Trigonometry Quizzes: Master Trig Ratios, Identities, and Equations

Question 1

Which trigonometric identity is used to simplify expressions involving sine and cosine?

Accepted Answer

Pythagorean identity

Answer

Double angle identity

Answer

Sum-to-product identity

Answer

Product-to-sum identity

Question 2

A tower is 100m high. From a point on level ground, the angle of elevation to the top of the tower is 30°. How far from the base of the tower is the point?

Accepted Answer

50√3 m

Answer

50m

Answer

100√3 m

Answer

100 m

Question 3

When is the sine function undefined?

Accepted Answer

Never

Answer

When the angle is 0°

Answer

When the angle is 90°

Answer

When the angle is 180°

Question 4

The sum-to-product identity for sine and cosine is:

Accepted Answer

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

Answer

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

Answer

tan(a + b) = (tan(a) + tan(b))/(1 - tan(a)tan(b))

Answer

cot(a + b) = (cot(a)cot(b) - 1)/(cot(a) + cot(b))

Question 5

Find the equivalent expression for sin(2θ).

Accepted Answer

2sinθcosθ

Answer

sin²θ + cos²θ

Answer

sin²θ - cos²θ

Answer

cos²θ - sin²θ

Question 6

A tall building casts a shadow of length 120 m when the angle of elevation to the top of the building is 30°. Calculate the height of the building.

Accepted Answer

80 m

Answer

60 m

Answer

100 m

Answer

120 m

Question 7

Identify the range of values for cosx.

Accepted Answer

-1 to 1

Answer

-√2 to √2

Answer

0 to 2

Answer

-2 to 2

Question 8

Identify the amplitude of the function y = 2sin(3x + π/2).

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 9

What is the value of cos(60°)?

Accepted Answer

√3/2

Answer

1/2

Answer

-1/2

Answer

1

Question 10

If the length of the adjacent side of a right-angled triangle is 6 cm and the hypotenuse is 15 cm, what is the length of the opposite side?

Accepted Answer

12 cm

Answer

9 cm

Answer

18 cm

Answer

10 cm

Question 11

Which trigonometric identity represents the Pythagorean Theorem?

Accepted Answer

sin² θ + cos² θ = 1

Answer

tan² θ + cot² θ = 1

Answer

sec² θ - 1 = tan² θ

Answer

csc² θ - 1 = cot² θ

Question 12

Identify the inverse function of the cosine function.

Accepted Answer

Arccosine

Answer

Arcsine

Answer

Arctangent

Answer

Arcsecant

Question 13

Specify the domain of the tangent function.

Accepted Answer

All real numbers except multiples of π/2

Answer

All real numbers

Answer

All positive real numbers

Answer

All negative real numbers

Question 14

What is the period of the cosine function?

Accepted Answer

2π

Answer

π

Answer

4π

Answer

6π

Question 15

What is the exact value of arctan(1)?

Accepted Answer

π/4

Answer

0

Answer

π/2

Answer

π

Question 16

In a right-angled triangle, the ratio of the length of the side adjacent to the acute angle to the length of the hypotenuse is known as the:

Accepted Answer

Cosine

Answer

Sine

Answer

Tangent

Answer

Cosecant

Question 17

Evaluate cos(π/3) + sin(π/4):

Accepted Answer

√3/2

Answer

0

Answer

1

Answer

√2/2

Question 18

Which of the following is a radian measure?

Accepted Answer

π/2

Answer

60°

Answer

300°

Answer

1 revolution

Question 19

Which of the following is not a trigonometric ratio?

Accepted Answer

Median

Answer

Sine

Answer

Cosine

Answer

Tangent

Question 20

Simplify the trigonometric identity: sin^2(θ) + cos^2(θ) = ?

Accepted Answer

1

Answer

0

Answer

2

Answer

θ

Question 21

Which of the following is the double-angle formula for cosine?

Accepted Answer

cos(2θ) = 2cos^2(θ) - 1

Answer

cos(2θ) = cos^2(θ) - sin^2(θ)

Answer

cos(2θ) = 1 - 2sin^2(θ)

Answer

cos(2θ) = 2cos(θ) - sin(θ)

Question 22

What is the value of sin(π/2)?

Accepted Answer

1

Answer

-1

Answer

0

Answer

Undefined

Question 23

What is the inverse trigonometric function of cosine?

Accepted Answer

cos^-1(x)

Answer

sin^-1(x)

Answer

tan^-1(x)

Answer

cot^-1(x)

Question 24

A tower has a height of 20 meters. From a point on the ground, the angle of elevation to the top of the tower is 30°. What is the horizontal distance between the point and the base of the tower?

Accepted Answer

10√3 m

Answer

10m

Answer

20m

Answer

20√3 m

Question 25

What is the period of the function y = cos(2x)?

