# ACT Math : How to find a missing side with tangent

## Example Questions

### Example Question #1 : Trigonometry

For the triangles in the figure given, which of the following is closest to the length of line NO?

9

10

8

7

6

9

Explanation:

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.

### Example Question #1 : How To Find A Missing Side With Tangent

Josh is at the state fair when he decides to take a helicopter ride. He looks down at about a 35 ° angle of depression and sees his house. If the helicopter was about 250 ft above the ground, how far does the helicopter have to travel to be directly above his house?

sin 35 ° = 0.57          cos 35 ° = 0.82          tan 35 ° = 0.70

304.88 ft

438.96 ft

205.00 ft

142.50 ft

357.14 ft

357.14 ft

Explanation:

The angle of depression is the angle formed by a horizontal line and the line of sight looking down from the horizontal.

This is a right triangle trig problem. The vertical distance is 250 ft and the horizontal distance is unknown. The angle of depression is 35°. We have an angle and two legs, so we use tan Θ = opposite ÷ adjacent. This gives an equation of tan 35° = 250/d where d is the unknown distance to be directly over the house.

### Example Question #1 : How To Find A Missing Side With Tangent

Consider the triangle  where . Find  to the nearest decimal place.

Note: The triangle is not necessarily to scale

Explanation:

To solve this equation, it is best to remember the mnemonic SOHCAHTOA which translates to Sin = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, and Tangent = Opposite / Adjacent. Looking at the problem statement, we are given an angle and the side opposite of the angle, and we are looking for the side adjacent to the angle. Therefore, we will be using the TOA part of the mnemonic. Inserting the values given in the problem statement, we can write . Rearranging, we get . Therefore

### Example Question #1 : How To Find A Missing Side With Tangent

A piece of wire is tethered to a  building at a  angle. How far back is this wire from the bottom of said building? Round to the nearest inch.

Explanation:

Begin by drawing out this scenario using a little right triangle:

Note importantly: We are looking for  as the the distance to the bottom of the building. Now, this is not very hard at all! We know that the tangent of an angle is equal to the ratio of the side adjacent to that angle to the opposite side of the triangle. Thus, for our triangle, we know:

Using your calculator, solve for :

This is . Now, take the decimal portion in order to find the number of inches involved.

### Example Question #4 : How To Find A Missing Side With Tangent

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

Explanation:

Recall that the tangent of an angle is the ratio of the opposite side to the adjacent side of that triangle. Thus, for this triangle, we can say:

Solving for , we get:

or

### Example Question #1 : How To Find A Missing Side With Tangent

In the right triangle shown above, let , and . What is the value of  Reduce all fractions.

Explanation:

First we need to find the value of . Use the mnemonic SOH-CAH-TOA which stands for:

.

Now we see at point  we are looking for the opposite and adjacent sides, which are  and  respectively.
Thus we get that

and plugging in our values and reducing yields:

### Example Question #6 : How To Find A Missing Side With Tangent

In a given right triangle , leg  and . Using the definition of , find the length of leg . Round all calculations to the nearest hundredth.

Explanation:

In right triangles, SOHCAHTOA tells us that , and we know that  and hypotenuse . Therefore, a simple substitution and some algebra gives us our answer.

Use a calculator or reference to approximate cosine.

Isolate the variable term.

Thus, .

### Example Question #7 : How To Find A Missing Side With Tangent

In a given right triangle , leg  and . Using the definition of , find the length of leg . Round all calculations to the nearest tenth.

Explanation:

In right triangles, SOHCAHTOA tells us that , and we know that  and hypotenuse . Therefore, a simple substitution and some algebra gives us our answer.

Use a calculator or reference to approximate cosine.

Isolate the variable term.

Thus, .

### Example Question #8 : How To Find A Missing Side With Tangent

In a given right triangle , leg  and . Using the definition of , find the length of leg . Round all calculations to the nearest tenth.