Find the exact value of the trigonometric expression given that sin(u) = − 3 5 , where 3𝜋/2 < u < 2𝜋, and cos(v) = 15 17 , where 0 < v < 𝜋/2. sin(u + v)

Answer 1

Daniela,

Whenever you see two sides of a 3-4-5 right triangle, you should use Pythagorean theorem to check the other angle as well.

Given 15 and 17, suppose 17 is the hypotenuse.

Then 17² = 15² + s²

s² = 17² - 15²

= 289 - 225 = 64

solving for s,

s= sqrt(64) = 8

so we know that v is part of an 8-15-17 right triangle

why was this a valuable calculation? Because now you know the fractions corresponding to all the trig functions for both u and v.

however, first we need to figure out which legs are negative, based on the constraints the problem has given.

3𝜋/2 < u < 2𝜋, 

since sin(u) = -3/5 and 3pi/2 < u 2pi, we can draw the angle in the fourth quadrant., and the x-coordinate is (-3).

similar argument shows that v is in the first quadrant, so both x and y coordinates are positive.

so, what is the formula for sin(u + v)?

sin(u+v) = sin(u)cos(v) +cos(u)sin(v)

= (-3/5)(15/17) + (4/5)(8/17)

at this point, you can do the arithmetic.