# ACT Math : Cosine

## Example Questions

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### Example Question #1 : Cosine

In the above triangle,  and . Find .

Explanation:

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

### Example Question #2 : Cosine

For the above triangle,  and . Find .

Explanation:

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

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

For the above triangle,  and . Find .

This triangle cannot exist.

This triangle cannot exist.

Explanation:

With right triangles, we can use SOH CAH TOA to solve for unknown side lengths and angles. For this problem, we are given the adjacent and hypotenuse sides of the triangle with relation to the angle. However, if we plug the given values into the formula for cosine, we get:

This problem does not have a solution. The sides of a right triangle must be shorter than the hypotenuse. A triangle with a side longer than the hypotenuse cannot exist. Similarly, the domain of the arccos function is . It is not defined at 1.3.

### Example Question #4 : Cosine

A  rope is thrown down from a building to the ground and tied up at a distance of   from the base of the building. What is the angle measure between the rope and the ground? Round to the nearest hundredth of a degree.

Explanation:

You can draw your scenario using the following right triangle:

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

or  degrees.

### Example Question #5 : Cosine

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

Explanation:

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

or .

### Example Question #6 : Cosine

A support beam (buttress) lies against a building under construction. If the beam is  feet long and strikes the building at a point  feet up the wall, what angle does the beam strike the building at? Round to the nearest degree.

Explanation:

Our answer lies in inverse functions. If the buttress is  feet long and is  feet up the ladder at the desired angle, then:

Thus, using inverse functions we can say that

Thus, our buttress strikes the buliding at approximately a  angle.

### Example Question #7 : Cosine

A stone monument stands as a tourist attraction. A tourist wants to catch the sun at just the right angle to "sit" on top of the pillar. The tourist lies down on the ground  meters away from the monument, points the camera at the top of the monument, and the camera's display reads "DISTANCE --  METERS". To the nearest  degree, what angle is the sun at relative to the horizon?

Explanation:

Our answer lies in inverse functions. If the monument is  meters away and the camera is  meters from the monument's top at the desired angle, then:

Thus, using inverse functions we can say that

Thus, our buttress strikes the buliding at approximately a  angle.

### 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?

8√3

2√3

4

2

√3

2√3

Explanation:

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 #9 : Cosine

Explanation:

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 #1 : Cosine

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