Solving Trig Functions

Question

Solve the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to two decimal places where appropriate.)

cos theta = -1/2

Find what theta =

List six specific solutions is =

Answer 1

This is a unit circle based question. It is asking you for what values of theta is cos(theta) = -1/2. Keep in mind when you have an ordered pair (x,y) in trig we think of ordered pairs as (cos(theta), sin(theta)). You want to find what angles have an "x" coordinate of -1/2. Remember ASTC or that cosine is negative when x is negative which is in quadrants II and III. We need to find what angles in the 2nd and 3rd quadrant yield a cosine value of -1/2.

There are certain angles you should know on the unit circle, namely 0, pi/6, pi/4, pi/3, pi/2 and their multiples. The angle (and its certain multiples) where cosine(theta) is -1/2 would be pi/3. Since we need cosine to be negative we need the pi/3 in the 2nd and 3rd quadrants which would be 2pi/3 and 4pi/3. The period (how long it takes for the unit circle to repeat or complete one period) is 2pi so we can add any 2pi multiple to those values to return back the same value.

For instance 2pi/3 would then be 8pi/3 14pi/3 20pi/3, etc.

4pi/3 would be 10pi/3 16pi/3 22pi/3, etc.