ACT Math : Cosine

Study concepts, example questions & explanations for ACT Math

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Example Questions

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

Soh_cah_toa

In the above triangle,  and . Find .

Possible Answers:

Correct answer:

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

Soh_cah_toa

For the above triangle,  and . Find .

Possible Answers:

Correct answer:

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 #3 : How To Find An Angle With Cosine

Soh_cah_toa

For the above triangle,  and . Find .

Possible Answers:

This triangle cannot exist.

Correct answer:

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 : How To Find An Angle With 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.

Possible Answers:

Correct answer:

Explanation:

You can draw your scenario using the following right triangle:

Theta3

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 : How To Find An Angle With Cosine

Theta6

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

Possible Answers:

Correct answer:

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 : How To Find An Angle With 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.

Possible Answers:

Correct answer:

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 : How To Find An Angle With 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?

Possible Answers:

Correct answer:

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

Triangle

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

4

2

2√3

8√3

Correct answer:

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 #1 : How To Find A Missing Side With Cosine

 

Possible Answers:

Correct answer:

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:

Triangle_3

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 ?

5

Possible Answers:

Correct answer:

Explanation:

Remember that 

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

Now, solve for .

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