Trigonometry Quizzes: Master Angles, Sine, Cosine, and Tangent

Question 1

Prove the trigonometric equality: cos²θ - sin²θ = 2cos²θ - 1

Accepted Answer

cos²θ - sin²θ = 2cos²θ - 1

Answer

cos²θ - sin²θ = cos²θ + sin²θ

Answer

cos²θ - sin²θ = 1 - 2sin²θ

Answer

cos²θ - sin²θ = 1 + 2sin²θ

Question 2

In a right-angled triangle, the sine of an angle is defined as the ratio of:

Accepted Answer

Opposite side to the hypotenuse

Answer

Adjacent side to the opposite side

Answer

Hypotenuse to the adjacent side

Answer

Opposite side to the adjacent side

Question 3

Which trigonometric equation can be used to find the length of the hypotenuse of a right-angled triangle?

Accepted Answer

h² = a² + b²

Answer

h = a² + b²

Answer

h = ab

Answer

h² = ab

Question 4

What is the derivative of the function y = sin(x)?

Accepted Answer

cos(x)

Answer

-cos(x)

Answer

sin(x)

Answer

-sin(x)

Question 5

Which function is the inverse of sine?

Accepted Answer

arcsin(x)

Answer

arccos(x)

Answer

arctan(x)

Answer

arccsc(x)

Question 6

Identify the amplitude of the function y = 2sin(x).

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 7

What is the range of the function y = tan(x)?

Accepted Answer

[-∞, -1] ∪ [1, ∞]

Answer

[-1, 1]

Answer

[0, ∞]

Answer

(-∞, ∞)

Question 8

Calculate the cosine of π/6 radians.

Accepted Answer

√3/2

Answer

0

Answer

1/2

Answer

1

Question 9

A ship sails 50 km north and then 120 km east. What is the magnitude of the ship's total displacement?

Accepted Answer

170 km

Answer

50 km

Answer

120 km

Answer

130 km

Question 10

State the Pythagorean identity.

Accepted Answer

sin²θ + cos²θ = 1

Answer

tan²θ + 1 = sec²θ

Answer

cot²θ - 1 = cosec²θ

Answer

sinθ + cosθ = tanθ

Question 11

Calculate the tangent of a 45° angle.

Accepted Answer

1

Answer

0

Answer

2

Answer

Undefined

Question 12

Identify the inverse function of sine.

Accepted Answer

Arcsine

Answer

Cosine

Answer

Tangent

Answer

Arccosine

Question 13

Determine the period of the sine function.

Accepted Answer

2π

Answer

π

Answer

π/2

Answer

1

Question 14

Identify the amplitude of the function y = 2sin(x).

Accepted Answer

2

Answer

1

Answer

π

Answer

2π

Question 15

Specify the domain of the sine function.

Accepted Answer

All real numbers

Answer

[-1, 1]

Answer

[0, 2π]

Answer

[-π/2, π/2]

Question 16

Evaluate: cos(π/6)

Accepted Answer

√3/2

Answer

0

Answer

1/2

Answer

1

Question 17

If the cosine of an angle is 0.6, then the sine of its complementary angle is:

Accepted Answer

0.4

Answer

0.6

Answer

0.8

Answer

0.2

Question 18

The area of a triangle with sides of length a, b, and c is given by:

Accepted Answer

1/2 bc sin A

Answer

s(s - a)(s - b)(s - c)

Answer

ab/2

Answer

abc/4

Question 19

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

Accepted Answer

Sine

Answer

Cosine

Answer

Tangent

Answer

Cotangent

Question 20

Which formula correctly expresses the sine of an angle?

Accepted Answer

Opposite / Hypotenuse

Answer

Adjacent / Hypotenuse

Answer

Opposite / Adjacent

Answer

Hypotenuse / Opposite

Question 21

What is the value of sin 90°?

Accepted Answer

1

Answer

0

Answer

-1

Answer

Undefined

Question 22

Which angle has a tangent value of √3?

Accepted Answer

60°

Answer

30°

Answer

45°

Answer

90°

Question 23

What is the period of the sine function?

Accepted Answer

2π

Answer

π

Answer

3π

Answer

4π

Question 24

What is the amplitude of the function f(x) = 2sin(3x + π/6)?

Accepted Answer

2

Answer

1

Answer

3

Answer

6

Question 25

Calculate the sine of a 30-degree angle in a right-angled triangle.

