Question
Question: How do you solve \({\cos ^2}x = \,\dfrac{3}{4}\,\)?...
How do you solve cos2x=43?
Solution
Hint : In this question first, we have the square root on both sides and then we have to find the angle whose cosine is equal to the square root of the value given in the right hand side. You can think that there is only one value but there can be multiple values of angle.
Complete step-by-step answer :
In this question, it is given that cos2x=43
Now taking the root on both sides, we get
⇒cosx=±23
Now, taking the inverse on both sides
⇒x=cos−1±23
Now, there are two cases
Case I:
⇒cos−123= An angle whose cosine is equal to 23.
Hence, x=2nπ±6π where n∈Z,Z represents the set of all integers.
Case II:
⇒cos−1−23=An angle whose cosine is equal to −23.
Hence, x=2nπ±65π where n∈Z,Z represents the set of all integers.
This means that after every one rotation we get the same value of cosx.
Therefore, the solution to expression cos2x=43 is 2nπ±6πand2nπ±65π where n∈Z,Z represents the set of all integers.
Note : There are multiple ways to do this question. We can also convert the cos2x into sec2x by using formulas and then sec2x into tan2x to solve this problem. We can also do this question by converting cos2x in terms of sin2x and then we can solve the equation. It is a question of trigonometric equations. Always be careful while doing the questions of trigonometric equations because there can be multiple solutions of a question. Sometimes we can also do the questions using graphs.