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

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