Simple Angle Equations Examples

Solve .

Lucky us: we already have our trig function isolated on one side.

The x-coordinate on the unit circle equals cosine.

Two angles between 0 and 2π have an x-coordinate of positive .

Trigonometric functions repeat themselves infinitely, so we need to list all of the possible correct angles. Just be careful about staring into the face of infinity; it has the teensy-weensy problem of driving people mad after a while.

Solve tan t = -1 on the interval [-2π, 2π].

Tangent is sine over cosine.

Looking at the unit circle, they will be equal when at angles.

Tangent is negative in the second and fourth quadrants.

n = 0:

The angles fall smack dab in the middle of our interval.

n = 1:

These are greater than 2π, so they aren't acceptable answers.

n = -1:

These are within the interval.

n = -2:

These are less than -2π, so no good.

Our answers are .

Solve on the interval .

Cosecant is the reciprocal of sine. That means we want the angles where

Sine and cosecant are the y-coordinate on the unit circle. They are negative in the third and fourth quadrants. is more than π, and is less than 2π.

The interval, , covers the left half of the unit circle, meaning the second and third quadrants. That means the only solution is:

Every other angle falls outside of the interval.