Question

How do you find the exact value of `cos62` using the sum and difference, double angle or half angle formulas?

Answer 1

`cos(62^@)=cos((7pi)/12)=(sqrt(2)-sqrt(6))/4`

Explanation:

Best way to start is to convert the angle measure from degrees to radians. A quick refresher on how to do this;

Radians = Degrees `xxpi/180`

Radians = `62xxpi/180=(62pi)/180=(7pi)/12`

`:.cos(62^@)=cos((7pi)/12)`

`cos((7pi)/12)=cos(pi/3+pi/4)`

The trigonometric identity that we are going to use is the angle sum identity for cosine;

`color(red)(cos(A+B)=cos(A)cos(B)-sin(A)sin(B))`

Entering in the appropriate angles;

`cos(pi/3+pi/4)=cos(pi/3)cos(pi/4)-sin(pi/3)sin(pi/4)`

Evaluating the trigonometric ratios yields;

`cos(pi/3+pi/4)=(1/2)(sqrt(2)/2)-(sqrt(3)/2)(sqrt(2)/2)`

Hence,

`cos(pi/3+pi/4)=sqrt(2)/4-sqrt(6)/4=(sqrt(2)-sqrt(6))/4`

I hope that is helpful,

If you like I have a youtube channel with heaps of videos on problems like this, you should check it out:

https://www.youtube.com/haroldwalden

:)