Q: A building casts a shadow of 12 meters when the angle of elevation of the sun is $$30^\circ. $$What is the height of the building?

$$12 \tan 30^\circ = 12 \times \frac{1}{\sqrt{3}} = 4\sqrt{3}$$ meters

Q: Which of the following is the correct identity for $$\tan(a+b)$$?

$$\tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}$$

Q: A ramp rises 2 meters over a horizontal distance of 8 meters. What is the angle of elevation $$\theta of $$the ramp?

$$\theta = \arctan \left( \frac{2}{8} \right ) = \arctan \left( \frac{1}{4} \right )$$

Q: A pilot is flying at an altitude of 5000 meters and sees an airport at an angle of depression of $$10^\circ. $$What is the horizontal distance to the airport?

$$\frac{5000}{\tan 10^\circ}$$ meters

Question 1

Which of the following is the correct formula for the cosine of the sum of two angles $$a$$ and $$b$$?

Accepted Answer

$$\cos(a+b) = \cos a \cdot \cos b - \sin a \cdot \sin b$$

Question 2

A ladder leans against a wall, forming an angle $$	heta$$ with the ground. If $$\sin 	heta = 0.6$$ and the ladder is 10 meters long, how high up the wall does the ladder reach?

Accepted Answer

$$6$$ meters

Question 3

Which of the following is the correct double angle formula for $$\sin 2a$$?

Accepted Answer

$$\sin 2a = 2 \sin a \cos a$$

Question 4

If $$	an x = 3$$, what is the value of $$\sec^2$ x$$?

Accepted Answer

$$10$$

Question 5

Given $$\cos 2a = \cos^2$ a - \sin^2$ a$$, which of the following is also a valid expression for $$\cos 2a$$?

Accepted Answer

$$2 \cos^2$ a - 1$$

Question 6

A building casts a shadow of 12 meters when the angle of elevation of the sun is $$30^\circ. $$What is the height of the building?

Accepted Answer

$$12 	an 30^\circ = 12 	imes \frac{1}{\sqrt{3}} = 4\sqrt{3}$$ meters

Question 7

Which of the following is the correct identity for $$	an(a+b)$$?

Accepted Answer

$$	an(a+b) = \frac{	an a + 	an b}{1 - 	an a 	an b}$$

Question 8

If $$\sin x = \frac{3}{5}$$ and $$x is in $$the first quadrant, what is $$\cos x$$?

Accepted Answer

$$\frac{4}{5}$$

Question 9

Which of the following is the correct definition of $$\csc x$$?

Accepted Answer

$$\csc x = \frac{1}{\sin x}$$

Question 10

A ramp rises 2 meters over a horizontal distance of 8 meters. What is the angle of elevation $$	heta of $$the ramp?

Accepted Answer

$$	heta = \arctan \left( \frac{2}{8} ight ) = \arctan \left( \frac{1}{4} ight )$$

Question 11

Which of the following is the correct expression for $$\cot x in $$terms of $$\sin x$$ and $$\cos x$$?

Accepted Answer

$$\cot x = \frac{\cos x}{\sin x}$$

Question 12

If $$\sin^2$ x + \cos^2$ x = 1$$, what is $$\sin^2$ x in $$terms of $$\cos^2$ x$$?

Accepted Answer

$$\sin^2$ x = 1 - \cos^2$ x$$

Question 13

A Ferris wheel has a radius of 20 meters. What is the height above the center of a passenger at an angle $$	heta$$ from the horizontal?

Accepted Answer

$$20 \sin 	heta$$

Question 14

Which of the following is the correct definition of $$\sec x$$?

Accepted Answer

$$\sec x = \frac{1}{\cos x}$$

Question 15

If $$	an x = \frac{1}{2}$$, what is $$\sin^2$ x + \cos^2$ x$$?

Accepted Answer

$$1$$

Question 16

Which of the following is the correct formula for $$\cos(a-b)$$?

Accepted Answer

$$\cos(a-b) = \cos a \cos b + \sin a \sin b$$

Question 17

A pilot is flying at an altitude of 5000 meters and sees an airport at an angle of depression of $$10^\circ. $$What is the horizontal distance to the airport?

Accepted Answer

$$\frac{5000}{	an 10^\circ}$$ meters

Question 18

Which of the following is the correct Pythagorean identity?

Accepted Answer

$$\sin^2$ x + \cos^2$ x = 1$$

Question 19

If $$	an^2$ x + 1 = \sec^2$ x$$, what is $$	an^2$ 45^\circ + 1$$?

Accepted Answer

$$2$$

Question 20

Which of the following is the correct expression for $$	an^2$ x in $$terms of $$\sec^2$ x$$?

Accepted Answer

$$	an^2$ x = \sec^2$ x - 1$$

Trigonometric Identities and Functions: Key Formulas and Relationships

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