Accepted Answer

1/2

Answer

1/4

Answer

1/6

Answer

1/8

Question 26

Which of the following is not a trigonometric function?

Accepted Answer

Quadratic

Answer

Sine

Answer

Cosine

Answer

Tangent

Question 27

Find the tangent of a 45-degree angle in a right-angled triangle.

Accepted Answer

1

Answer

0

Answer

2

Answer

3

Question 28

Which of the following is the Pythagorean identity?

Accepted Answer

sin²θ + cos²θ = 1

Answer

sin²θ - cos²θ = 1

Answer

tan²θ + cot²θ = 1

Answer

tan²θ - cot²θ = 1

Question 29

Calculate the measure of the angle whose cosine is 0 in a right-angled triangle.

Accepted Answer

0°

Answer

30°

Answer

45°

Answer

90°

Question 30

How are the sine and cosine functions related to the unit circle?

Accepted Answer

They represent the y- and x-coordinates of points on the unit circle.

Answer

They are the inverses of each other.

Answer

They are always equal in value.

Answer

They are only defined for angles between 0 and π/2.

Question 31

In the first quadrant, as the angle increases from 0 to 90 degrees, what happens to the cosine value?

Accepted Answer

The cosine value decreases from 1 to 0.

Answer

The cosine value increases from 0 to 1.

Answer

The cosine value stays the same at 1.

Question 32

Which angle does the inverse sine function calculate?

Accepted Answer

The angle whose sine is a given value

Answer

The angle whose cosine is a given value

Answer

The angle whose tangent is a given value

Question 33

For the trigonometric function y = sin(x), what is its maximum vertical distance from the x-axis?

Accepted Answer

1

Answer

0

Answer

2

Answer

-1

Question 34

Which of the following identities for cosine is correct?

Accepted Answer

cos^2(x) + sin^2(x) = 1

Answer

sin^2(x) - cos^2(x) = 1

Answer

cos^2(x) - sin^2(x) = 0

Answer

sin^2(x) + cos^2(x) = 0

Question 35

What is the cosine of 0 degrees?

Accepted Answer

1

Answer

0

Answer

√2/2

Answer

-1

Question 36

A ladder is leaning against a wall. The base of the ladder is 3 m from the wall, and the top of the ladder is 4 m above the ground. What is the length of the ladder?

Accepted Answer

5 m

Answer

4 m

Answer

6 m

Answer

3 m

Question 37

What is the length of the hypotenuse of a right-angled triangle with sides of 5 cm and 12 cm?

Accepted Answer

13 cm

Answer

10 cm

Answer

15 cm

Question 38

Which trigonometric ratio represents the ratio of the length of the opposite side to the length of the hypotenuse of a right-angled triangle?

Accepted Answer

Sine

Answer

Cosecant

Answer

Cosine

Answer

Tangent

Question 39

What is the numerical value of sin(30°) in a right-angled triangle?

Accepted Answer

1/2

Answer

1

Answer

√2/2

Answer

√3/2

Question 40

What is the value of sin 90 degrees?

Accepted Answer

1

Answer

0

Answer

√2 / 2

Answer

-1

Question 41

A surveyor measures the angle of elevation to the top of a building to be 35 degrees.  The surveyor is standing 50 meters away from the building.  What is the height of the building?

Accepted Answer

50 tan 35 degrees

Answer

50 sin 35 degrees

Answer

50 / tan 35 degrees

Answer

50 cos 35 degrees

Question 42

A triangle has sides of length 5 cm, 12 cm, and 13 cm. Is this a right-angled triangle?

Accepted Answer

Yes

Answer

No

Answer

Cannot be determined

Answer

Only if the 13 cm side is the hypotenuse

Question 43

What is the value of tan(45°)?

Accepted Answer

1

Answer

√3/2

Answer

0

Answer

√2

Question 44

In a right-angled triangle, the hypotenuse is 10 cm long, and one of the acute angles is 30°. What is the length of the side opposite the 30° angle?

Accepted Answer

5 cm

Answer

10 cm

Answer

5√3 cm

Answer

20 cm

Question 45

In a right-angled triangle, what is the sine of an angle?

Accepted Answer

The length of the side opposite the angle divided by the length of the hypotenuse

Answer

The length of the side adjacent to the angle divided by the length of the hypotenuse

Answer

The length of the hypotenuse divided by the length of the side opposite the angle

Answer

The length of the side opposite the angle divided by the length of the side adjacent to the angle

Question 46

Which trigonometric ratio is equivalent to sin(60°)?