Accepted Answer

2π

Answer

π

Answer

π/2

Answer

π/4

Question 26

What is the amplitude of the function y = 3sin(x - π/3)?

Accepted Answer

3

Answer

1

Answer

2

Answer

-3

Question 27

What is the inverse function of y = cos(x)?

Accepted Answer

y = cos⁻¹(x)

Answer

y = sin(x)

Answer

y = tan(x)

Answer

y = sec(x)

Question 28

In a right-angled triangle, which trigonometric ratio represents the ratio of the length of the side opposite the angle to the length of the hypotenuse?

Accepted Answer

Sine

Answer

Cosine

Answer

Tangent

Answer

Cotangent

Question 29

Which trigonometric identity is used to express (sin θ / cos θ)?

Accepted Answer

tan θ

Answer

cot θ

Answer

sec θ

Answer

cosec θ

Question 30

To determine the value of θ if sin θ = 0.75, which equation should be used?

Accepted Answer

θ = sin^-1(0.75)

Answer

θ = cos^-1(0.75)

Answer

θ = tan^-1(0.75)

Answer

θ = cot^-1(0.75)

Question 31

A ladder is placed against a wall such that its foot is 4 m from the wall and the top reaches a height of 6 m. Determine the length of the ladder.

Accepted Answer

10 m

Answer

5 m

Answer

8 m

Answer

12 m

Question 32

Which expression is equivalent to 2 cos^2 θ - 1?

Accepted Answer

cos 2θ

Answer

sin 2θ

Answer

tan 2θ

Answer

cot 2θ

Question 33

Find the amplitude of the function y = 2 cos 3x.

Accepted Answer

2

Answer

1

Answer

3

Answer

6

Question 34

Identify the trigonometric identity commonly utilized to simplify expressions involving sin^2 and cos^2.

Accepted Answer

Pythagorean identity

Answer

Double angle identity

Answer

Half angle identity

Answer

Product-to-sum identity

Question 35

What is the trigonometric value of tan(π/2)?

Accepted Answer

Undefined

Answer

0

Answer

1

Answer

-1

Question 36

Which transformation shifts the trigonometric function h(x) = cos(x) horizontally to the right by π/2 units?

Accepted Answer

h(x) = cos(x + π/2)

Answer

h(x) = cos(x - π/2)

Answer

h(x) = sin(x - π/2)

Answer

h(x) = sin(x + π/2)

Question 37

Identify the correct double-angle formula for sine.

Accepted Answer

sin(2θ) = 2sin(θ)cos(θ)

Answer

sin(2θ) = sin(θ) + sin(θ)

Answer

sin(2θ) = cos(θ) - cos(θ)

Answer

sin(2θ) = 2cos(θ)

Question 38

A ship's journey consists of sailing 10 km east, followed by 15 km north. Calculate the ship's distance from its starting point.

Accepted Answer

√250 km

Answer

18 km

Answer

25 km

Answer

5√2 km

Question 39

Determine the height of a building given that the angle of elevation from a point 50 meters away is measured to be 30°.

Accepted Answer

43.3 meters

Answer

25 meters

Answer

86.6 meters

Answer

Cannot be determined with the given information

Question 40

Identify the inverse function of sine.

Accepted Answer

arcsine

Answer

cosine

Answer

tangent

Answer

secant

Question 41

Determine the range of the function y = 2tan(x).

Accepted Answer

[-∞, ∞]

Answer

[-2, 2]

Answer

[-1, 1]

Answer

[0, ∞)

Question 42

Which trigonometric ratio is defined as the ratio of the length of the adjacent side to the length of the hypotenuse?

Accepted Answer

Cosine

Answer

Secant

Answer

Sine

Answer

Cotangent

Question 43

Find the value of sin(30°).

Accepted Answer

1/2

Answer

0

Answer

1

Answer

-1

Question 44

Which trigonometric identity allows us to express (cos A - sin A)² in terms of trigonometric functions?

Accepted Answer

Pythagorean identity

Answer

Half-angle identity

Answer

Double-angle identity

Answer

Sum-to-product identity

Question 45

A flagpole casts a shadow that is 20 feet long when the angle of elevation to the sun is 30°. What is the height of the flagpole?

Accepted Answer

11.55 feet

Answer

25 feet

Answer

15 feet

Answer

20 feet

Question 46

Which transformation shifts the graph of y = sin x vertically upwards by 3 units?

Accepted Answer

y = sin x + 3

Answer

y = 3sin x

Answer

y = sin 3x

Answer

y = sin(x + 3)

Question 47

What is the domain of the function y = cot x?

Accepted Answer

x ≠ nπ, where n is an integer

Answer

x ≠ π/2

Answer

All real numbers

Answer

x ≠ 0