Trigonometric Functions: Master Sine, Cosine, and Tangent

Question 1

What is the range of possible values for the sine, cosine, and tangent functions?

Accepted Answer

[-1, 1]

Answer

[-π/2, π/2]

Answer

[0, 1]

Question 2

Which of the following accurately defines the range of possible values for the cosine function?

Accepted Answer

-1 to 1

Answer

-∞ to ∞

Answer

0 to 1

Answer

-∞ to 0

Question 3

Which of the following is the trigonometric identity for the tangent function?

Accepted Answer

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

Answer

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

Answer

tan(x) = sin(x) + cos(x)

Answer

tan(x) = cos(x) - sin(x)

Question 4

Find the cosine of a 90° angle.

Accepted Answer

0

Answer

1

Answer

undefined

Answer

2

Question 5

Which of the following is the Pythagorean identity?

Accepted Answer

sin²(x) + cos²(x) = 1

Answer

sin²(x) - cos²(x) = 1

Answer

tan²(x) + cot²(x) = 1

Answer

tan²(x) - cot²(x) = 1

Question 6

Determine the angle whose sine is equal to 0.5.

Accepted Answer

30°

Answer

45°

Answer

60°

Answer

90°

Question 7

Which of the following best describes the graph of the sine function?

Accepted Answer

A wave

Answer

A straight line

Answer

A circle

Answer

A parabola

Question 8

Derive the equation of a sine function with an amplitude of 5 and a period of 2π.

Accepted Answer

y = 5sin(2πx)

Answer

y = 5cos(2πx)

Answer

y = 2πsin(5x)

Answer

y = 2πcos(5x)

Question 9

How does the unit circle help us visualize and define trigonometric functions?

Accepted Answer

By mapping angles to points on its circumference, the unit circle provides a visual representation of trigonometric function values.

Answer

The unit circle is used to find the area of a triangle given its angle.

Answer

The unit circle is a physical circle with a radius of 1 unit.

Answer

The unit circle helps convert degrees to radians.

Question 10

Find the cosine of an angle measuring 45 degrees.

Accepted Answer

√2/2

Answer

1/2

Answer

1

Answer

0

Question 11

Calculate the tangent of an angle measuring 60 degrees.

Accepted Answer

√3

Answer

1

Answer

2

Answer

0

Question 12

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

Accepted Answer

Opposite side to the length of the hypotenuse

Answer

Opposite side to the length of the adjacent side

Answer

Adjacent side to the length of the opposite side

Answer

Adjacent side to the length of the hypotenuse

Question 13

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

Accepted Answer

Adjacent side to the length of the hypotenuse

Answer

Opposite side to the length of the hypotenuse

Answer

Opposite side to the length of the adjacent side

Answer

Adjacent side to the length of the opposite side

Question 14

Describe the graph of the tangent function for one complete period. How does it relate to the graphs of the sine and cosine functions?

Accepted Answer

The tangent function's graph has vertical asymptotes at x = π/2 and x = 3π/2.

Answer

The tangent function's graph is a straight line.

Answer

The tangent function's graph is identical to the sine function's graph.

Answer

The tangent function's graph is identical to the cosine function's graph.

Question 15

Which function is the inverse of the sine function?

Accepted Answer

Arcsine

Answer

Cosine

Answer

Tangent

Answer

Arctangent

Question 16

What is the period of the cosine function?

Accepted Answer

360°

Answer

90°

Answer

180°

Answer

720°

Question 17

Which identity below accurately represents the tangent function?

Accepted Answer

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

Answer

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

Answer

tan(x) = sin(x) + cos(x)

Answer

tan(x) = 1 / cos(x)

Question 18

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

Accepted Answer

Sine

Answer

Cosine

Answer

Tangent

Answer

Cosecant

Question 19

Which trigonometric function represents the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle?

Accepted Answer

Cosine

Answer

Sine

Answer

Tangent

Answer

Cotangent

Question 20

In a right triangle with an angle measuring 30 degrees, what is the value of the sine of that angle?

Accepted Answer

1/2

Answer

0

Answer

1

Answer

2

Question 21

Which of the following is the inverse function of the tangent function?

Accepted Answer

Arctangent

Answer

Arcsine

Answer

Arccosine

Answer

Inverse tangent

Question 22

What is the period of the sine function?

Accepted Answer

2π

Answer

π

Answer

1

Answer

0

Question 23

Identify the graph of the cosine function.

Accepted Answer

A wave that oscillates between -1 and 1.

Answer

A straight line that slopes upwards.

Answer

A parabola that opens upwards.

Answer

A circle.

Question 24

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

Accepted Answer

2

Answer

1

Answer

-2

Answer

π

Question 25

In a right triangle with angle θ, define the sine, cosine, and tangent functions.

Accepted Answer

sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent

Answer

sin θ = opposite / radius, cos θ = adjacent / radius, tan θ = opposite / hypotenuse

Answer

sin θ = adjacent / opposite, cos θ = hypotenuse / adjacent, tan θ = hypotenuse / opposite

Answer

sin θ = hypotenuse / opposite, cos θ = opposite / adjacent, tan θ = adjacent / hypotenuse

Question 26

Which of the following graphs represents the function y = sin(x)?

