Trigonometry Quizzes: Master Trig Functions, Identities, and Applications

Question 1

Simplify the trigonometric expression using the Pythagorean Identity: (sin²θ + cos²θ)² - 2sin²θcos²θ

Accepted Answer

1

Answer

sin²θ + cos²θ

Answer

cos²θ - sin²θ

Answer

2sin²θcos²θ

Question 2

The function f(x) = tan(2x - π/4) + 1 is a transformation of the parent function y = tan(x). Describe the transformation that has been applied to the parent function.

Accepted Answer

The graph of f(x) is shifted π/4 units to the right and 1 unit up compared to the graph of y = tan(x).

Answer

The graph of f(x) is shifted π/4 units to the left and 1 unit down compared to the graph of y = tan(x).

Answer

The graph of f(x) is stretched vertically by a factor of 2 compared to the graph of y = tan(x).

Answer

The graph of f(x) is shifted π/2 units to the right compared to the graph of y = tan(x).

Question 3

In the context of trigonometric functions, what is the period of the sine function?

Accepted Answer

2π

Answer

π

Answer

1

Answer

0

Question 4

What is the value of sin(90°)?

Accepted Answer

1

Answer

0

Answer

-1

Answer

undefined

Question 5

Determine the period of the sine function:

Accepted Answer

2π

Answer

π

Answer

1

Answer

4π

Question 6

Which of these functions is NOT a trigonometric function?

Accepted Answer

Exponential

Answer

Sine

Answer

Cosine

Answer

Tangent

Question 7

Calculate the amplitude of the function y = 2cos(3x):

Accepted Answer

2

Answer

1

Answer

3

Answer

6

Question 8

Identify the equation of the horizontal asymptote of the function y = tan(x):

Accepted Answer

The function has no horizontal asymptote.

Answer

y = 0

Answer

y = 1

Answer

y = -1

Question 9

Find the value of sin(π/6):

Accepted Answer

√3/2

Answer

0

Answer

1/2

Answer

1

Question 10

Identify the domain of the function y = cot(x):

Accepted Answer

All real numbers except odd multiples of π/2

Answer

All real numbers

Answer

All real numbers except multiples of π

Answer

Only multiples of π/2

Question 11

Which of the following fundamental trigonometric identities is correct?

Accepted Answer

sin²x + cos²x = 1

Answer

sin(x) - cos(x) = 1

Answer

tanx = sinx / cosx

Answer

cotx = tanx

Question 12

Solve the trigonometric equation sinx = 0.5 for 0° ≤ x ≤ 360°:

Accepted Answer

x = 30°, x = 150°

Answer

x = 60°, x = 120°

Answer

x = 45°, x = 135°

Answer

x = 70°, x = 250°

Question 13

Which of the following is NOT an application of trigonometry?

Accepted Answer

Chemistry

Answer

Navigation

Answer

Architecture

Answer

Medical imaging

Question 14

What is the inverse function of the sine function?

Accepted Answer

arcsin

Answer

arccos

Answer

arctan

Answer

arccot

Question 15

Find the period of the function f(x) = cos(3x):

Accepted Answer

2π/3

Answer

π/3

Answer

π

Answer

2π

Question 16

The sine function is defined as the ratio of which two sides of a right triangle?

Accepted Answer

Opposite side to hypotenuse

Answer

Adjacent side to opposite side

Answer

Hypotenuse to adjacent side

Answer

Opposite side to adjacent side

Question 17

Which trigonometric identity relates the squares of sine and cosine to 1?

Accepted Answer

sin²(x) + cos²(x) = 1

Answer

tan(x) = sin²(x)/cos(x)

Answer

cot(x) = cos(x)/tan(x)

Answer

sec(x) = tan(x)/cos(x)

Question 18

Which trigonometric function is the inverse of sine?

Accepted Answer

Arcsine

Answer

Arccosine

Answer

Arctangent

Answer

Arccotangent

Question 19

What is the range of values of the function y = tangent of x?

Accepted Answer

(-∞, ∞)

Answer

(-1, 1)

Answer

(0, ∞)

Answer

(-π/2, π/2)

Question 20

Which trigonometric function represents the ratio of the opposite side to the hypotenuse?

Accepted Answer

Sine

Answer

Cosine

Answer

Tangent

Answer

Cotangent

Question 21

Which of the following trigonometric identities is correct?

