### All ACT Math Resources

## Example Questions

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

If angle A measures 30 degrees and the hypotenuse is 4, what is the length of AB in the given right triangle?

**Possible Answers:**

√3

2

2√3

4

8√3

**Correct answer:**

2√3

Cosine A = Adjacent / Hypotenuse = AB / AC = AB / 4

Cosine A = AB / 4

Cos (30º) = √3 / 2 = AB / 4

Solve for AB

√3 / 2 = AB / 4

AB = 4 * (√3 / 2) = 2√3

### Example Question #7 : Cosine

**Possible Answers:**

**Correct answer:**

To solve this problem you need to make the triangle that the problem is talking about. Cosine is equal to the adjacent side over the hypotenuse of a right triangle

So this is what our triangle looks like:

Now use the pythagorean theorem to find the other side:

Sine is equal to the opposite side over the hypotenuse, the opposite side is 12

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

The hypotenuse of right triangle HLM shown below is long. The cosine of angle is . How many inches long is ?

**Possible Answers:**

**Correct answer:**

Remember that

Then, we can set up the equation using the given information.

Now, solve for .

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

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

**Possible Answers:**

**Correct answer:**

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

Solving for , we get:

or

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

A man has a rope that is long, attached to the top of a small building. He pegs the rope into the ground at an angle of . How far away from the building did he walk horizontally to attach the rope to the ground? Round to the nearest inch.

**Possible Answers:**

**Correct answer:**

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

We know that the cosine of an angle is equal to the ratio of the side *adjacent* 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 #3 : How To Find A Missing Side With Cosine

In the right triangle shown above, what is the ?

**Possible Answers:**

**Correct answer:**

Use SOH-CAH-TOA to solve for the sine of a given angle. This stands for:

.

From our triangle we see that at point , the adjacent side is side and the hypotenuse doesn't depend upon position, it's always . Thus we get that

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

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 #4 : How To Find A Missing Side With Cosine

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.

Use a calculator or reference to approximate cosine.

Isolate the variable term.

Thus, .

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

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 Cosine

Edgar is standing at the top of a -foot long slide looking down the slope. He knows the angle the top of the slide makes with the vertical ladder he just climbed is . How far, to the nearest foot, did Edgar climb to the top of the ladder?

**Possible Answers:**

**Correct answer:**

Edgar is effectively standing on top of a right triangle, since the angle from the vertical ladder to the ground can be assumed to be . In this case, the cosine function will help us out, so long as we remember our **SOHCAHTOA** mnemonic.

We can solve for , since the problem allows us to round.

Thus, Edgar climbed feet, rounded to the nearest foot.

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