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Trigonometry : Solving Triangles

Study concepts, example questions & explanations for Trigonometry

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Example Questions

Example Question #1 : Finding Angles

A 15 foot plank has one end on the ground and one end 6 feet off the ground. What is the measure of the angle formed by the plank and the ground?

Possible Answers:

$\approx 50^{\circ}$

$\approx 70^{\circ}$

$\approx 10^{\circ}$

$\approx 24^{\circ}$

Correct answer:

$\approx 24^{\circ}$

Explanation:

The length of the plank becomes the hypotenuse of the triangle, while the distance between the plank and the ground becomes the length of one side. To solve for the angle between the plank and the ground, you must find the value of $sin (\angle )$ . The sine of the angle is the value of the opposite side over the hypotenuse, which are values that we know.

$sin(\angle )= \frac{opposite}{hypotenuse} = \frac{6}{15}$

$\angle =sin^{-1} \frac{6}{15} = 24^{\circ}$

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Example Question #1 : Finding Angles

Two angles in a triangle are $30^\circ$ and $70^\circ$ . What is the measure of the 3rd angle?

Possible Answers:

$100^\circ$

$90^\circ$

There is not enough information to determine the angle measure.

$80^\circ$

$70^\circ$

Correct answer:

$80^\circ$

Explanation:

The sum of the angles of a triangle is 180˚.

Thus, since the sum of our two angles is 100˚, our missing angle must be,

180 - 100 = 80 .

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Example Question #2 : Finding Angles

If the hypotenuse of a right triangle has a length of 6, and the length of a leg is 2, what is the angle between the hypotenuse and the leg?

Possible Answers:

$4.87^\circ$

$19.47^\circ$

$13.54^\circ$

$32.12^\circ$

$70.53^\circ$

Correct answer:

$70.53^\circ$

Explanation:

The leg must be an adjacent side to the hypotenuse.

Therefore, we can use inverse cosine to solve for the angle.

First write the equation for sine of an angle.

$cos(\theta)=\frac{\textup{Adjacent}}{\textup{Hypotenuse}}$

$\theta=cos^{-1}\left(\frac{\textup{Adjacent}}{\textup{Hypotenuse}}\right)$

Substitute the lengths given and solve for the angle.

$\theta=cos^{-1}\left(\frac{\textup{2}}{\textup{6}}\right)=70.53^\circ$

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Example Question #3 : Finding Angles

A skateboard ramp made so that the rider can gain sufficient speed before a jump is 15 feet high and the ramp is 17 feet long. What is the measure of the angle $\theta$ between the ramp and the ground?

Possible Answers:

$\theta=70.4^{\circ}$

$\theta=62^{\circ}$

$\theta=20^{\circ}$

$\theta=34.2^{\circ}$

None of the other answers.

Correct answer:

$\theta=62^{\circ}$

Explanation:

For the angle in question we have the opposite side and the hypotenuse given to us. We can use the sine function.

$\sin\theta= \frac{opp}{hyp}\rightarrow\sin\theta=(\frac{15}{17})$

Use the inverse sin to find the measure of an angle between these sides:

$\theta =\sin^{-1}(\frac{15}{17})=62^{\circ}$

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Example Question #2 : Finding Angles

What angle does the ramp make with the bottom of the stair?

Screen_shot_2014-02-15_at_3.16.35_pm

Possible Answers:

33.7^o

30^o

45.0^o

$\small 28.9^o$

56.3^o

Correct answer:

33.7^o

Explanation:

$\small tan(c)=\frac{c}{a}=\frac{2}{3}$

$\small tan(c)=\frac{2}{3}$

$\small c=tan^{-1}(\frac{2}{3})$

c=33.7^o

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Example Question #1 : Solving Triangles

Given that angle A is 23.0^o, side a is 1.43 in., and side b is 3.62 in., what is the angle of B?

