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:

First solve using the quadratic formula:

This gives two potential solutions:

The only value for where sine is 1 is .

Using a calculator, we get

Adding that to 360 givesus the angle's positive value,

That's just one instance where the sine is -0.75. We also need to find the other angle below the x-axis by adding .

So our three values for theta are

First, solve for using the quadratic formula:

This gives two solutions:

this is outside of the range of cosine so it will not work.

Consulting the unit circle tells us that or . To get our final answers, just divide these by 4:

First, solve for by using the quadratic formula:

This gives two solutions:

this is outside of the range for cosine, so that does not work as a solution 

To solve for theta, take the inverse of cosine of both sides:

 according to the calculator. That's just one potential value, though. The other angle that would have a cosine of positive 0.6 would be 53.13 degrees below the x-axis in quadrant IV, so subtract from 360:

That gives us two values for , so to get theta we have to subtract 1:

First solve for using the quadratic formula:

One answer is this is outside the range for cosine, so it does not work as a solution

The other answer is

To solve for theta, take the inverse cosine using a calculator:

This is just one answer for theta, in quadrant II. Cosine is also negative in quadrant III, so we want to find the angle there with the same cosine. This would be , or

To solve, use the quadratic formula:

This gives two solutions.

The first is:

Using a calculator gives us

This is just one potential value, the one in quadrant I. Tangent is also positive in quadrant III, and we can get this angle by adding 180:

The second solution from the quadratic formula is:

Using a calculator gives us , which we can add to 360 to get as a positive value, .

This is just one potential value, the one in quadrant IV. Tangent is also negative in quadrant II, and we can get this angle by subtracting 180:

Dividing all four of these angles by 3 gives us