Use the given conditions to find the exact values

Question

Use the given conditions to find the exact values of sin(2u), cos(2u), and tan(2u) using the double-angle formulas.

cos(u) = − 4/5 ,     π/2 < u < π

sin(2u)	=
cos(2u)	=
tan(2u)	=

Answer 1

First of all, identify the quadrant where angle "u" is at.

I see that angle u is in the second quadrant between π/2 and π. Thats why cos(u) is negative, but sine should be possitev (+), and tangent its negative (-)

So let's proceed to find the information we need..

We have cos(u)= -4/5

In the unit circle cosine represents the x-coordinate and sine represents y-coordinate. I will set r as 1 arbitrary. So,

x=-4/5

r=1

Let's find y

we know that x²+y²=r²

Solve for y² ⇒ y²= r²-x²

Let's substitute the values that we have.

y² = 1² - (-4/5)²

y² =1 - (16/25)

y² = 9/25 apply square root property to solve for y

y = SQRT(9/25)

y = 3/5

Now we have everything we need.

cos(u) = x = -4/5

sin(u) = y = 3/5

tan(u) = sin(u)/cos(u) = y/x = (3/5)/(-4/5) ⇒ tan(u) = -3/4

Apply formulas of double angles

cos(2u) = cos2(u) - sin2 (u) = (-4/5)2 - (3/5)2 = 16/25 - 9/25 = 7/25 ⇒ cos(2u) = 7/25

sin(2u) = 2 sin(u) cos(u) = 2 (3/5) (-4/5) = -24/25 ⇒ sin(2u) = -24/25

tan(2u)= sin(2u)/cos(2u) = (-24/25)/(7/25) = -24/7 ⇒ tan(2u) = -24/7