How to find a missing side with cosine - ACT Math

Card 0 of 99

Question

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

Answer

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

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Question

$\textup{If}\ cos(\theta)=\frac{16}{20}$

$\textup{then}\ sin(\theta)=$

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

$cos(\theta)=\frac{adj}{hyp}$

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

$sin(\theta)=\frac{opp}{hyp}$

$sin(\theta)=\frac{12}{20}=\frac{3}{5}$

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Question

The hypotenuse of right triangle HLM shown below is long. The cosine of angle is $\frac{2}{5}$ . How many inches long is ?

Answer

Remember that $\cos\theta=\frac{adjacent}{hypotenuse}$

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

$\cos H=\frac{HL}{19}=\frac{2}{5}$

Now, solve for .

$HL=\frac{38}{5}=7.6$

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Question

A man has a rope that is $40\textup{ feet}$ long, attached to the top of a small building. He pegs the rope into the ground at an angle of $14.5$^{\circ}$$ . How far away from the building did he walk horizontally to attach the rope to the ground? Round to the nearest inch.

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:

$cos(14.5) =\frac{x}{40}$

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.

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Question

What is $\small x$ in the right triangle above? Round to the nearest hundredth.

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:

$\small cos(9.75)=\frac{x}{375}$

Solving for $\small x$ , we get:

$\small 375*cos(9.75)=x$

$\small x=369.58352214677914$ or $\small 369.58$

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Question

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 $75 ^{\circ}$ to the vertical, how far above the ground to the nearest foot is the landing gear?

(Ignore the height of the plane itself).

Answer

The plane itself is effectively at the top of a right triangle, with topmost angle $75^{\circ}$ and hypotenuse feet. If this is the case, then SOHCAHTOA tells us that $\textup{cos} (75)^{\circ} = \frac{\textup{adjacent}}{\textup{hypotenuse}} = \frac{x}{43}$ .

Now, solve for the adjacent:

$x = \textup{cos}(75^{\circ}) \cdot 43 \approx 11.13$

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

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Question

In the right triangle shown above, what is the $\cos\left (A \right )$ ?

Answer

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

$\sin = \frac{\textup{opposite}}{\textup{hypotenuse}}$

$\cos = \frac{\textup{adjacent}}{\textup{hypotenuse}}$

$\tan = \frac{\textup{opposite}}{\textup{adjacent}}$ .

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 $cos(A)=\frac{b}{c}$

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Question

In a given right triangle $\Delta ABC$ , hypotenuse and $\angle A = 42^{\circ}$ . Using the definition of $\cos$ , find the length of leg . Round all calculations to the nearest tenth.

Answer

In right triangles, SOHCAHTOA tells us that $\cos A =\frac{\textup{adjacent}}{\textup{hypotenuse}}$ , and we know that $\angle A = 42^{\circ}$ and hypotenuse . Therefore, a simple substitution and some algebra gives us our answer.

$\cos 42^{\circ} = \frac{AB}{25}$

$.7 = \frac{AB}{25}$ Use a calculator or reference to approximate cosine.

Isolate the variable term.

Thus, .

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Question

In a given right triangle $\Delta ABC$ , hypotenuse and $\angle A = 66^{\circ}$ . Using the definition of $\cos$ , find the length of leg . Round all calculations to the nearest hundredth.

Answer

In right triangles, SOHCAHTOA tells us that $\cos A = \frac{\textup{adjacent}}{\textup{hypotenuse}}$ , and we know that $A = 66^{\circ}$ and hypotenuse . Therefore, a simple substitution and some algebra gives us our answer.

$\cos 66^{\circ} = \frac{AB}{8}$

$.41 = \frac{AB}{8}$ Use a calculator or reference to approximate cosine.

Isolate the variable term.

Thus, .

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Question

In a given right triangle $\Delta ABC$ , hypotenuse and $\angle C = 75^{\circ}$ . Using the definition of $\cos$ , find the length of leg . Round all calculations to the nearest tenth.

Answer

In right triangles, SOHCAHTOA tells us that $\cos C = \frac{\textup{adjacent}}{\textup{hypotenuse}}$ , and we know that $A = 75^{\circ}$ and hypotenuse . Therefore, a simple substitution and some algebra gives us our answer.

$\cos 75^{\circ} = \frac{CB}{42}$

$.3 = \frac{CB}{42}$ Use a calculator or reference to approximate cosine.

Isolate the variable term.

Thus, .

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Question

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?

Answer

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.

