ACT Math : How to find a missing side with 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 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:

2√3

2

√3

4

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 : 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 #1 : 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 .

Example Question #102 : Trigonometry

Cos75

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

Possible Answers:

Correct answer:

Explanation:

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 #1 : 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:

Explanation:

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

Cos30

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 #101 : Trigonometry

Right triangle

In the right triangle shown above, what is the ?

Possible Answers:

Correct answer:

Explanation:

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 

Right triangle

Example Question #3 : 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:

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 #1 : 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:

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 #2 : How To Find A Missing Side With Cosine

An airline pilot must know the exact vertical height of his plane above the runway to know when to extend the landing gear under the nose. If the nose of the plane is  feet away from the ground and the plane is descending at an angle of  to the vertical, how far above the ground to the nearest  foot is the landing gear?

(Ignore the height of the plane itself).

Possible Answers:

Correct answer:

Explanation:

The plane itself is effectively at the top of a right triangle, with topmost angle  and hypotenuse  feet. If this is the case, then SOHCAHTOA tells us that .

Now, solve for the adjacent:

Thus, our plane's nose is approximately  feet from the runway.

Example Question #7 : How To Find A Missing Side With Cosine

Edgar is standing at the top of a 35-foot long slide. He knows that the angle between the top of the slide and the ladder that he climbed to reach the top is 68 degrees. If the ladder meets the ground at a right angle, how far did Edgar climb?

Possible Answers:

Correct answer:

Explanation:

Edgar is standing on top of a right triangle because the angle from the vertical ladder to the ground is 90 degrees. To solve this question, you must know SOHCAHTOA. This acronym can be broken into three parts to solve for the sine, cosine, and tangent.

In order to solve for the missing side, you need to choose the trigonometric function that includes the side you need to find and the side that you know, relative to the angle that you know. In this case, you know the hypotenuse, so you would not use the tangent function; furthermore, you are looking for the side that is adjacent to the 68-degree angle. Thus, you need the function that incorporates adjacent and hypotenuse—the cosine function.

Typically, you would use a calculator at this point to calculate the cosine function; however, based on the answer choices provided, you can stop at this point.

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