more than times here this type of questions is given not only this one also this type of questions is here given that time use the same method it is a concept it is a concept and concept and shortcut method shortcut so very very important here the answer is second plus it is finished so remaining now I'm taking here remaining so here same similarly other question now I'm taking here other question second one so look So it is finished already concept.

[00:45:59]Look here 1 + sin theta by 1 - sin a 1 + sin a 1 - sin a plus 1 - sin a 1 + sin here look here s is given that time divided by the cos 1x cos 1x cos theta 1x cos theta is mean theta second theta sin by cos tan theta tan a here a here nothing but a is theta so s is given that I'm divided by the cos 1x cos 1 by cos 1x cos is mean Second second a plus sin a by cos tan tan a remaining plus here look here s is given s is given here also sign is given divided by the cos 1 by cos is mean say a minus here sorry here minus here minus here minus so here so here plus 1 by cos 1x cos is mean se sec se a - sin A by 10 A 10.

[00:47:00]Look here + 10 A - 10 A cancel the remaining se 2 A. The answer is 2 second A. 2 second A is a what? B option. It is a very very important shortcut trick. Okay. This is a the easy the 2 seconds or 5 seconds that we get the answer. Okay. This is very very important shortcut method. But lot of students here step by step four marks question, six mark question. But this is a not your academy or not 10th class exam. Okay, this is IAT or all competitive exam. Here use the smart way or shortcut method in 5 seconds or 10 seconds we get the answer. Right.

[00:47:37]Remaining. So remaining remaining here. So remaining it is finished. So remaining.

[00:47:45]So listen. So here now I'm taking here listen.

[00:47:50]So remaining second the minus tan theta is equal 1x2 find the second theta + tan theta so theta minus tan theta is equal 1x2 theta + tan theta is equal to how much it is easy theta minus tan theta is equal to 1x2 that time theta plus tan theta is equal to directly answer 2 1x2 reciprocal 1x2 reciprocal 1x2 reciprocal so 1x2 reciprocal is what 2x1 2x1 is nothing but two so minus is given that plus second theta minus tan theta is equal 1 by 2 theta + tan theta is equal to 2 option B is a correct answer option B is a correct answer sir how how here using this method look here I am taking the three trigonometry identities three trigonometry identities what is the trigonometry three identities look here the trigonometry three identities the first one the three trigonometry identities the first one sin square theta sin square theta plus cos² theta is equal to 1. So first identity. So second one is a second square theta minus tan square theta is equal to 1. So second square theta minus tan square theta is equal to 1. So third one third one cosec square theta. So cosec square theta minus square theta is equal to 1.

[00:49:11]So here three trigonometry identities very very important. So look here here se tan is relation given se tan. So answer is two. So fix here answer is two. But how? This is a short method.

[00:49:24]But how? Here. So prove that here. So here second tan is given. Se theta second square theta. Second square theta minus tan square theta is equal how much? 1. So this is a squareus b² formula. So second theta plus second theta.

[00:49:42][sighs] Second theta plus tan theta and remaining second theta minus tan theta is equal to 1. Why second square theta minus tan square theta is equal to 1. So use that trigonometry identity second theta + tan theta second theta minus tan theta. So a square - b square formula a + b into a minus b. So it is clear. So second theta second theta plus tan theta it is as it is right. But second theta minus tan theta. Second theta minus tan theta is equal to how much? 1 by 2. So 1x 2. Now I'm taking here second theta minus tan theta 1x2 is equal to 1. So 2 1 are 2. 2 1 are 2. So directly answer second theta + tan theta is equal to 1 2. Answer is a second theta + tan theta is equal to how much? 2. Right? So sometimes this question is given. Sometimes here different question is suppose um cosecant theta cosecant theta minus c theta cosec theta minus c theta cosec theta minus c theta cosec theta minus c theta is equal to 2 cosec theta minus c theta is equal to 2 this is a question new question again so that time answer is directly coc theta ask the question is coc theta coc theta plus theta is equal to how much that time 2 reciprocal 2 reciprocal is how much one B answer is one B this type of question here more than times you are asking in exam purpose so it is finished so remaining so remaining here so it is a finish so remaining here uh look here so remaining here now I'm taking here tan theta so here this is very very important in every competitive exam so more than times you're asking tan theta plus second theta minus one tan theta minus second theta + 1 how So it is a easy method. Now I'm here using so first to observe the question tan theta plus second theta minus one tan theta minus second theta plus one how much here the first observe the question here so now I'm here observe the tan theta first given tan theta that time write the sin theta so sorry so here tan theta is given write the second theta so tan theta is given write the second theta second theta first first given tan theta write the Second theta plus plus so sec is plus write the tan tan theta. Okay easy it is complete. So tan theta here tan theta is given here second theta is given first tan is given where the second theta plus plus second theta is given where the tan theta. So second theta is can be written as 1 by 1 by cos theta plus tan theta is can be written as sin theta by cos theta. So 1x cos okay 1x cos theta plus look here both are same denominator cos theta cos theta same the cos theta is equal to remaining denominator is a same so denominator is the same so numerator addition 1 + sin theta the answer is 1 + sin theta by cos theta tan theta + theta minus 1 tan theta - theta + 1 value is how much 1 + sin theta by cos theta 1 + sin theta by cos theta 1 + sin theta 1 + sin theta by cos theta 1 + sin theta by cos theta. It is a B option is a it is a smart method.

