Find the exact values of sin 2𝜃, cos 2𝜃, and tan 2𝜃 for the given value of 𝜃.

Question

cos 𝜃 = 4 5 ; 0° < 𝜃 < 90°

sin2𝜃= ?

cos 2𝜃=?

tan 2𝜃=?

Answer 1

I'm assuming you meant that cos(x) = 4/5. I'm also using x in place of θ as that is easier to type.

a) We know that sin(2x) = 2sin(x)cos(x).

In this case, we're not given sin(x), but we know that on the unit circle that sin²x + cos²x = 1. This means that sin²x = 1 - cos²x, which means sin(x) = √(1-cos²(x)).

Since we're given that cos(x) = 4/5, we know by our formula that sin(x) = √(1 - (4/5)² = √1 - 16/25 = √(9/25) = 3/5.

So in this case, sin(2x) = 2(3/5)(4/5) = 24/5.

b) In this case, we have a few options, and ideally, so we have to do the least amount of work, we can just use an identity that only involves cos(x). In this case, it is most convenient to choose the identity:

cos(2x) = 2cos²(x) - 1

So in this case cos(2x) = 2(4/5)² - 1 = 2(16/25) -1 = 32/25 - 25/25 = 7/25.

c) Finally, tan(2x) = 2tan(x)/ 1 - tan²x.

We aren't given tan(x), but we do know that tan x = sin(x) / cos(x). In this case, tan(x) = 3/5 / 4/5 = 3/4.

Now that we know tan(x), we can solve for tan(2x):

tan(2x) = 2(3/4) / 1 - (3/4)² = 3/2 / 1 - 9/16 = 3/2 / 7/16

Multiplying by 16/16 = 1, we get our final answer is tan(2x) = 24/7.