ACT Math : How to find a missing side with tangent

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #4 : Trigonometry

For the triangles in the figure given, which of the following is closest to the length of line NO?

Screen_shot_2013-06-03_at_5.56.50_pm

Possible Answers:

9

7

10

8

6

Correct answer:

9

Explanation:

First, solve for side MN. Tan(30°) = MN/16√3, so MN = tan(30°)(16√3) = 16. Triangle LMN and MNO are similar as they're both 30-60-90 triangles, so we can set up the proportion LM/MN = MN/NO or 16√3/16 = 16/x. Solving for x, we get 9.24, so the closest whole number is 9.

Example Question #1 : How To Find A Missing Side With Tangent

Josh is at the state fair when he decides to take a helicopter ride. He looks down at about a 35 ° angle of depression and sees his house. If the helicopter was about 250 ft above the ground, how far does the helicopter have to travel to be directly above his house?

sin 35 ° = 0.57          cos 35 ° = 0.82          tan 35 ° = 0.70

Possible Answers:

205.00 ft

304.88 ft

438.96 ft

142.50 ft

357.14 ft

Correct answer:

357.14 ft

Explanation:

The angle of depression is the angle formed by a horizontal line and the line of sight looking down from the horizontal.

This is a right triangle trig problem. The vertical distance is 250 ft and the horizontal distance is unknown. The angle of depression is 35°. We have an angle and two legs, so we use tan Θ = opposite ÷ adjacent. This gives an equation of tan 35° = 250/d where d is the unknown distance to be directly over the house.

Example Question #2 : How To Find A Missing Side With Tangent

Consider the triangle Find_missing_side_with_tangent where \displaystyle A = 55^{\circ}. Find \displaystyle x to the nearest decimal place.

 

Note: The triangle is not necessarily to scale

Possible Answers:

\displaystyle 12.9

\displaystyle 11.0

\displaystyle 6.3

None of the other answers

\displaystyle 15.7

Correct answer:

\displaystyle 6.3

Explanation:

To solve this equation, it is best to remember the mnemonic SOHCAHTOA which translates to Sin = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, and Tangent = Opposite / Adjacent. Looking at the problem statement, we are given an angle and the side opposite of the angle, and we are looking for the side adjacent to the angle. Therefore, we will be using the TOA part of the mnemonic. Inserting the values given in the problem statement, we can write \displaystyle TAN\left ( 55\right ) = \frac{9}{x}. Rearranging, we get \displaystyle x = \frac{9}{TAN\left ( 55\right )}. Therefore \displaystyle x = 6.3

Example Question #12 : Tangent

A piece of wire is tethered to a  building at a \displaystyle 37$^{\circ}$ angle. How far back is this wire from the bottom of said building? Round to the nearest inch.

Possible Answers:

Correct answer:

Explanation:

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

Tan30

Note importantly: We are looking for \displaystyle \small x as the the distance to the bottom of the building. Now, this is not very hard at all! We know that the tangent of an angle is equal to the ratio of the side adjacent to that angle to the opposite side of the triangle. Thus, for our triangle, we know:

\displaystyle \small tan(37) =\frac{30}{x}

Using your calculator, solve for \displaystyle x:

\displaystyle \small x=\frac{30}{tan(37)}

This is \displaystyle \small 39.8113446486123. Now, take the decimal portion in order to find the number of inches involved.

\displaystyle \small 0.8113446486123 * 12 = 9.7361357833476

Thus, rounded, your answer is \displaystyle \small 39 feet and \displaystyle \small 10 inches.

Example Question #11 : Trigonometry

Tan50

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

Possible Answers:

\displaystyle 35.82

\displaystyle 44.77

\displaystyle 23.26

\displaystyle 22.41

\displaystyle 26.21

Correct answer:

\displaystyle 26.21

Explanation:

Recall that the tangent of an angle is the ratio of the opposite side to the adjacent side of that triangle. Thus, for this triangle, we can say:

\displaystyle \small tan(27.45)=\frac{x}{50.45}

Solving for \displaystyle \small x, we get:

\displaystyle \small 50.45*tan(27.45)=x

\displaystyle \small x=26.20667657491611 or \displaystyle \small 26.21

Example Question #13 : Tangent

Right triangle

In the right triangle shown above, let \displaystyle a = 6\displaystyle \:b=8, and \displaystyle c = 10. What is the value of \displaystyle \tan(B)? Reduce all fractions.

