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.

sin(u) = −3/5,    3π/2 < u < 2π

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

Answer 1

First let's notice that 3π/2 < u < 2π denotes the 4^th quadrant.

It is important to keep in mind that only cosine and secant are positive in this quadrant

Next, let's draw a triangle with angle u being an angle that isn't the right angle and set up the 3/5 ratio accordingly. We know it is the sine so we will make 3 the opposite side and 5 the hypotenuse.

We can use Pythagorean Theorem to solve for the adjacent side or we can notice that it is a special 3-4-5 right triangle. Either way, the missing adjacent side is 4.

Since we are evaluating the double angles, we will have to first convert the functions into functions with angles of just u instead of 2u.

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

cos(2u)=cos²(u)-sin²(u) (this is just one of three options, you can use any of the three, I just chose this one)

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

tan(2u) can be done with sin(2u)/cos(2u) but I will do it with a formula to be consistent and show work

tan(2u)=2tan(u)/[1-tan²(u)]=2(-3/4)/[1-(-3/4)²]=(-3/2)/[1-(9/16)]=(-3/2)/(7/16)=(-3/2)×(16/7) (by keep change flip)=-24/7

Hope this helps!

Answer 2

sin u = -3/5 and angle is in 4th quadrant

so opp = -3

adj = √5² - 3² = 4

hyp = 5

cos u = 4/5

tan u = -3/4

So use the following formula , substitute values of sinu, cos u, tan u to find the answer

Sin 2u = 2 sin u cos u

Cos 2u = cos²u = sin²u

tan 2u = sin 2u/cos 2u