Q: Given a right triangle with legs of length 3 and 4, what is the measure of the angle opposite the side of length 3 (rounded to 3 decimal places)?

$\theta = \arctan\left(\frac{3}{4}\right) \approx 36.870^\circ$

Q: A hiking path rises from an elevation of 3500 feet to 4000 feet over a horizontal distance of 4000 feet. What is the angle of inclination of the path?

$\theta = \arctan\left(\frac{4000 - 3500}{4000}\right) = \arctan\left(\frac{500}{4000}\right) \approx 7.125^\circ$

Q: A right triangle has an angle of $31^\circ$ and a hypotenuse of 8.0. What is the length of the side opposite the $31^\circ$ angle?

$8.0 \times \sin(31^\circ) \approx 4.123$

Q: Which trigonometric ratio would you use to find the length of the side adjacent to a given angle in a right triangle if you know the hypotenuse?

$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

Q: If you are given two side lengths of a right triangle, which method should you use to find the missing angle?

Use the inverse trigonometric function: $\theta = \arctan\left(\frac{\text{opposite}}{\text{adjacent}}\right)$

Q: Given a right triangle with sides 5, 12, and 13, what is the measure of the angle opposite the side of length 12?

$\theta = \arcsin\left(\frac{12}{13}\right) \approx 67.380^\circ$

Q: A right triangle has a hypotenuse of 10 and one angle of $45^\circ$. What is the length of each leg?

$10 \times \sin(45^\circ) = 10 \times \frac{\sqrt{2}}{2} \approx 7.071$

Question 1

A right triangle has one leg of length 5 and a hypotenuse of length 13. What is the length of the other leg?

Accepted Answer

$\sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$

Question 2

Given a right triangle with legs of length 3 and 4, what is the measure of the angle opposite the side of length 3 (rounded to 3 decimal places)?

Accepted Answer

$	heta = \arctan\left(\frac{3}{4}ight) \approx 36.870^\circ$

Question 3

A hiking path rises from an elevation of 3500 feet to 4000 feet over a horizontal distance of 4000 feet. What is the angle of inclination of the path?

Accepted Answer

$	heta = \arctan\left(\frac{4000 - 3500}{4000}ight) = \arctan\left(\frac{500}{4000}ight) \approx 7.125^\circ$

Question 4

A right triangle has an angle of $31^\circ$ and a hypotenuse of 8.0. What is the length of the side opposite the $31^\circ$ angle?

Accepted Answer

$8.0 	imes \sin(31^\circ) \approx 4.123$

Question 5

A coastal cliff stands 280 ft tall and is 2.2 km from the shore. A sailor observes the top of the cliff at an angle of elevation of $3.2^\circ$. What is the distance from the sailor to the top of the cliff (in kilometers)?

Accepted Answer

$\sqrt{2.2^2 + \left(\frac{280}{1000}\right)^2} \approx 2.247$ km

Question 6

Which trigonometric ratio would you use to find the length of the side adjacent to a given angle in a right triangle if you know the hypotenuse?

Accepted Answer

$\cos(	heta) = \frac{	ext{adjacent}}{	ext{hypotenuse}}$

Question 7

If you are given two side lengths of a right triangle, which method should you use to find the missing angle?

Accepted Answer

Use the inverse trigonometric function: $	heta = \arctan\left(\frac{	ext{opposite}}{	ext{adjacent}}ight)$

Question 8

A right triangle has legs of length 7 and 24. What is the length of the hypotenuse?

Accepted Answer

$\sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25$

Question 9

Given a right triangle with sides 5, 12, and 13, what is the measure of the angle opposite the side of length 12?

Accepted Answer

$	heta = \arcsin\left(\frac{12}{13}ight) \approx 67.380^\circ$

Question 10

A right triangle has an angle $x$ and its complement $y$. Which of the following is true about their sine and cosine values?

Accepted Answer

$\sin(x) = \cos(y)$

Question 11

A surveyor stands 2.2 km from a lighthouse that is 280 ft tall. If the angle of elevation to the top of the lighthouse is $3.2^\circ$, what is the horizontal distance from the surveyor to the base of the lighthouse (in kilometers)?

Accepted Answer

$2.2$ km

Question 12

Which equation would you use to find the missing side $a$ in a right triangle if you know the hypotenuse $c$ and the angle $	heta$ opposite $a$?

Accepted Answer

$a = c \sin(	heta)$

Question 13

A right triangle has a hypotenuse of 10 and one angle of $45^\circ$. What is the length of each leg?

Accepted Answer

$10 	imes \sin(45^\circ) = 10 	imes \frac{\sqrt{2}}{2} \approx 7.071$

Question 14

If a right triangle has legs of length 8 and 15, what is the measure of the angle opposite the side of length 8?

Accepted Answer

$	heta = \arctan\left(\frac{8}{15}ight) \approx 28.072^\circ$

Question 15

A right triangle has a hypotenuse of 20 and one leg of 12. What is the length of the other leg?

Accepted Answer

$\sqrt{20^2 - 12^2} = \sqrt{400 - 144} = \sqrt{256} = 16$

Question 16

Which trigonometric function relates the side opposite an angle to the side adjacent in a right triangle?

Accepted Answer

$	an(	heta) = \frac{	ext{opposite}}{	ext{adjacent}}$

Question 17

A right triangle has an angle $x$ and its complement $y$. If $x = 31^\circ$, what is $y$?

Accepted Answer

$y = 90^\circ - 31^\circ = 59^\circ$

Solving Right Triangles and Applications of Trigonometry

Study Guide - Practice Questions

Study Guide - Flashcards