$\textup{SOH: }\textup{Sine}=\frac{\textup{opposite}}{\textup{hypotenuse}}$

$\textup{CAH: }\textup{Cosine}=\frac{\textup{adjacent}}{\textup{hypotenuse}}$

$\textup{TOA: }\textup{Tangent}=\frac{\textup{opposite}}{\textup{adjacent}}$

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.

$\cos 68^{\circ}=\frac{x}{35}$

$35\cos 68^{\circ}=\frac{x}{35}\cdot 35$

$35\cos 68^{\circ}=x$

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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Question

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

Answer

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

Compare your answer with the correct one above

Question

$\textup{If}\ cos(\theta)=\frac{16}{20}$

$\textup{then}\ sin(\theta)=$

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

$cos(\theta)=\frac{adj}{hyp}$

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

$sin(\theta)=\frac{opp}{hyp}$

$sin(\theta)=\frac{12}{20}=\frac{3}{5}$

Compare your answer with the correct one above

Question

The hypotenuse of right triangle HLM shown below is long. The cosine of angle is $\frac{2}{5}$ . How many inches long is ?

Answer

Remember that $\cos\theta=\frac{adjacent}{hypotenuse}$

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

$\cos H=\frac{HL}{19}=\frac{2}{5}$

Now, solve for .

$HL=\frac{38}{5}=7.6$

Compare your answer with the correct one above

Question

A man has a rope that is $40\textup{ feet}$ long, attached to the top of a small building. He pegs the rope into the ground at an angle of $14.5$^{\circ}$$ . How far away from the building did he walk horizontally to attach the rope to the ground? Round to the nearest inch.

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:

$cos(14.5) =\frac{x}{40}$

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.

Compare your answer with the correct one above

Question

What is $\small x$ in the right triangle above? Round to the nearest hundredth.

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:

$\small cos(9.75)=\frac{x}{375}$

Solving for $\small x$ , we get:

$\small 375*cos(9.75)=x$

$\small x=369.58352214677914$ or $\small 369.58$

Compare your answer with the correct one above

Question

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 $75 ^{\circ}$ to the vertical, how far above the ground to the nearest foot is the landing gear?

(Ignore the height of the plane itself).

Answer

The plane itself is effectively at the top of a right triangle, with topmost angle $75^{\circ}$ and hypotenuse feet. If this is the case, then SOHCAHTOA tells us that $\textup{cos} (75)^{\circ} = \frac{\textup{adjacent}}{\textup{hypotenuse}} = \frac{x}{43}$ .

Now, solve for the adjacent:

$x = \textup{cos}(75^{\circ}) \cdot 43 \approx 11.13$

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

Compare your answer with the correct one above

Question

In the right triangle shown above, what is the $\cos\left (A \right )$ ?

Answer

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

$\sin = \frac{\textup{opposite}}{\textup{hypotenuse}}$

$\cos = \frac{\textup{adjacent}}{\textup{hypotenuse}}$

$\tan = \frac{\textup{opposite}}{\textup{adjacent}}$ .

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 $cos(A)=\frac{b}{c}$

Compare your answer with the correct one above

Question

In a given right triangle $\Delta ABC$ , hypotenuse and $\angle A = 42^{\circ}$ . Using the definition of $\cos$ , find the length of leg . Round all calculations to the nearest tenth.

Answer

In right triangles, SOHCAHTOA tells us that $\cos A =\frac{\textup{adjacent}}{\textup{hypotenuse}}$ , and we know that $\angle A = 42^{\circ}$ and hypotenuse . Therefore, a simple substitution and some algebra gives us our answer.

$\cos 42^{\circ} = \frac{AB}{25}$

$.7 = \frac{AB}{25}$ Use a calculator or reference to approximate cosine.

Isolate the variable term.

Thus, .

Compare your answer with the correct one above

Question

In a given right triangle $\Delta ABC$ , hypotenuse and $\angle A = 66^{\circ}$ . Using the definition of $\cos$ , find the length of leg . Round all calculations to the nearest hundredth.

Answer

In right triangles, SOHCAHTOA tells us that $\cos A = \frac{\textup{adjacent}}{\textup{hypotenuse}}$ , and we know that $A = 66^{\circ}$ and hypotenuse . Therefore, a simple substitution and some algebra gives us our answer.

$\cos 66^{\circ} = \frac{AB}{8}$

$.41 = \frac{AB}{8}$ Use a calculator or reference to approximate cosine.

Isolate the variable term.

Thus, .

Compare your answer with the correct one above

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