Q: A building casts a shadow of 20 meters when the angle of elevation of the sun is $$30^\circ. $$What is the height of the building?

$$h = 20 \tan 30^\circ = 20 \times \frac{1}{\sqrt{3}} \approx 11.55$$ meters

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

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

Q: A ramp rises 1 meter for every 4 meters of horizontal distance. What is the angle $$\theta$$ the ramp makes with the ground?

$$\theta = \arctan\left(\frac{1}{4}\right) \approx 14.0^\circ$$

Q: A surveyor measures the angle of elevation to the top of a tower as $$53^\circ$$ from a point 40 meters from the base. What is the height of the tower?

$$h = 40 \tan 53^\circ \approx 53.1$$ 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

Given $$\sin 2a = 2 \sin a \cos a$$, what is $$\sin 60^\circ$$ using this identity?

Accepted Answer

$$\sin 60^\circ = 2 \sin 30^\circ \cos 30^\circ$$

Question 3

A ladder leans against a wall making an angle $$	heta$$ with the ground. If $$	an 	heta = \frac{3}{4}$$, what is $$\sin 	heta$$?

Accepted Answer

$$\sin 	heta = \frac{3}{5}$$

Question 4

Which of the following expresses the double angle formula for cosine?

Accepted Answer

$$\cos 2a = \cos^2$ a - \sin^2$ a$$

Question 5

If $$	an x = 2$$, what is $$\sec^2$ x$$?

Accepted Answer

$$\sec^2$ x = 5$$

Question 6

Which of the following is the correct definition of the cotangent function?

Accepted Answer

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

Question 7

A building casts a shadow of 20 meters when the angle of elevation of the sun is $$30^\circ. $$What is the height of the building?

Accepted Answer

$$h = 20 	an 30^\circ = 20 	imes \frac{1}{\sqrt{3}} \approx 11.55$$ meters

Question 8

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

Accepted Answer

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

Question 9

If $$\sin x = \frac{4}{5}$$ and $$x is in $$the first quadrant, what is $$\cos x$$?

Accepted Answer

$$\cos x = \frac{3}{5}$$

Question 10

Which of the following is the correct definition of the secant function?

Accepted Answer

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

Question 11

A ramp rises 1 meter for every 4 meters of horizontal distance. What is the angle $$	heta$$ the ramp makes with the ground?

Accepted Answer

$$	heta = \arctan\left(\frac{1}{4}ight) \approx 14.0^\circ$$

Question 12

Which of the following is the correct Pythagorean identity for sine and cosine?

Accepted Answer

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

Question 13

If $$	an x = 1$$, what is the value of $$x in $$degrees between $$0^\circ$$ and $$90^\circ$$?

Accepted Answer

$$x = 45^\circ$$

Question 14

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 15

A surveyor measures the angle of elevation to the top of a tower as $$53^\circ$$ from a point 40 meters from the base. What is the height of the tower?

Accepted Answer

$$h = 40 	an 53^\circ \approx 53.1$$ meters

Question 16

If $$\sin x = 0.6$$ and $$x is in $$the first quadrant, what is $$\csc x$$?

Accepted Answer

$$\csc x = \frac{1}{0.6} \approx 1.67$$

Question 17

Which of the following is the correct relationship between tangent and secant?

Accepted Answer

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

Question 18

If $$\cos x = 0.8$$ and $$x is in $$the first quadrant, what is $$\sec x$$?

Accepted Answer

$$\sec x = \frac{1}{0.8} = 1.25$$

Question 19

A tree is 15 meters tall and casts a shadow of 9 meters. What is the angle of elevation of the sun?

Accepted Answer

$$	heta = \arctan\left(\frac{15}{9}ight) \approx 59.0^\circ$$

Trigonometric Identities and Functions: Key Formulas and Relationships

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