### All Trigonometry Resources

## Example Questions

### Example Question #1 : Quadratic Formula With Trigonometry

Solve the following equation for .

**Possible Answers:**

No solution exists

**Correct answer:**

; Use the double angle identity for cosine.

; Move everything to the left side of the equation.

; This is a quadratic-like expression that cannot be factored. We must use the quadratic formula. It may be helpful to see this if you replace with , so it becomes:

Recall the quadratic formula

plug in .

We now have

; Separate this into two equations and take the inverse sine.

or

The first equation gives us . Using the unit circle as we did in previous problems, we can find a second answer from this which is . The second equation will not give us a solution.

### Example Question #2 : Quadratic Formula With Trigonometry

What are the zeros of the function listed above for the interval .

**Possible Answers:**

**Correct answer:**

When the quadratic formula is applied to the function, it yields

So those are the zeros for sine, but sine has a minimum of -1, so -2 is out. For -1/2, sine achieves that twice in a cycle, at π+π/6 and 2π-π/6. So while -π/6 is true, it is not correct since it is not in the given interval.

Therefore on the given interval the zeros are:

### Example Question #3 : Quadratic Formula With Trigonometry

Solve the following trigonometric equation:

for

**Possible Answers:**

The equation does not have a solution.

**Correct answer:**

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.

### Example Question #4 : Quadratic Formula With Trigonometry

Solve each equation over the domain (answer in degrees).

**Possible Answers:**

**Correct answer:**

Rearrange the problem,

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

### Example Question #5 : Quadratic Formula With Trigonometry

Solve the equation over the interval

**Possible Answers:**

**Correct answer:**

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 .

### Example Question #61 : Trigonometric Equations

Solve for , giving your answer as a positive angle measure:

**Possible Answers:**

No solution

**Correct answer:**

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 .

### Example Question #7 : Quadratic Formula With Trigonometry

Solve for :

.

Give your answer as a positive angle measure.

**Possible Answers:**

**Correct answer:**

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 .

### Example Question #8 : Quadratic Formula With Trigonometry

Solve for :

**Possible Answers:**

**Correct answer:**

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:

.

### Example Question #9 : Quadratic Formula With Trigonometry

Solve for :

**Possible Answers:**

**Correct answer:**

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 :

### Example Question #1 : Quadratic Formula With Trigonometry

Solve for :

**Possible Answers:**

**Correct answer:**

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

Certified Tutor