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

Question

sin(u) = −3/5,    3𝜋/2 < u < 2𝜋

sin(2u)

cos(2u)

tan(2u)

Answer 1

I think it REALLY helps to sketch a triangle. Given that ∠u is in Q4 (3𝜋/2 < u < 2𝜋) and that the opposite side is -3 with the hypotenuse equal to 5 (sin(u) = opp/hyp = −3/5) here is a sketch:

Then you can use the Pythagorean Theorem to solve for a.

a = √(5² - (-3)²) = 4

Since a = 4 then we have:

sin(u) = -3/5

cos(u) = 4/5

tan(u) = -3/4

Now, just plug those values into the double angle formulas:

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

cos(2x) = cos²(x) – sin²(x) or cos(2x) = 2cos²(x) – 1 or cos(2x) = 1 – 2sin²(x)

tan(2u) = (2tan(u))/(1 - tan²(u))

Answer 2

sinu=-3/5, cosu=4/5 for quadrant3

sin2u=2sinucosu= 2(-3/5)(4/5)= -24/25=-.96

cos2u=cos^2(u)-sin^2(u)=16/25-9/25=7/25=.28

tan2u=sin2u/cos2u= -24/7= about -3.4286

check the answer

u=360-cos^-1(.8)=about 360- 36.9=323.1

2u=646.2

cos2u=cos646.2= cos(647.2-360)=cos287.2=.about .29