### All ACT Math Resources

## Example Questions

### Example Question #72 : Trigonometry

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?

**Possible Answers:**

4

4√2

3

3√3

4√3

**Correct answer:**

4

sin(30º) = ½

sine = opposite / hypotenuse

½ = opposite / 8

Opposite = 8 * ½ = 4

### Example Question #1 : 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?

**Possible Answers:**

24

18

12 * 3^{1/2}

12 * 2^{1/2}

15

**Correct answer:**

24

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 #2 : 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 .

**Possible Answers:**

**Correct answer:**

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 #3 : How To Find A Missing Side With Sine

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

**Possible Answers:**

**Correct answer:**

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

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

**Possible Answers:**

**Correct answer:**

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 #5 : 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.

**Possible Answers:**

**Correct answer:**

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.

Thus, rounded, your answer is feet and inches.

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

Below is right triangle with sides . What is ?

**Possible Answers:**

**Correct answer:**

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 #7 : 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.

**Possible Answers:**

**Correct answer:**

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 Sine

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

**Possible Answers:**

**Correct answer:**

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 hundredth.

**Possible Answers:**

**Correct answer:**

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

Isolate the variable term.

Thus, .

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