Since can be written as:

. We can't have .

Therefore . This means that where k is an integer.

since . We have x=0 is the only number that satisfies this property.

Rearrange the problem,

Over the interval 0 to 360 degrees, cosx = 1/2 at 60 degrees and 300 degrees. 

First, get the equation in terms of one trig function. We can do this by substituting in using the Pythagorean Identity for .

Then we have .

Bring all the terms to one side to find that .

We can factor this quadratic to .

This means that .

The only angle value for which this is true is . 

First, re-write the equation so that it is equal to zero:

Now we can use the quadratic formula to solve for x. In this case, the coefficients a, b, and c are a=1, b=2, and c=-3:

simplify

This gives two potential answers:

and

Sine must be between -1 and 1, so there are no values of x that would give a sine of -3. The only solution that works is . The only angle measure that has a sine of 1 is .

Use the quadratic formula to solve for x. In this case, the coefficients a, b, and c are a=4, b=1, and c=-1:

simplify

the square root of 17 is about 4.123. This gives two potential answers:

. We can solve for x by evaluating both and . The first gives an answer of . Add this to 360 to get that as a positive angle measure, . If this has a sine of -0.64, so does its reflection over the y-axis, which is .

The second gives an answer of . If that has a sine of 0.39, then so does its reflection over the y-axis, which is .

There are multiple solution paths. We could subtract 1 from both sides and use the quadratic formula with and . Or we could solve using inverse opperations:

divide both sides by 2

take the square root of both sides

The unit circle tells us that potential solutions for are .

To get our final solution set, divide each by 3, giving:

.

 This problem has multiple solution paths, including subtracting 5 from both sides and using the quadratic formula with . We can also solve using inverse opperations:

subtract 2 from both sides

divide both sides by 4

take the square root of both sides

If the sine of an angle is , that angle must be one of . Since the angle is , we can get theta by subtracting :

To solve, use the quadratic formula with and where x would normally be:

This gives us two potential answers:

since this number is greater than 1, it is outside of the domain for cosine and won't give us any solutions.

Consulting the unit circle, the cosine is when

To start solving, first realize that this is a quadratic with "x" as :

We can solve using the quadratic formula:

One potential solution:

Taking the square root gives: , but 2 is outside the range of cosine, so that won't work.

The other potential solution:

Taking the square root gives:

Consulting the unit circle,

To solve, first use the quadratic formula:

This gives us two potential solutions:

that is outside the range of sine, so it won't work

We can continue solving by taking the inverse sine:

using a calculator gives us

, which we can convert to a positive angle measure by adding to 360:

This is just one answer for . The other angle that would work would be below , or

Since those are values for , to get our final answers divide by 2: