### All ACT Math Resources

## Example Questions

### Example Question #1 : Trigonometric Operations

If , what is if is between and ?

**Possible Answers:**

**Correct answer:**

Recall that .

Therefore, we are looking for or .

Now, this has a reference angle of , but it is in the third quadrant. This means that the value will be negative. The value of is . However, given the quadrant of our angle, it will be .

### Example Question #11 : Sine

What is the sine of the angle formed between the origin and the point if that angle is formed with one side of the angle beginning on the -axis and then rotating counter-clockwise to ?

**Possible Answers:**

**Correct answer:**

You can begin by imagining a little triangle in the fourth quadrant to find your reference angle. It would look like this:

Now, to find the sine of that angle, you will need to find the hypotenuse of the triangle. Using the Pythagorean Theorem, , where and are leg lengths and is the length of the hypotenuse, the hypotenuse is , or, for our data:

The sine of an angle is:

For our data, this is:

Since this is in the fourth quadrant, it is negative, because sine is negative in that quadrant.

### Example Question #61 : Trigonometry

What is the sine of the angle formed between the origin and the point if that angle is formed with one side of the angle beginning on the -axis and then rotating counter-clockwise to ?

**Possible Answers:**

**Correct answer:**

You can begin by imagining a little triangle in the third quadrant to find your reference angle. It would look like this:

Now, to find the sine of that angle, you will need to find the hypotenuse of the triangle. Using the Pythagorean Theorem, , where and are leg lengths and is the length of the hypotenuse, the hypotenuse is , or, for our data:

The sine of an angle is:

For our data, this is:

Since this is in the third quadrant, it is negative, because sine is negative in that quadrant.

### Example Question #63 : Trigonometry

If , what is the value of if ?

**Possible Answers:**

**Correct answer:**

Recall that the triangle appears as follows in radians:

Now, the sine of is . However, if you rationalize the denominator, you get:

Since , we know that must be represent an angle in the third quadrant (where the sine function is negative). Adding its reference angle to , we get:

Therefore, we know that:

, meaning that

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