Accepted Answer

A sinusoidal curve

Answer

A straight line

Answer

A circle

Answer

A parabola

Question 27

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

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 28

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

Accepted Answer

2π

Answer

π

Answer

π/2

Answer

π/4

Question 29

The formula for the sine of an angle is:

Accepted Answer

sin(θ) = opposite / hypotenuse

Answer

cos(θ) = adjacent / hypotenuse

Answer

tan(θ) = opposite / adjacent

Answer

cot(θ) = hypotenuse / adjacent

Question 30

Which trigonometric function is used to find the ratio of the length of the opposite side to the length of the hypotenuse?

Accepted Answer

Sine

Answer

Cosine

Answer

Tangent

Answer

Cotangent

Question 31

The inverse of the sine function is called:

Accepted Answer

Arcsine

Answer

Arccosine

Answer

Arctangent

Answer

Arccotangent

Question 32

What is the period of the sine function?

Accepted Answer

2π

Answer

π

Answer

3π

Answer

4π

Question 33

If y = 2sin(x), what is the amplitude of the function?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 34

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

Accepted Answer

Opposite side to hypotenuse

Answer

Adjacent side to hypotenuse

Answer

Adjacent side to opposite side

Question 35

At what angle is the tangent undefined?

Accepted Answer

90 degrees

Answer

0 degrees

Answer

30 degrees

Answer

60 degrees

Question 36

What is a practical application of trigonometry?

Accepted Answer

Solving problems involving heights and distances

Answer

Solving equations involving polynomials

Answer

Calculating the volume of a sphere

Answer

Calculating the area of a circle

Question 37

What is the Pythagorean identity for cosine?

Accepted Answer

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

Answer

sin(x) + cos(x) = 1

Answer

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

Answer

sin(x) - cos(x) = 1

Question 38

What is the value of sine for an angle of -30 degrees in a unit circle?

Accepted Answer

-1/2

Answer

0

Answer

1/2

Answer

1

Question 39

In a right triangle, the adjacent side is 3 and the opposite side is 4. What is the tangent of the angle opposite the 4-unit side?

Accepted Answer

4/3

Answer

2

Answer

3/4

Answer

1

Question 40

In a right triangle, the opposite side measures 3 and the adjacent side measures 4. What is the value of tan(θ)?

Accepted Answer

3/4

Answer

1/4

Answer

1/3

Answer

4/3

Question 41

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

Accepted Answer

sin²(θ) + cos²(θ) = 1

Answer

sin(θ) + cos(θ) = 1

Answer

sin(θ) - cos(θ) = 1

Answer

sin(θ) / cos(θ) = 1

Question 42

In a right triangle with a 30-degree angle and a hypotenuse of 10 cm, what is the length of the side opposite the 30-degree angle?

Accepted Answer

5 cm

Answer

10 cm

Answer

12 cm

Answer

7 cm

Question 43

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

Accepted Answer

√3 / 2

Answer

-√3 / 2

Answer

0.5

Answer

1

Question 44

Which of the following represents the domain of the tangent function?

Accepted Answer

All real numbers except odd multiples of π/2

Answer

All real numbers

Answer

Only angles between 0 and 90 degrees

Answer

All positive real numbers

Question 45

If the tangent of an angle is 2, what is the approximate measure of the angle (in degrees)?

Accepted Answer

63.43 degrees

Answer

90 degrees

Answer

30 degrees

Answer

45 degrees

Question 46

What is a key characteristic of the graph of the sine function?

Accepted Answer

It oscillates between -1 and 1.

Answer

It is a straight line.

Answer

It has a period of 1.

Answer

It is always increasing.

Question 47

If the cosine of an angle is negative, in which quadrant(s) could the angle lie?

Accepted Answer

Quadrant II or III

Answer

Quadrant I

Answer

Quadrant IV

Answer

Cannot be determined without more information

Question 48

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

Accepted Answer

The length of the side opposite the angle to the length of the hypotenuse

Answer

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

Answer

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

Question 49

What is the cosine of 30 degrees?

Accepted Answer

√3 / 2

Answer

√2 / 2

Answer

1/2

Answer

1

Question 50

If the tangent of an angle is 1, what is the measure of the angle?

Accepted Answer

45 degrees

Answer

90 degrees

Answer

60 degrees

Answer

30 degrees

Question 51

In a right triangle, which trigonometric function is represented by the ratio of the opposite side to the adjacent side?

Accepted Answer

Tangent

Answer

Cosecant

Answer

Sine

Answer

Cosine

Question 52

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

Accepted Answer

√3 / 2

Answer

√2 / 2

Answer

1/2

Answer

1

Question 53

A surveyor measures the angle of elevation to the top of a building to be 30 degrees. If the surveyor is standing 50 meters away from the base of the building, how tall is the building?

Accepted Answer

50√3 / 3 meters

Answer

50 meters

Answer

50√2 meters

Answer

100 meters

Question 54

What is the value of tan(90 degrees)?

Accepted Answer

Undefined

Answer

1

Answer

∞

Answer

0

Question 55

A ramp is 12 meters long and rises 3 meters vertically. What is the angle of elevation of the ramp?

Accepted Answer

arcsin(1/4)

Answer

arccos(1/4)

Answer

arctan(1/4)

Answer

arcsin(4)