# How do you find the values of sin 2? and cos 2? when cos? = 12/13?

We need to find the values of sin2Θ and cos2Θ

### Solution

Given

We are given that cosΘ= 12/13

Find out

Using the given value cosΘ= 12/13 we need to evaluate the values of sin2Θ and cos2Θ

We know that

Now if we consider the right angles triangle we know the value of adjacent side and hypotemuse side. Let us find out the remaining opposite side by Pythagoras theorem.

According to Pythagoras theorem

Hypotunese2= Side2 + Side2

Let the unknown side that is opposite be = x

132= x2 + 122

x2 = 132 – 122

x2 = 169 – 144

x2 = 25

x= √25

x= 5 units

Hence the other side that is opposite side = 5 units

So now

sinΘ= opposite / hypotenuse = 5/13

tan Θ= opposite / adjacent = 5 / 12

cos Θ= adjacent/ hypotenuse = 12/13

First let us find out the value of sin2Θ

We know that sin2Θ = 2SinΘCosΘ

Substituting the known values we get

sin2Θ =2SinΘCosΘ

2 X 5 /13 X 12/13= 120 / 169

Now we know that cos2Θ= Cos2Θ – Sin2Θ

So cos2Θ= (12/13)2 – (5/13)2

cos2Θ= 144 / 169 – 25 / 169

cos2Θ=119 / 169