[00:53:08]It is a here this is a very very uh long method here but don't use the long method here. Simple method tan is given write the second theta here plus plus second is given write the tan here simple. Okay. So suppose here minus is given write the minus don't don't observe this one only these two conditions it is very very important two ratios observe this one. Answer is this one complete. So remaining it is remaining. So finish.

[00:53:38]So finally here trigonometry box. Okay.

[00:53:42]Trigonometry box. Now I'm using the trigonometry box. Some questions. It's a short method. Trigonometry. It is a short method. Don't use the long method.

[00:53:51]Trigonometry box. It is very very important. Trigonometry. Trigonometry box. Trigonometry box. So shortcut box.

[00:54:00]Shortcut box. Okay, very very important.

[00:54:02]Okay, so trigonometry shortcut box most most important. So trigonometry shortcut box. Look here trigonometry shortcut box. So now I'm simple method. Now I'm taking here for example opposite side.

[00:54:16]Opposite side OP opposite side remaining. So remaining this is a adjacent side. Adjacent side adjacent side. So remaining here uh remaining is a hypotenus hypotenus side. So hy is me hyper side. So look here listen. So this is very very important. 0° 90° 30° 60° 45°. It is very very important method. Look here. So now I'm taking your smart way shortcut method. It is a easy method. Don't okay look here. So here don't confuse here. Lot of students in 10th class trigonometry box. Okay.

[00:54:57]Lot of students here confused. But here this is simple method. Okay, use the simple method listen. So how to use the 0° first? Now I take the 0° here 0° 90° 30° 60° 45°. Now I'm taking here. So 0° 90° 30° 60° 45° opposite. Now I'm taking simple 0 here 0. This is a one. It is a trick here trick. 01 01 is always hypotenus is also one. Reverse here 01 reverse right here. Rever 1 0 1 1 0 0 1 is 1 reverse right 1 0 1 0 is also 1 remaining 1 <unk>3 1<unk>3 take the shortcut here 1 <unk>3 1 < is also two 0 1 1 0 1 1 roo<unk>3 2 roo<unk>3 1 is also two remaining here one one so shortcut here take the here root two this is most most important so this is a most most important concept here uh look here so here suppose sin 30 value how much question you are asking sin 30 is how much don't know sin 30 is don't know sin 30 value sin 30 observe here sin 30 where 30° is what here sin 30 sin theta formula sin theta formula Opposite side by adjacent side. SR not sin 30. Sin 30. Opposite is how much?

[00:56:33]30. Here observe the 30. Opposite is how much? One. One.

[00:56:39]Opposite is how much? 1 1 by adjacent is how much? 30. Opposite is 1. Adjacent is how much? Roo<unk>3. Sin 30 is how much?

[00:56:47]1 by <unk>3. So sorry. Sorry. Sin 30.

[00:56:50]Sin 30 is a sin 30. So sin 30 0 0 90 30 60 45 sin 30 opposite side by sorry formula here actually formula here sin 30 asking sin 30 so sin theta formula is nothing but opposite side by hypotenus sorry so sin 30 sin 30 opposite is how much 30 opposite is how much one by hypotenus 30 observe the hypotenus is how much? 2 so 1x2 answer is sin 30 value is how much 1x2 suppose sin 60 sin 60 or suppose here tan tan theta now I'm taking tan 60 tan 60° tan 60° is nothing but opposite side by tan theta formula tan theta formula opposite by adjacent side so tan 60 tan 60° opposite opposite 60 go to the 60 60 opposite is how <unk>3 <unk>3 by adjacent adjacent is how much? 1. So roo<unk>3 by 1 is nothing but 1 root3.

