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

Question

sec(u) = −2, 𝜋/2 < u < 𝜋

sin(2u)

cos(2u)

tan(2u)

Answer 1

Given that the secant of the angle is -2, then we can say that the cosine is -1/2. (secant and cosine are reciprocal functions. And given that the angle lies between π/2 and π (𝜋/2 < u < 𝜋) we can sketch the following triangle:

since cosine is adjacent/hypotenuse and since cos(u) = -1/2 then the adjacent side must be -1 and the hypotenuse must be 2 and the angle is in Q2.

We can use the Pythagorean Theorem to solve for "y"

y = √(2² - 1²) = √3

which means sin(u) = √3/2 and tan(u) = -√3

Since we are being asked to find double angles, we must use the double angle identities:

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

cos(2u) = cos²(u) – sin²(u)

tan(2u) = (2tan(u))/(1 – tan²(u))

So now just plug in the values:

sin(2u) = 2(√3/2)(-1/2) = -√3/2

cos(2u) = (-1/2)² – (√3/2)² = (1/4) - (3/4) = -2/4 = -1/2

tan(2u) = (2(-√3)/(1 – (-√3)²) = (-2√3)(1 - 3) = √3