Solve the equation below.

Question

For the equation below, solve for the following. Do not use a calculator. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)

sin θ = - rad3/2

(a)    all degree solutions (Let k be any integer.)

θ =

(b)    θ if 0° ≤ θ ≤ 360°

θ =

Answer 1

If sinΦ = -√3/2, (it's negative) and

sinΦ = opposite side / hypotenuse

then our opposite side is negative and we are looking for an angle in Quadrant III and Quadrant IV.

Let's take the inverse sine of both sides

sin^-1(sinΦ) =sin^-1 -√3/2

Φ =sin^-1 -√3/2

Φ = -60 or 300º in Quadrant IV (Reference angle is 60° away from the x-axis)

The angle 60° away from the x-axis in Quadrant III is 180°+60 = 240°

Your solution is 240° and 300°

Note: If you need to convert these degrees to radians, just put the degree with a π (pi), divide by 180, and simplify the fraction

240° = 240π/180 Then simplify the fraction 4π/3

300° = 300π/180 Then simplify the fraction 5π/3