Trigonometric Functions: Definition, Evaluation, and Equations

Question 1

Evaluate sin(π/2) precisely.

Accepted Answer

1

Answer

0

Answer

-1

Answer

Undefined

Question 2

Identify the correct trigonometric identity.

Accepted Answer

sin²x + cos²x = 1

Answer

sin²x + cos²x = 0

Answer

sin²x - cos²x = 1

Answer

tan²x + 1 = sin²x

Question 3

Solve the equation sin(x) = 1/2 for the interval 0 ≤ x ≤ 2π.

Accepted Answer

π/6, 5π/6

Answer

π/4, 3π/4

Answer

π/3, 2π/3

Answer

0, π/2

Question 4

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

Accepted Answer

2π

Answer

π/2

Answer

π

Answer

4π

Question 5

Determine the period of the cosine function.

Accepted Answer

2π

Answer

π

Answer

4π

Answer

6π

Question 6

Identify the Pythagorean identity.

Accepted Answer

sin² θ + cos² θ = 1

Answer

tan² θ + 1 = sec² θ

Answer

cosec² θ - 1 = cot² θ

Answer

sin θ + cos θ = 1

Question 7

Determine the amplitude of the function f(x) = 2cos(3x) + 1.

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 8

Which of the following is NOT an application of trigonometric functions?

Accepted Answer

computing the volume of a sphere

Answer

solving triangles

Answer

calculating projectile motion

Answer

finding the height of a building

Question 9

Which trigonometric identity can be used to prove the double-angle formula for cosine?

Accepted Answer

cos(A - B) = cos A cos B + sin A sin B

Answer

cos(A + B) = cos A cos B - sin A sin B

Answer

sin(A + B) = sin A cos B + cos A sin B

Answer

tan(A + B) = (tan A + tan B)/(1 - tan A tan B)

Question 10

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

Accepted Answer

2

Answer

1

Answer

3

Answer

6

Question 11

What is the inverse function of the sine function?

Accepted Answer

arcsin(x)

Answer

arccos(x)

Answer

arctan(x)

Answer

arccot(x)

Question 12

Define the sine of an angle.

Accepted Answer

Opposite side divided by the hypotenuse

Answer

Hypotenuse divided by the opposite side

Answer

Adjacent side divided by the opposite side

Answer

Adjacent side divided by the hypotenuse

Question 13

What is the period of the sine function?

Accepted Answer

2π

Answer

π

Answer

360°

Answer

720°

Question 14

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

Accepted Answer

2

Answer

1

Answer

3

Answer

6

Question 15

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

Accepted Answer

(-∞, ∞)

Answer

[-1, 1]

Answer

[0, 1]

Answer

[-∞, 0]

Question 16

Which trigonometric identity is universally valid?

Accepted Answer

sin²x + cos²x = 1

Answer

tanx = sinx/cosx

Answer

cotx = cosx/sinx

Answer

sin(x + y) = sinx + siny

Question 17

What is the range of values for the cosine function?

Accepted Answer

[-1, 1]

Answer

[-1, 0]

Answer

[0, 1]

Answer

[0, π]

Question 18

Which identity represents the sum of the sine and cosine functions?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 19

Determine the value of tan(π/4).

Accepted Answer

√2/2

Answer

0

Answer

1

Answer

2

Question 20

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

Accepted Answer

3

Answer

1

Answer

2

Answer

6

Question 21

Sketch the graph of y = 2cos(x - π/4) + 1. Determine its amplitude, period, and phase shift.

Accepted Answer

Amplitude: 2, Period: 2π, Phase shift: π/4

Answer

Amplitude: 1, Period: π, Phase shift: π/2

Answer

Amplitude: 2, Period: π, Phase shift: π/4

Question 22

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

Accepted Answer

Sine

Answer

Tangent

Answer

Cosine

Question 23

Which trigonometric equation has a solution in the interval [0, π]?

Accepted Answer

sin(x) = 1/2

Answer

cot(x) = -1

Answer

tan(x) = 0

Answer

cos(x) = -2

Question 24

Determine the value of cosine of π/4 radians.

Accepted Answer

√2/2

Answer

1

Answer

0

Answer

1/2

Question 25

Identify the equation that represents the tangent function.

Accepted Answer

y = tan(x)

Answer

y = cot(x)

Answer

y = cos(x)

Answer

y = sin(x)

Question 26

Which function is the inverse of the cosine function?

Accepted Answer

Arccosine

Answer

Secant

Answer

Tangent

Answer

Sine

Question 27

Calculate the value of tangent of π/3 radians.

Accepted Answer

√3

Answer

0

Answer

1

Question 28

Which trigonometric identity is used to derive the Pythagorean identity?