Possible Answers:

\displaystyle \frac{3}{5}

\displaystyle \frac{4}{3}

\displaystyle \frac{5}{3}

\displaystyle \frac{3}{4}

\displaystyle \frac{4}{5}

Correct answer:

\displaystyle \frac{4}{3}

Explanation:

Right triangle

First we need to find the value of \displaystyle \tan(B). Use the mnemonic SOH-CAH-TOA which stands for:
\displaystyle \sin = \frac{opposite}{hypotenuse}

\displaystyle \cos = \frac{adjacent}{hypotenuse}

\displaystyle \tan = \frac{opposite}{adjacent}.

Now we see at point \displaystyle B we are looking for the opposite and adjacent sides, which are \displaystyle b and \displaystyle a respectively.
Thus we get that 

\displaystyle \tan(B) = \frac{b}{a} 

and plugging in our values and reducing yields:

\displaystyle \frac{4}{3}

Example Question #2921 : Act Math

In a given right triangle \displaystyle \Delta ABC, leg \displaystyle AB = 3.87 and \displaystyle \angle A = 61^{\circ}. Using the definition of \displaystyle \tan, find the length of leg \displaystyle CB. Round all calculations to the nearest hundredth.

Possible Answers:

\displaystyle 6.97

\displaystyle 4.58

\displaystyle 11.16

\displaystyle 2.48

\displaystyle 5.88

Correct answer:

\displaystyle 6.97

Explanation:

In right triangles, SOHCAHTOA tells us that , and we know that \displaystyle \angle A = 61^{\circ} and hypotenuse \displaystyle AC = 3.87. Therefore, a simple substitution and some algebra gives us our answer.

\displaystyle \tan 61^{\circ} = \frac{CB}{3.87}

\displaystyle 1.80 = \frac{CB}{3.87} Use a calculator or reference to approximate cosine.

\displaystyle 6.97= CB Isolate the variable term.

 

Thus, \displaystyle 6.97= CB.

Example Question #1 : How To Find A Missing Side With Tangent

In a given right triangle \displaystyle \Delta ABC, leg \displaystyle AB = 125 and \displaystyle \angle A = 50.5^{\circ}. Using the definition of \displaystyle \tan, find the length of leg \displaystyle CB. Round all calculations to the nearest tenth.

Possible Answers:

\displaystyle 150

\displaystyle 174.4

\displaystyle 168

\displaystyle 183

\displaystyle 166.2

Correct answer:

\displaystyle 150

Explanation:

In right triangles, SOHCAHTOA tells us that , and we know that \displaystyle \angle A = 50.5^{\circ} and hypotenuse \displaystyle AC = 125. Therefore, a simple substitution and some algebra gives us our answer.

\displaystyle \tan 50.5^{\circ} = \frac{CB}{125}

\displaystyle 1.2 = \frac{CB}{125} Use a calculator or reference to approximate cosine.

\displaystyle 150= CB Isolate the variable term.

 

Thus, \displaystyle 150= CB.

Example Question #12 : Tangent

In a given right triangle \displaystyle \Delta ABC, leg \displaystyle AB = 300 and \displaystyle \angle A = 27^{\circ}. Using the definition of \displaystyle \tan, find the length of leg \displaystyle CB. Round all calculations to the nearest tenth.

Possible Answers:

\displaystyle 127

\displaystyle 150

\displaystyle 237

\displaystyle 183

\displaystyle 166

Correct answer:

\displaystyle 150

Explanation:

In right triangles, SOHCAHTOA tells us that , and we know that \displaystyle \angle A = 27^{\circ} and leg\displaystyle AB = 300. Therefore, a simple substitution and some algebra gives us our answer.

\displaystyle \tan 27^{\circ} = \frac{CB}{300}

\displaystyle .5 = \frac{CB}{300} Use a calculator or reference to approximate cosine.

\displaystyle 150= CB Isolate the variable term. 

Thus, \displaystyle 150= CB.

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