### All ACT Math Resources

## Example Questions

### Example Question #1 : Trigonometry

What is the tangent of C in the given right triangle??

**Possible Answers:**

**Correct answer:**

Tangent = Opposite / Adjacent

### Example Question #1 : Tangent

Consider a right triangle with an inner angle .

If

and

what is ?

**Possible Answers:**

**Correct answer:**

The tangent of an angle *x *is defined as

Substituting the given values for cos *x *and sin *x*, we get

### Example Question #3 : Trigonometry

Triangle ABC shown is a right triangle. If the tangent of angle C is , what is the length of segment BC?

**Possible Answers:**

**Correct answer:**

Use the definition of the tangent and plug in the values given:

tangent C = Opposite / Adjacent = AB / BC = 3 / 7

Therefore, BC = 7.

### Example Question #1 : How To Find The Tangent Of An Angle

If the sine of an angle equals , and the cosine of the same angle equals , what is the tangent of the angle?

**Possible Answers:**

**Correct answer:**

The cosine of the angle is and since that is a reduced fraction, we know the hypotenuse is and the adjacent side equals .

The sine of the angle equals , and since the hyptenuse is already we know that we must multiply the numerator and denominator by to get the common denominator of . Therefore, the opposite side equals .

Since , the answer is .

### Example Question #1 : How To Find An Angle With Tangent

For the above triangle, and . Find .

**Possible Answers:**

This triangle cannot exist.

**Correct answer:**

With right triangles, we can use SOH CAH TOA to solve for unknown side lengths and angles. For this problem, we are given the opposite and adjacent sides of the triangle with relation to the angle. With this information, we can use the tangent function to find the angle.

### Example Question #2 : How To Find An Angle With Tangent

In the above triangle, and . Find .

**Possible Answers:**

**Correct answer:**

With right triangles, we can use SOH CAH TOA to solve for unknown side lengths and angles. For this problem, we are given the opposite and adjacent sides of the triangle with relation to the angle. With this information, we can use the tangent function to find the angle.

### Example Question #3 : How To Find An Angle With Tangent

For the above triangle, and . Find .

**Possible Answers:**

This triangle cannot exist.

**Correct answer:**

With right triangles, we can use SOH CAH TOA to solve for unknown side lengths and angles. For this problem, we are given the opposite and adjacent sides of the triangle with relation to the angle. With this information, we can use the tangent function to find the angle.

### Example Question #4 : How To Find An Angle With Tangent

A laser is placed at a distance of from the base of a building that is tall. What is the angle of the laser (presuming that it is at ground level) in order that it point at the top of the building?

**Possible Answers:**

**Correct answer:**

You can draw your scenario using the following right triangle:

Recall that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side of the triangle. You can solve for the angle by using an inverse tangent function:

or .

### Example Question #5 : How To Find An Angle With Tangent

What is the value of in the right triangle above? Round to the nearest hundredth of a degree.

**Possible Answers:**

**Correct answer:**

Recall that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side of the triangle. You can solve for the angle by using an inverse tangent function:

or .

### Example Question #1 : Tangent

**Possible Answers:**

10

6

8

7

9

**Correct answer:**

9

First, solve for side MN. Tan(30^{°}) = MN/16√3, so MN = tan(30^{°})(16√3) = 16. Triangle LMN and MNO are similar as they're both 30-60-90 triangles, so we can set up the proportion LM/MN = MN/NO or 16√3/16 = 16/x. Solving for x, we get 9.24, so the closest whole number is 9.