Accepted Answer

cos²x + sin²x = 1

Answer

sin²x - cos²x = 1

Answer

tan²x + 1 = sec²x

Answer

cot²x - 1 = csc²x

Question 29

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

Accepted Answer

2

Answer

3

Answer

4

Answer

1

Question 30

If cos(θ) = 1/2, what is the value of θ in the interval [0, 2π)?

Accepted Answer

π/3 or 5π/3

Answer

π/2

Answer

π/6

Answer

π/4

Question 31

If tanθ = -1, and θ is in the interval [0, 2π), what is the value of cosθ?

Accepted Answer

-√2/2

Answer

√2/2

Answer

1/2

Answer

0

Question 32

What is the value of sin(π/3)?

Accepted Answer

√3/2

Answer

0

Answer

1

Answer

2

Question 33

If cos(θ) = 0.5, what is the value of θ?

Accepted Answer

π/3

Answer

π/4

Answer

π/6

Answer

π/2

Question 34

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

Accepted Answer

π

Answer

1

Answer

2π

Answer

π/2

Question 35

Which of the following is an identity?

Accepted Answer

sin²θ + cos²θ = 1

Answer

cotθ = 1/tanθ

Answer

tanθ = sinθ/cosθ

Answer

sinθ + cosθ = 1

Question 36

What is the amplitude of the function y = 2sin(3x + π/4)?

Accepted Answer

2

Answer

π/4

Answer

1

Answer

3

Question 37

Which of the following is the graph of y = cos(x)?

Accepted Answer

[Image of the graph of y = cos(x)]

Answer

[Image of the graph of y = tan(x)]

Answer

[Image of the graph of y = sin(x)]

Question 38

What is the sum formula for sine?

Accepted Answer

sin(α + β) = sinαcosβ + cosαsinβ

Answer

sin(α + β) = tanα + tanβ

Answer

sin(α + β) = sinα + sinβ

Answer

sin(α + β) = cosαcosβ + sinαsinβ

Question 39

Which of the following is the derivative of tan(x)?

Accepted Answer

sec²(x)

Answer

cos²(x)

Answer

cot²(x)

Answer

sin²(x)

Question 40

What is the integral of cos(x)?

Accepted Answer

sin(x) + C

Answer

tan(x) + C

Answer

cot(x) + C

Answer

cos(x) + C

Question 41

Which of the following is an application of trigonometric functions?

Accepted Answer

Solving real-world problems involving triangles

Answer

Finding the volume of a sphere

Answer

Converting temperature scales

Answer

Calculating the area of a circle

Question 42

What is the value of sin(60°)?

Accepted Answer

√3/2

Answer

√2/2

Answer

1/2

Answer

1

Question 43

Which of the following represents the period of the function f(x) = sin(2x)?

Accepted Answer

π

Answer

2π

Answer

π/2

Answer

4π

Question 44

A right triangle has an angle of 30° and a hypotenuse of 10 cm. What is the length of the side opposite the 30° angle?

Accepted Answer

5 cm

Answer

10√3 cm

Answer

5√3 cm

Answer

10 cm

Question 45

What is the value of tan(45°)?

Accepted Answer

1

Answer

√2/2

Answer

0

Answer

√3/2

Question 46

Which of the following trigonometric functions has a period of 2π?

Accepted Answer

Cosine

Answer

Cosecant

Answer

Tangent

Answer

Secant

Question 47

If sin(θ) = 1/2 and θ is in the first quadrant, what is the value of cos(θ)?

Accepted Answer

√3/2

Answer

1/2

Answer

-√3/2

Answer

-1/2

Question 48

What is the general solution of the equation sin(x) = 1/2?

Accepted Answer

x = π/6 + 2πk or x = 5π/6 + 2πk, where k is an integer.

Answer

x = π/4 + 2πk or x = 3π/4 + 2πk, where k is an integer.

Answer

x = π/3 + 2πk or x = 2π/3 + 2πk, where k is an integer.

Question 49

A surveyor measures the angle of elevation to the top of a building to be 20°. If the surveyor is standing 100 meters away from the base of the building, what is the height of the building?

Accepted Answer

100 tan(20°) meters

Answer

100 cos(20°) meters

Answer

100 sin(20°) meters

Answer

100/tan(20°) meters

Question 50

What is the amplitude of the function f(x) = 3sin(x)?

Accepted Answer

3

Answer

2π

Answer

π

Answer

1

Question 51

Which of the following trigonometric identities is NOT true?

Accepted Answer

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

Answer

1 + tan^2(x) = sec^2(x)

Answer

1 + cot^2(x) = csc^2(x)

Answer

tan(x) = sin(x)/cos(x)