Cosine - ACT Math
Card 0 of 567
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
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
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The hypotenuse of right triangle HLM shown below is 
long. The cosine of angle 
is 
. How many inches long is 
?
The hypotenuse of right triangle HLM shown below is
Remember that 
Then, we can set up the equation using the given information.
Now, solve for 
.
Remember that
Then, we can set up the equation using the given information.
Now, solve for
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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.
A man has a rope that is
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.
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
Thus, rounded, your answer is
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What is 
in the right triangle above? Round to the nearest hundredth.
What is
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 
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
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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).
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
(Ignore the height of the plane itself).
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.
The plane itself is effectively at the top of a right triangle, with topmost angle
Now, solve for the adjacent:
Thus, our plane's nose is approximately
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In the right triangle shown above, what is the 
?
In the right triangle shown above, what is the
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 
Use SOH-CAH-TOA to solve for the sine of a given angle. This stands for:
From our triangle we see that at point
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In a given right triangle 
, hypotenuse 
and 
. Using the definition of 
, find the length of leg 
. Round all calculations to the nearest tenth.
In a given right triangle
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, 
.
In right triangles, SOHCAHTOA tells us that
Thus,
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In a given right triangle 
, hypotenuse 
and 
. Using the definition of 
, find the length of leg 
. Round all calculations to the nearest hundredth.
In a given right triangle
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, 
.
In right triangles, SOHCAHTOA tells us that
Thus, 
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In a given right triangle 
, hypotenuse 
and 
. Using the definition of 
, find the length of leg 
. Round all calculations to the nearest tenth.
In a given right triangle
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, 
.
In right triangles, SOHCAHTOA tells us that
Thus,
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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?
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?
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.
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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In the above triangle, 
and 
. Find 
.
In the above triangle,
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.
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.
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For the above triangle, 
and 
. Find 
.
For the above triangle,
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.
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.
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For the above triangle, 
and 
. Find 
.
For the above triangle,
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.
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
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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_._
A
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.
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:
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What is the value of 
in the right triangle above? Round to the nearest hundredth of a degree.
What is the value of
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 
.
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:
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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.
A support beam (buttress) lies against a building under construction. If the beam is
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.
Our answer lies in inverse functions. If the buttress is
Thus, using inverse functions we can say that
Thus, our buttress strikes the buliding at approximately a
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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?
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
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.
Our answer lies in inverse functions. If the monument is
Thus, using inverse functions we can say that
Thus, our buttress strikes the buliding at approximately a
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If 
and 
, what is the value of 
?
If
Based on this data, we can make a little triangle that looks like:
This is because 
.
Now, this means that 
must equal 
. (Recall that the cosine function is negative in the second quadrant.) Now, we are looking for:
or 
. This is the cosine of a reference angle of:
Looking at our little triangle above, we can see that the cosine of 
is 
.
Based on this data, we can make a little triangle that looks like:
This is because
Now, this means that
Looking at our little triangle above, we can see that the cosine of
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What is the cosine of the angle formed between the origin and the point 
if that angle is formed with one side of the angle beginning on the 
-axis and then rotating counter-clockwise to ? Round to the nearest hundredth.
What is the cosine of the angle formed between the origin and the point
Recall that when you calculate a trigonometric function for an obtuse angle like this, you always use the 
-axis as your reference point for your angle. (Hence, it is called the "reference angle.") Now, it is easiest to think of this like you are drawing a little triangle in the second quadrant of the Cartesian plane. It would look like:
So, you first need to calculate the hypotenuse:
So, the cosine of an angle is:
or, for your data, 
.
This is approximately 
. Rounding, this is 
. However, since 
is in the second quadrant your value must be negative: 
.
Recall that when you calculate a trigonometric function for an obtuse angle like this, you always use the
So, you first need to calculate the hypotenuse:
So, the cosine of an angle is:
This is approximately
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What is the cosine of the angle formed between the origin and the point 
if that angle is formed with one side of the angle beginning on the 
-axis and then rotating counter-clockwise to ? Round to the nearest hundredth.
What is the cosine of the angle formed between the origin and the point
Recall that when you calculate a trigonometric function for an obtuse angle like this, you always use the 
-axis as your reference point for your angle. (Hence, it is called the "reference angle.")
Now, it is easiest to think of this like you are drawing a little triangle in the third quadrant of the Cartesian plane. It would look like:
So, you first need to calculate the hypotenuse. You can do this by using the Pythagorean Theorem, 
, where 
and 
are the lengths of the legs of the triangle and 
the length of the hypotenuse. Rearranging the equation to solve for 
, you get:
Substituting in the given values:
So, the cosine of an angle is:
or, for your data, 
.
This is approximately 
. Rounding, this is 
. However, since 
is in the third quadrant your value must be negative: 
.
Recall that when you calculate a trigonometric function for an obtuse angle like this, you always use the
Now, it is easiest to think of this like you are drawing a little triangle in the third quadrant of the Cartesian plane. It would look like:
So, you first need to calculate the hypotenuse. You can do this by using the Pythagorean Theorem,
Substituting in the given values:
So, the cosine of an angle is:
This is approximately
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