### All ACT Math Resources

## Example Questions

### Example Question #1 : How To Find Positive Tangent

For triangle , what is the cotangent of angle ?

**Possible Answers:**

**Correct answer:**

The cotangent of the angle of a triangle is the adjacent side over the opposite side. The answer is

### Example Question #2 : How To Find Positive Tangent

What is the tangent 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 ? Round to the nearest hundredth.

**Possible Answers:**

**Correct answer:**

Recall that when you calculate a trigonometric function for an obtuse angle like this, you always use the -axis as your reference point for your angle. (Hence, it is called the "reference angle.")

Now, it is easiest to think of this like you are drawing a little triangle in the third quadrant of the Cartesian plane. It would look like:

So, the tangent of an angle is:

or, for your data, .

This is . Rounding, this is . Since is in the third quadrant, your value must be positive, as the tangent function is positive in that quadrant.

### Example Question #2 : How To Find Positive Tangent

What is the tangent 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 ? Round to the nearest hundredth.

**Possible Answers:**

**Correct answer:**

Recall that when you calculate a trigonometric function for an obtuse angle like this, you always use the -axis as your reference point for your angle. (Hence, it is called the "reference angle.")

Now, it is easiest to think of this like you are drawing a little triangle in the third quadrant of the Cartesian plane. It would look like:

So, the tangent of an angle is:

or, for your data, , or . Since is in the third quadrant, your value must be positive, as the tangent function is positive in this quadrant.

### Example Question #23 : Trigonometry

A ramp is being built at an angle of from the ground. It must cover horizontal feet. What is the length of the ramp? Round to the nearest hundredth of a foot.

**Possible Answers:**

**Correct answer:**

Based on our information, we can draw this little triangle:

Since we know that the tangent of an angle is , we can say:

This can be solved using your calculator:

or

Now, to solve for , use the Pythagorean Theorem, , where and are the legs of a triangle and is the triangle's hypotenuse. Here, , so we can substitute that in and rearrange the equation to solve for :

Substituting in the known values:

, or approximately . Rounding, this is .

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