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

## Example Questions

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

You have a 30-60-90 triangle. If the hypotenuse length is 8, what is the length of the side opposite the 30 degree angle?

3√3

4√3

3

4√2

4

4

Explanation:

sin(30º) = ½

sine = opposite / hypotenuse

½ = opposite / 8

Opposite = 8 * ½ = 4

### Example Question #2 : How To Find A Missing Side With Sine

If a right triangle has a 30 degree angle, and the opposite leg of the 30 degree angle has a measure of 12, what is the value of the hypotenuse?

12 * 31/2

18

12 * 21/2

15

24

24

Explanation:

Use SOHCAHTOA. Sin(30) = 12/x, then 12/sin(30) = x = 24.

You can also determine the side with a measure of 12 is the smallest side in a 30:60:90 triangle. The hypotenuse would be twice the length of the smallest leg.

### Example Question #3 : How To Find A Missing Side With Sine

The radius of the above circle is  is the center of the circle. . Find the length of chord .

Explanation:

We can solve for the length of the chord by drawing a line the bisects the angle and the chord, shown below as .

In this circle, we can see the triangle  has a hypotenuse equal to the radius of the circle (), an angle  equal to half the angle made by the chord, and a side  that is half the length of the chord.  By using the sine function, we can solve for .

The length of the entire chord is twice the length of , so the entire chord length is .

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

The above circle has a radius of  and a center at . . Find the length of chord .

Explanation:

We can solve for the length of the chord by drawing a line the bisects the angle and the chord, shown below as .

In this circle, we can see the triangle  has a hypotenuse equal to the radius of the circle (), an angle  equal to half the angle made by the chord, and a side  that is half the length of the chord.  By using the sine function, we can solve for .

The length of the entire chord is twice the length of , so the entire chord length is .

### Example Question #5 : How To Find A Missing Side With Sine

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

Explanation:

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

Solving for , we get:

or

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

A man has set up a ground-level sensor to look from the ground to the top of a  tall building. The sensor must have an angle of  upward to the top of the building. How far is the sensor from the top of the 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 top of the building. We know that the sine of an angle is equal to the ratio of the side opposite to that angle to the hypotenuse 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 #7 : How To Find A Missing Side With Sine

Below is right triangle  with sides . What is ?

Explanation:

To find the sine of an angle, remember the mnemonic SOH-CAH-TOA.
This means that

.

We are asked to find the . So at point  we see that side  is opposite, and the hypotenuse never changes, so it is always . Thus we see that

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

In a given right triangle , hypotenuse  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 #9 : How To Find A Missing Side With Sine

In a given right triangle , hypotenuse  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 #10 : How To Find A Missing Side With Sine

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