Accepted Answer

sin²θ + cos²θ = 1

Answer

sinθ/cosθ = tanθ

Answer

cos²θ - sin²θ = 1

Answer

tanθ/sinθ = cosθ

Question 22

What is the amplitude of the function y = 2sin(x)?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 23

A Ferris wheel has a diameter of 50 meters. If you start at the bottom of the wheel and rotate counterclockwise through an angle of 120°, how high above the ground will you be?

Accepted Answer

35.36 meters

Answer

25 meters

Answer

43.30 meters

Answer

50 meters

Question 24

What is the inverse function of the cosine function?

Accepted Answer

Arccosine

Answer

Sine

Answer

Cosine

Answer

Arcsine

Question 25

In a right triangle, the ratio of the length of the opposite side to the length of the hypotenuse is called the:

Accepted Answer

Sine

Answer

Cosine

Answer

Tangent

Answer

Cosecant

Question 26

Which of the following trigonometric identities is always true?

Accepted Answer

sin²θ + cos²θ = 1

Answer

tanθ = sinθ / cosθ

Answer

secθ = 1 / sinθ

Answer

cotθ = tanθ

Question 27

A ladder 10 feet long is leaning against a wall. If the base of the ladder is 6 feet from the wall, what is the measure of the angle between the ladder and the ground?

Accepted Answer

45°

Answer

30°

Answer

60°

Answer

75°

Question 28

If sinA = 0.707, what is the value of cos(π - A)?

Accepted Answer

-0.707

Answer

0.707

Answer

0

Answer

1

Question 29

Identify the range of the sine function.

Accepted Answer

[-1, 1]

Answer

[0, 1]

Answer

[-π/2, π/2]

Answer

[π/2, 3π/2]

Question 30

Determine the period of the cosine function.

Accepted Answer

2π

Answer

π

Answer

π/2

Answer

3π/2

Question 31

In a right triangle with a hypotenuse of 10 and an angle of 30 degrees, find the length of the opposite side.

Accepted Answer

5√3

Answer

5

Answer

10√3

Answer

10

Question 32

Identify the inverse function of cosine.

Accepted Answer

arccos

Answer

arcsin

Answer

arctan

Answer

arccot

Question 33

A ship sails due east at 18 knots. A strong wind blows from the north at 12 knots. Calculate the magnitude and direction of the ship's resultant velocity.

Accepted Answer

22.2 knots, 33.7° north of east

Answer

22.2 knots, 33.7° east of north

Answer

30 knots, 45° north of east

Answer

18 knots, 0° north of east

Question 34

Calculate the value of sin(π/2).

Accepted Answer

1

Answer

0

Answer

-1

Answer

Undefined

Question 35

Identify the identity used to determine the cosine of an angle given its sine.

Accepted Answer

cos²θ + sin²θ = 1

Answer

tanθ = sinθ/cosθ

Answer

cotθ = cosθ/sinθ

Answer

secθ = 1/cosθ

Question 36

Determine the period of the trigonometric function y = cos(2x).

Accepted Answer

2π

Answer

π

Answer

π/2

Answer

π/4

Question 37

Identify the notation for the inverse cosine function.

Accepted Answer

cos⁻¹(x)

Answer

sin⁻¹(x)

Answer

tan⁻¹(x)

Answer

cot⁻¹(x)

Question 38

Determine the amplitude of the function y = 2sin(3x + π/4).

Accepted Answer

2

Answer

1

Answer

3

Answer

π/4

Question 39

Calculate the value of tan(π/6).

Accepted Answer

1/√3

Answer

√3

Answer

1/√2

Answer

√2

Question 40

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

Accepted Answer

(-∞, ∞)

Answer

[-π/2, π/2]

Answer

[0, ∞)

Answer

[-1, 1]

Question 41

Which trigonometric identity relates the sine, cosine, and tangent functions?

Accepted Answer

Pythagorean identity

Answer

Double-angle identity

Answer

Half-angle identity

Answer

Sum-to-product identity

Question 42

Which of the following identities is equivalent to sin²(x) + cos²(x) = 1?

Accepted Answer

tan²(x) + 1 = sec²(x)

Answer

cot²(x) - 1 = cosec²(x)

Answer

cos²(x) - sin²(x) = tan²(x)

Answer

sin²(x) - cos²(x) = cosec²(x)

Question 43

What is the period of the function y = 2cos(3x)?

