Q: A surveyor stands on one side of a river and measures a $$90^\circ$$ angle at point $$C$$, then walks $$300$$ meters to point $$B$$ and measures an angle $$\beta = 25^\circ to $$point $$A$$ directly across the river. What is the width of the river (distance $$b$$), rounded to the nearest meter?

$$b = 300 \cdot \tan 25^\circ \approx 140$$ meters

Q: A tree casts a shadow $$48$$ feet long. If the angle of elevation from the tip of the shadow to the top of the tree is $$65^\circ$$, what is the height of the tree?

$$h = 48 \cdot \tan 65^\circ \approx 103$$ feet

Question 1

Given a right triangle with sides $$a$$, $$b$$, and hypotenuse $$c$$, and angle $$	heta$$ opposite side $$b$$, which of the following correctly expresses $$\sin 	heta in $$terms of the triangle's sides?

Accepted Answer

$$\sin 	heta = \frac{b}{c}$$

Question 2

A surveyor stands on one side of a river and measures a $$90^\circ$$ angle at point $$C$$, then walks $$300$$ meters to point $$B$$ and measures an angle $$\beta = 25^\circ to $$point $$A$$ directly across the river. What is the width of the river (distance $$b$$), rounded to the nearest meter?

Accepted Answer

$$b = 300 \cdot 	an 25^\circ \approx 140$$ meters

Question 3

If $$\sin A = \cos B in a $$right triangle, what is the relationship between angles $$A$$ and $$B$$?

Accepted Answer

$$A$$ and $$B$$ are complementary angles

Question 4

Given the triangle with sides $$7$$, $$10$$, and $$b as $$shown, what is the value of $$b$$?

Accepted Answer

$$b = \sqrt{10^2$ - 7^2$} = \sqrt{51}$$

Question 5

A tree casts a shadow $$48$$ feet long. If the angle of elevation from the tip of the shadow to the top of the tree is $$65^\circ$$, what is the height of the tree?

Accepted Answer

$$h = 48 \cdot 	an 65^\circ \approx 103$$ feet

Question 6

Which of the following is the correct definition of $$	an 	heta$$ for a right triangle with sides $$a$$, $$b$$, and hypotenuse $$c$$?

Accepted Answer

$$	an 	heta = \frac{b}{a}$$

Question 7

If $$\sin 35^\circ = x$$, what is $$\cos 55^\circ$$?

Accepted Answer

$$\cos 55^\circ = x$$

Question 8

A pilot flies 2 miles on a bearing of $$N45^\circ W$$, then turns $$90^\circ$$ and flies 1 mile southwest. What bearing should the control tower use to locate the aircraft?

Accepted Answer

$$N75^\circ W$$

Question 9

Given a right triangle with legs $$a = 7$$, $$b = 4$$, and hypotenuse $$c$$, what is the value of $$c$$?

Accepted Answer

$$c = \sqrt{7^2$ + 4^2$} = \sqrt{65}$$

Question 10

If the angle of depression from the top of a $$1000-$$foot cliff to two cars in the valley below are $$10^\circ$$ and $$15^\circ$$, how far apart are the cars (to the nearest foot)?

Accepted Answer

$$d = 1000 \left( \cot 10^\circ - \cot 15^\circ \right) \approx 3,047$$ feet

Question 11

Which of the following is the correct mnemonic for remembering the definitions of sine, cosine, and tangent in a right triangle?

Accepted Answer

SOH-CAH-TOA

Question 12

Given a right triangle with $$\angle A = 35^\circ$$ and side opposite $$A is$$ $$8$$, what is the length of the hypotenuse $$c$$?

Accepted Answer

$$c = 8 / \sin 35^\circ \approx 13.95$$

Question 13

If $$	an 	heta = \frac{y}{x}$$, which of the following is true for a point $$(x, y) on $$the terminal side of $$	heta on $$the circle $$x^2$$ + $$y^2$$ = $$r^2$$?

Accepted Answer

$$	an 	heta = \frac{y}{x}$$

Question 14

Which of the following is the correct expression for $$\sec 	heta in $$terms of the sides of a right triangle?

Accepted Answer

$$\sec 	heta = \frac{c}{a}$$

Question 15

A ladder leans against a wall, making a $$70^\circ$$ angle with the ground. If the foot of the ladder is $$5$$ feet from the wall, how long is the ladder?

Accepted Answer

$$L = 5 / \cos 70^\circ \approx 14.6$$ feet

Right Triangle Trigonometry and Applications

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