trigonometry

Question

if sin of theta equals 3/5, cosine of theta is <0, sin of alpha equals -8/15 and cos of alpha <0, determine the exact values for sin(theta plus alpha) and cos(theta plus alpha)  

Answer 1

The first thing to do is to figure out the values of cos(θ) and cos(α). Let's start with θ. Draw a right triangle and label one of the non-right angles θ. Since sine is opposite/hypotenuse and sin(θ) = 3/5, you can label the side opposite θ as 3 and the hypotenuse as 5. Use the Pythagorean Theorem to figure out the triangle's third side:

 

a² + b² = c²

3² + b² = 5²

9 + b² = 25

b² = 16

b = 4

 

Now you can label the triangle's third side as 4. Since cosine is adjacent/hypotenuse and cos(θ) < 0, this means that cos(θ) = -4/5.

 

Let's do the same thing for α. Draw another right triangle and label one non-right angle α. As above, since sin(α) = -8/15, label the opposite side 8 and the hypotenuse 15. Use the Pythagorean Theorem to figure out the triangle's third side:

 

8² + b² = 15²
64 + b² = 225
b² = 161
b = √161

 

Now you can label the triangle's third side as √161. Since cosine is adjacent/hypotenuse and cos(α) < 0, this means that cos(α) = -√161/15.

 

(By the way, I think it's possible that you meant tan(α) = -8/15, in which case the triangle is a little different and the numbers work out much more nicely. In that case, the opposite side is 8, the adjacent side is 15, and the Pythagorean Theorem gives us c² = 8² + 15² = 64 + 225 = 289, so the hypotenuse is 17. If that's the case, make sure you use sin(α) = 8/17 and cos(α) = -15/17 for the next part.)

 

Now that you know sine and cosine of θ and α, you can plug those values into the Trigonometric Addition Formulas to find exact values for sine and cosine of (θ+α):

 

sin(θ+α) = sin(θ)cos(α) + sin(α)cos(θ)

cos(θ+α) = cos(θ)cos(α) - sin(θ)sin(α)