[00:58:00]So answer is tan 60 value is how much?

[00:58:02]<unk>3 sin 30 that is 1x2 tan 60 is a roo<unk>3 suppose sometimes so here second 45 suppose second data formula sec is reverse reciprocal sec is a cost cos theta formula adjacent by hypotenus. So this is reverse hypotenus by adjacent. So look here uh 45 go to the 45 45 here. So sec 45 hypotenus is how much here? <unk>2 <unk>2 by adjacent is how much here adjacent one. So by one so second 45 second 45 second 45 value is how much? <unk>2 by 1 nothing but <unk>2. Always second 45 is a roo<unk>2 also. Second 45 is also <unk>2 sin 31 by 2 tan 60 roo<unk>3 5 <unk>3 okay this is very very important smart method here it is a simple trigonometry box shortcut box it is easy lot of students here confuse here trigonometry box okay this is a simple shortcut trigonometry box very very important trigonometry box here okay right so some questions here uh in exam purpose this type of questions here exam purpose how to give the questions here look here so now I'm taking here trigonometry box. Okay. All right.

[00:59:19]So here listen uh suppose your question is given this question.

[00:59:26]Okay. This question is given. So what is here? Uh so very important co theta + tan theta cosec theta minus sin theta cosec theta minus second theta. So for example first one I'm taking sin 45 sin 45 cos 45 tan 45 cos 45 second 45 cosecond 45 so your sin 45 cos 45 1x <unk>2 tan 45 cos 45 1 second 45 cosec 45 <unk>2 this is very very important so this is very important so now I'm taking theta is equal to 45° so c 45 C 45 C 45 plus tan 45 smart method here. So remaining cosec cosec 45 minus sin 45 remaining cos 45 minus sec 45 right. So c 45 is how much here? C 45 value 1 plus tan 45 is also 1. So cosec 45 is how much? Cosec 45 <unk>2 minus sin 45 1x <unk>2. So remaining for cos cos 45 cos 45 1 by <unk>2 so second 45 <unk>2 so look here 1 + 1 2 so <unk>2 into roo<unk>2<unk>2 square<unk>2 square is mean 2 - 1 so here <unk>2 so here roo<unk>2<unk>2 square 1 - 2 by <unk>2 so here directly so here roo<unk>2 2 cancel 2 - 1 2 - 1 is mean 1 so here 2 - 1 is - So 1 into - 1 - 1 answer is minus one. Option C is a correct answer. Here use the theta is equal to 45. We get okay theta theta is equal to you're substituting here theta is equal to 45.

[01:01:17]We get the answer a simple method. It is a short method right. So remaining remaining here. So very important here remaining. So remaining here uh remaining here suppose some questions here given.

[01:01:34]So here suppose suppose here suppose here look here this question is observe here this questions here. Okay. What is this question here?

[01:02:03]R now I'm taking here suppose now I'm taking here suppose sin 45° question you're asking this type of question plus cos 45° is how much in exam purpose sin 45 is a 1x roo<unk>2 plus cos 45 is a 1x roo<unk>2 so <unk>2<unk>2 same therefore 1 + 1 2 so here <unk>2 by so here roo<unk>2 same roo<unk>2 into <unk>2 <unk>2 by divided by <unk>2. Why the 2 is can be written as <unk>2 into <unk>2 roo<unk>2 square remaining two. So <unk>2<unk>2 cancel answer is <unk>2. The answer is <unk>2 sin 45 + cos 45 value is how much? 1.

[01:02:44]The most important shortcut here. So very very important. Sometimes your question asking sin 30° by cos cos sin 30° sin 30°. So you're asking the question here exam purpose. So sin 30° sin 30° by cos 60° dash we asking option A 1 option B 0 option C 2 option B minus 2 this type of question is given sin 30° by cos 60° how much here so here so simple directly so 30° + 60° add here 30 + 60 is how much 90 so look here 30° + 60° 90 so here divided by by So here sin. So here sin 30 by cos 60 here. So here both are adding here 90° here divided by divided by is mean always answer is one divided is given 30 60 add here 90. So here we will get the one option a is a correct answer. It is a shortcut trick. So suppose for example sin 30° minus so cos cos 60°. So here look here 30 60 30 60 add the how much here 90 but different is how much here minus different is minus that time answer is zero very very important so here 30 60 add here 90 so minus we will get the zero 30 60 divided by here also 30 60 90 we will get the answer one very very very important concept here okay right so okay okay thank you so much to everyone by thank