Solving Trig Equations Intro

Question

Find all solutions to 2sin(Θ)=-√(3) where 0≤Θ<2π 

 

Θ= ,

 

Give your answers as exact values, as a list separated by commas. 

Answer 1

2sin(Θ)=-√(3) where 0≤Θ<2π

 

Solve the equation for sin(Θ)

 

sin(Θ) = -√(3)/2

 

Immediately, one should recognize these as side lengths of a special right triangle. The 30-60-90 triangle have side lengths 1, √(3), 2.

(the side with length 1 is across from the 30º angle, the side with length √(3) is across from the 60º angle, the side with length 2 is across from the 90º angle)

 

sin is referred to as opposite over hypotenuse. If we were to draw an angle on the coordinate plane, and then draw a vertical line to connect it to the x-axis, the side opposite of the angle is the y value, and the hypotenuse is referred to as the radius (because this represents the radius of the unit circle). Therefore, sin = y/r

 

The radius will always be positive, so since sin is negative, that means that y is negative. The two triangles will be located in quadrants 3 (the bottom left) and 4 (the bottom right).

 

The angles that produce triangles with -√(3) as its y and 2 as its r are: 240º and 300º. Expressed in radians, 4π/3 and 5π/3