Screen shot 2014 02 15 at 3.41.09 pm

Possible Answers:

70^o

81.5^o

83.6^o

55^o

90^o

Correct answer:

81.5^o

Explanation:

$\small \small \small \frac{sin(23.0^o)}{1.43\: in.}=\frac{sin(B)}{3.62\: in.}$

$\small \small \small \sin(B)=\frac{3.62 in.\times \sin(23.0^o)}{1.43 in.}$

$\small \small \small \sin(B)=0.99$

$\small B=sin^{-1}(0.99)$

$\small B=81.5^o$

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Example Question #367 : Trigonometry

Artemis wants to build a ramp to make the entrance to their home more accessible. The angle between the ramp and the ground cannot be more than $7^\circ$ steep. Artemis has feet of space in their yard that the ramp can take up, and the distance between the ground and the house entrance is feet high. Will Artemis be able to build a ramp that complies with the $\leq7^\circ$ standard?

Possible Answers:

Yes

Correct answer:

Explanation:

Begin the problem by visualizing a diagram of the situation:

Screen shot 2020 08 09 at 3.36.03 pm

We can use inverse trig to solve for the unknown angle $\theta$ .

$\tan\theta=\frac{3}{12}$

$\theta=\tan^-^1\left (\frac{3}{12} \right )$

$\theta=14.04^\circ$

Because this angle is larger than $7^\circ$ , this ramp would not comply with standards.

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Example Question #368 : Trigonometry

If the hypotenuse of a right triangle has a length of 42.29 meters, and the length of a leg is 12.88 meters, what is the angle between the hypotenuse and the leg?

Possible Answers:

$15.15^\circ$

$14.64^\circ$

$74.85^\circ$

$99.99^\circ$

Correct answer:

$74.85^\circ$

Explanation:

The leg must be an adjacent side to the hypotenuse.

Therefore, we can use inverse cosine to solve for the angle.

First write the equation for sine of an angle.

$cos(\theta)=\frac{\textup{Adjacent}}{\textup{Hypotenuse}}$

$\theta=cos^{-1}\left(\frac{\textup{Adjacent}}{\textup{Hypotenuse}}\right)$

Substitute the lengths given and solve for the angle.

$\theta=cos^-^1\left (\frac{12.88}{49.29} \right )=74.85^\circ$

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Example Question #1 : Finding Sides

If the height of the stair is 2 ft, and the length of the stair is 3 ft, how long must the ramp be to cover the stair?

Screen_shot_2014-02-15_at_2.24.54_pm

Possible Answers:

5 ft

10 ft

2.5 ft

13 ft

3.6 ft

Correct answer:

3.6 ft

Explanation:

Use the Pythagorean triangle to solve for the third side of the triangle.

a^2+b^2=c^2

2^2+3^2=13

$\therefore c^2=13$

$c=\sqrt{13}$

Simplify and you have the answer:

c=3.6 ft

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Example Question #2 : Finding Sides

A surveyor looks up to the top of a mountain at an angle of 35 degrees. If the surveyor is 2400 feet from the base of the mountain, how tall is the mountain (to the nearest tenth of a foot)?

Possible Answers:

3427.6 feet

1966.0 feet

1376.6 feet

2135.8 feet

1680.5 feet

Correct answer:

1680.5 feet

Explanation:

This is a typical angle of elevation problem.

From the question, we can infer that the ratio of the mountain height to the surveyor's horizontal distance from the mountain is equal to the tangent of 35 degrees - the mountain height is the "opposite side", while the horizontal distance is the "adjacent side". In other words:

$tan (35^{\circ}) = \frac{mountain\ height}{ 2400\ ft}$

Solving for the mountain height gives a vertical distance of 1680.5 feet.

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Trigonometry : Solving Triangles

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 : Finding Angles

Example Question #1 : Finding Angles

Example Question #2 : Finding Angles

Example Question #3 : Finding Angles

Example Question #2 : Finding Angles

Example Question #1 : Solving Triangles

Example Question #367 : Trigonometry

Example Question #368 : Trigonometry

Example Question #1 : Finding Sides

Example Question #2 : Finding Sides

All Trigonometry Resources

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