Q: Which of the following is the correct expression for the area $$A of $$the opening of a rain gutter bent at an angle $$\theta as $$described in the study materials?

$$A(\theta) = 25 \sin \theta (\cos \theta + 1)$$

Q: Which of the following is the correct reciprocal identity for $$\csc \theta$$?

$$\csc \theta = \dfrac{1}{\sin \theta}$$

Q: Which of the following is the correct quotient identity for $$\tan \theta$$?

$$\tan \theta = \dfrac{\sin \theta}{\cos \theta}$$

Question 1

A right triangle has two angles of $$45^\circ. If $$the legs are each 1 unit long, what is the exact value of $$\sin 45^\circ$$?

Accepted Answer

$$\dfrac{\sqrt{2}}{2}$$

Question 2

Given a $$30^\circ$$-$$60^\circ$$-$$90^\circ$$ triangle with hypotenuse 2, what is the length of the side opposite the $$30^\circ$$ angle?

Accepted Answer

$$1$$

Question 3

Find the exact value of $$	an 60^\circ.$$

Accepted Answer

$$\sqrt{3}$$

Question 4

Which of the following is the correct expression for the area $$A of $$the opening of a rain gutter bent at an angle $$	heta as $$described in the study materials?

Accepted Answer

$$A(	heta) = 25 \sin 	heta (\cos 	heta + 1)$$

Question 5

A surveyor wants to find the width of a river. She walks 300 meters from point C to point B, turns $$90^\circ$$, and measures an angle $$	heta = 30^\circ to $$point A across the river. What is the width of the river, rounded to the nearest meter?

Accepted Answer

$$173$$ meters

Question 6

If $$\sin 	heta = \dfrac{1}{2}$$ and $$	heta is an $$acute angle, what is $$	heta in $$degrees?

Accepted Answer

$$30^\circ$$

Question 7

A ceilometer is set up 275 feet from a light projector. If the angle of elevation to the base of the cloud is $$75^\circ$$, what is the height of the cloud cover to the nearest foot?

Accepted Answer

$$1026$$ feet

Question 8

Find the exact value of $$\sec 45^\circ.$$

Accepted Answer

$$\sqrt{2}$$

Question 9

A statue sits atop a building. From 500 feet away, the angle of elevation to the base is $$43.2^\circ$$ and to the top is $$46.1^\circ. $$What is the height of the statue (to the nearest foot)?

Accepted Answer

Question 10

Which trigonometric function would you use to find the height of a building if you know the distance from the building and the angle of elevation to the top?

Accepted Answer

$$	an$$

Question 11

If $$	an 	heta = 1$$, what is $$	heta in $$degrees, where $$0^\circ \u003c 	heta \u003c 90^\circ$$?

Accepted Answer

$$45^\circ$$

Question 12

Evaluate $$\sin 60^\circ$$ exactly.

Accepted Answer

$$\dfrac{\sqrt{3}}{2}$$

Question 13

A rain gutter is constructed by bending up the sides of a 15-inch wide aluminum sheet at an angle $$	heta. $$For which value of $$	heta is $$the area of the opening maximized?

Accepted Answer

$$60^\circ$$

Question 14

Which of the following is the correct reciprocal identity for $$\csc 	heta$$?

Accepted Answer

$$\csc 	heta = \dfrac{1}{\sin 	heta}$$

Question 15

If $$\cos 	heta = \dfrac{1}{2}$$ and $$	heta is an $$acute angle, what is $$	heta in $$degrees?

Accepted Answer

$$60^\circ$$

Question 16

A calculator in degree mode is used to find $$\cos(53^\circ). $$What is the approximate value to two decimal places?

Accepted Answer

$$0.60$$

Question 17

Which of the following is the correct quotient identity for $$	an 	heta$$?

Accepted Answer

$$	an 	heta = \dfrac{\sin 	heta}{\cos 	heta}$$

Question 18

If $$\sin 	heta = \dfrac{\sqrt{3}}{2}$$ and $$	heta is an $$acute angle, what is $$	heta in $$degrees?

Accepted Answer

$$60^\circ$$

Question 19

A right triangle has legs of length 1 and hypotenuse $$\sqrt{2}. $$What is the value of $$\cos 45^\circ$$?

Accepted Answer

$$\dfrac{\sqrt{2}}{2}$$

Question 20

If $$	an(	heta) = \dfrac{1}{\sqrt{3}}$$ and $$	heta is an $$acute angle, what is $$	heta in $$degrees?

Accepted Answer

$$30^\circ$$

Trigonometric Functions of Acute Angles: Exact Values, Calculator Approximations, and Applications

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