Only 10% students solve this trigonometry math olympiad question | tanx=? |

[00:00:01]Hello everyone. Welcome to Russell's Classroom. Today we are solving interesting trigonometric maths Olympiad question, which is sin x + cos x is equal to 1 over 5. tan x is equal to what? How to solve this interesting maths problem or maths solution?

[00:00:18]I solve this question easy method and I apply here is different formula. Now our question which is sin x + cos x. I take our question is equal to 1 over 5. This is our question.

[00:00:38]Now at this moment I use both side whole square.

[00:00:42]I take here is whole square.

[00:00:44]Now you can apply here is a + b whole square, which is a + b bracket power is two.

[00:00:53]It will be a square + 2ab + b square.

[00:01:00]If I apply this maths formula here, so here is a is equal to sin x. So sin square x + b square. b is cos square x.

[00:01:12]+ 2ab. 2a is sin x and b is cos x.

[00:01:21]Then this is 1 over 25. 5 to the power 2, this is 25.

[00:01:27]So we find out a nice equation, but at this moment this is sin square x and cos square x. So we know that according to trigonometric rules, sin square theta + cos square theta it will be 1. If I apply this maths formula here, so this expression it will be 1 + and this is 2 sin x cos x. So we know that 2 sin theta + times cos theta it will be sin 2 theta.

[00:02:02]sin 2 theta So, if I apply this math formula here, so it will be x, so it will be sin 2 x.

[00:02:11]Then, this is 1 over 25.

[00:02:17]Now, I move on this one in this side, it will be sin 2 x is equal to 1 over 25 minus 1. I just move on this one in this side, it will be negative 1.

[00:02:35]Now, this is sin 2 x and this is 25, so I take this 25 here, least common multiple is 25.

[00:02:46]25 divide 25, it will be 1. 1 * 1, this is 1.

[00:02:52]Minus this one divide 25, it will be 25.

[00:02:55]25 * 1, this is 25.

[00:03:00]Now, at this moment, you can say this expression, it will be sin 2 x is equal to minus 24 over 25.

[00:03:14]Now, our target tan x is equal to what?

[00:03:16]How do I simplify for tan x in this equation?

[00:03:19]So, you know that sin 2 x, it is a trigonometric rules.

[00:03:26]Uh it will be 2 tan x over 1 + tan squared x.

[00:03:35]I apply this math formula here.

[00:03:39]So, if I apply this math formula here, uh you can say this sin 2 x, and I mean this expression, it will be 2 tan x over 1 + tan squared x is equal to minus 24 over 25.

[00:04:03]Now, at this moment this is tan x, this is tan x. So, let tan x is equal to m or a, as you wish.

[00:04:15]I take here is m. Tan x is equal to m.

[00:04:18]So, I substitute this value here.

[00:04:21]So, this is 2 m over 1 plus m squared is equal to minus 24 over 25.

[00:04:38]Now, if I do here is cross multiplication, this is uh 25 * 2. This is 50 m is equal to 24 * 1. This is minus 24.

[00:04:53]Plus minus, this is minus.

[00:04:55]Plus minus, this is minus 24 m squared.

[00:05:02]Then, you can say this exponential equation, it will be 50 m.

[00:05:09]And if I move on this value on this side, it will be positive 24 m squared. And this value it will be positive 24 is equal to zero.

[00:05:21]Then, this exponential equation, if I divide both side by two, this is 12 m squared. And if I divide this value by two, I divide both side by two.

[00:05:38]So, it will be 25 m.

[00:05:42]And this value by two, it will be 20 12 is equal to zero. So, we are find out a nice quadratic equation.

[00:05:50]So, at this moment I do middle factor here. So, how to solve this exponential equation for m?

[00:05:57]This 12 * 12, you can say it will be 12 * 12 it will be 144.

[00:06:07]So, you know that 144 it will be 16 * 9.

[00:06:11]16 * 9 it will be 144. And 16 + 9 this is 25. So, you can say easily it will be 12 m squared plus this is 16 m plus 9 m plus 12 is equal to zero.

[00:06:35]Then at this moment you can say here is uh 4 m is common. So, if I take 4 m is common, this divide this it will be 3 m plus 4.

[00:06:48]Then if I take here is 3 is common it will be 3 m 12 divide 3 this is 4 is equal to zero. Now, at this moment uh you can say this is 3 m plus 4 this is 3 m plus 4. Then 3 m plus 4 is common. So, if I take 3 m plus 4 is common so, this divide this it will be 4 m.

[00:07:09]And this divide this it will be 3 is equal to zero.

[00:07:13]So, we have find out here is two cases.

[00:07:15]Uh our first case you can say easily it will be 3 m plus 4 is equal to zero.

[00:07:23]Or you can say here m is equal to minus 4 over 3.

[00:07:30]And others case it will be 4 m plus 3 is equal to zero.

[00:07:36]Or you can say here m is equal to minus 3 over 4.

[00:07:41]I move on this three in this side with the be three, then I divide both side by four. It will be minus three over four.

[00:07:48]So, m is equal to this, but remember that our target tan x is equal to y. So, we will substitute m is equal to tan x.

[00:07:55]So, you can say here uh this is tan x is equal to minus four over three or this is tan x is equal to minus three over four.

[00:08:10]So, this is our final answer in this tricky maths Olympiad question. Thank you all of you enjoyed this maths question. Please subscribe to my channel for other interesting video. Goodbye.

[00:08:19]Take care everyone.