Accepted Answer

cos(30°)

Answer

tan(60°)

Answer

sin(30°)

Answer

cos(60°)

Question 47

If sin(θ) = 3/5 and θ is an acute angle, what is the value of cos(θ)?

Accepted Answer

4/5

Answer

3/4

Answer

5/3

Answer

5/4

Question 48

In a right-angled triangle, what is the tangent of an angle?

Accepted Answer

The length of the side opposite the angle divided by the length of the side adjacent to the angle

Answer

The length of the hypotenuse divided by the length of the side adjacent to the angle

Answer

The length of the side opposite the angle divided by the length of the hypotenuse

Answer

The length of the side adjacent to the angle divided by the length of the side opposite the angle

Question 49

A kite is flying at a height of 50 meters and is attached to a string that makes an angle of 45° with the ground. What is the length of the string?

Accepted Answer

50√2 meters

Answer

100 meters

Answer

50 meters

Answer

25 meters

Question 50

What is the value of tan 60 degrees?

Accepted Answer

√3

Answer

√2

Answer

1/√3

Answer

1

Question 51

A ladder is 5 meters long and leans against a wall. The ladder makes a 60-degree angle with the ground. How high up the wall does the ladder reach?

Accepted Answer

5√3/2 meters

Answer

5 meters

Answer

5√2/2 meters

Answer

5/2 meters

Question 52

A line passes through the point (0, 1) and has a gradient of tan 45 degrees. What is the equation of this line?

Accepted Answer

y = x + 1

Answer

y = x - 1

Answer

y = -x - 1

Answer

y = -x + 1

Question 53

A right-angled triangle has a 30-degree angle. The side opposite this angle is 5 cm long. What is the length of the hypotenuse?

Accepted Answer

10 cm

Answer

5√2 cm

Answer

2.5 cm

Answer

5√3 cm

Question 54

What is the value of sin 60 degrees?

Accepted Answer

√3/2

Answer

√2/2

Answer

1/2

Answer

1

Question 55

A ladder is placed against a wall. Its base is 3 meters from the wall and the top reaches 4 meters up the wall. What angle does the ladder make with the ground?

Accepted Answer

53.1 degrees

Answer

45 degrees

Answer

36.9 degrees

Question 56

Which trigonometric identity is correct for tan θ?

Accepted Answer

sin θ / cos θ

Answer

1 / sin θ

Answer

cos θ / sin θ

Answer

1 / cos θ

Question 57

A triangle has angles of 45 degrees, 45 degrees, and 90 degrees. What is the ratio of the hypotenuse's length to the length of one of its legs?

Accepted Answer

√2 : 1

Answer

2 : 1

Answer

√3 : 1

Answer

1 : 1

Question 58

A ship travels 10 km east and then 5 km north. What is the ship's bearing from its starting point?

Accepted Answer

26.6 degrees North of East

Answer

63.4 degrees North of East

Answer

26.6 degrees North of West

Answer

63.4 degrees North of West

Question 59

What is the value of cos 0 degrees?

Accepted Answer

1

Answer

√2/2

Answer

0

Answer

-1

Question 60

A ramp is 5 meters long and makes an angle of 15 degrees with the ground. How high is the ramp at its highest point?

Accepted Answer

1.3 meters

Answer

2.5 meters

Answer

4.8 meters

Answer

5 meters

Question 61

Which of the following is the correct equation for the sine rule?

Accepted Answer

a/sin A = b/sin B = c/sin C

Answer

tan A = sin A / cos A

Answer

sin² A + cos² A = 1

Answer

a² = b² + c² - 2bc cos A

Question 62

A triangle has sides of length 5 cm, 12 cm, and 13 cm. What is the cosine of the angle opposite the side of length 12 cm?

Accepted Answer

5/13

Answer

5/12

Answer

12/13

Answer

12/5

Question 63

What is the value of tan 45 degrees?

Accepted Answer

1

Answer

√2/2

Answer

√3/2

Answer

0

Question 64

A ladder leaning against a wall makes an angle of 60 degrees with the ground. If the ladder is 10 meters long, how high up the wall does it reach?

Accepted Answer

5√3 meters

Answer

10 meters

Answer

10√3 meters

Answer

5 meters