Accepted Answer

(2π)/3

Answer

(π)/3

Answer

2π

Answer

π

Question 44

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

Accepted Answer

[-∞, -1) ∪ (1, ∞)

Answer

[-1, 1]

Answer

(-∞, ∞)

Answer

[-π/2, π/2]

Question 45

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

Accepted Answer

y = arccos(x)

Answer

y = sin(x)

Answer

y = tan(x)

Answer

y = cot(x)

Question 46

What is the amplitude of the function y = 3cos(2x + π/3)?

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 47

The range of the function y = -cos(x) + 1 is:

Accepted Answer

[-2, 2]

Answer

[-1, 1]

Answer

[0, 2]

Answer

[1, 2]

Question 48

The inverse function of y = tan(x) is:

Accepted Answer

y = arctan(x)

Answer

y = sin^-1(x)

Answer

y = cos^-1(x)

Answer

y = cot^-1(x)

Question 49

To evaluate sin(π/4) + cos(π/3), the result is:

Accepted Answer

√2/2

Answer

0

Answer

1

Answer

√3/2

Question 50

In a right triangle, the tangent of an angle is defined as the ratio of which two sides?

Accepted Answer

Opposite side / Adjacent side

Answer

Adjacent side / Hypotenuse

Answer

Hypotenuse / Opposite side

Answer

Opposite side / Hypotenuse

Question 51

For the trigonometric function y = sin(x), what is its period?

Accepted Answer

2π

Answer

π

Answer

π/2

Answer

1

Question 52

What trigonometric function is used to calculate the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle?

Accepted Answer

sine

Answer

cosine

Answer

cotangent

Answer

tangent

Question 53

Identify the domain of the function y = cot(x):

Accepted Answer

All real numbers except multiples of π

Answer

All real numbers

Answer

All real numbers except 0

Answer

The set of all integers

Question 54

Calculate the value of sine at an angle of π/2 (pi/2) radians.

Accepted Answer

1

Answer

undefined

Answer

0

Answer

-1

Question 55

Find the inverse function of cosine.

Accepted Answer

arccos(x)

Answer

sin⁻¹(x)

Answer

tan⁻¹(x)

Question 56

Calculate the value of tangent at an angle of π/4 (pi/4) radians.

Accepted Answer

1

Answer

π/4

Answer

0

Answer

undefined

Question 57

What is the value of sin(90°) in a unit circle?

Accepted Answer

1

Answer

-1

Answer

1/2

Answer

0

Question 58

Identify the period of the cosine function.

Accepted Answer

2π

Answer

π

Answer

4π

Answer

6π

Question 59

Describe the transformation that occurs to the graph of y = cos(x) when the function becomes y = 2cos(x).

Accepted Answer

Vertical stretch by a factor of 2

Answer

Horizontal stretch by a factor of 2

Answer

Vertical shrink by a factor of 2

Answer

Horizontal shrink by a factor of 2

Question 60

Determine the amplitude of the function y = -3sin(2x + 30°).

Accepted Answer

3

Answer

1

Answer

-3

Answer

2

Question 61

Which trigonometric identity is used to calculate the value of sin(x + y)?

Accepted Answer

sin(x + y) = sin(x)cos(y) + cos(x)sin(y)

Answer

cos(x - y) = sin(x)cos(y) + cos(x)sin(y)

Answer

cos(x + y) = cos(x)cos(y) - sin(x)sin(y)

Answer

sin(x - y) = sin(x)cos(y) - cos(x)sin(y)

Question 62

In trigonometry, which ratio is commonly used to find the height of a tree?

Accepted Answer

Tangent

Answer

Sine

Answer

Cotangent

Answer

Cosine

Question 63

In a right triangle with sides a, b, and c, under which condition can we use the Pythagorean theorem to find the length of side c?

Accepted Answer

a2 + b2 = c2

Answer

c2 - b2 = a2

Answer

c2 + a2 = b2

Answer

a2 - b2 = c2

Question 64

What is the domain of the function y = sec(x)?

Accepted Answer

All real numbers except x = (2n + 1)π/2, where n is an integer

Answer

The interval [-1, 1]

Answer

All real numbers