Trigonometry Quizzes: Test Your Skills and Master the Unit

Question 1

In a right triangle, what is the tangent of a 45-degree angle?

Accepted Answer

1

Answer

0

Answer

2

Answer

Cannot be determined

Question 2

In a right triangle, what is the sine of an angle defined as?

Accepted Answer

Opposite side divided by Hypotenuse

Answer

Adjacent side divided by Hypotenuse

Answer

Opposite side divided by Adjacent side

Question 3

In a triangle with sides a, b, and c, and angle θ opposite side c, which formula represents the Law of Cosines for finding side c?

Accepted Answer

c² = a² + b² - 2ab cos(θ)

Answer

a² = b² + c² - 2bc sin(θ)

Answer

c² = a² + b² + 2ab sin(θ)

Question 4

Which trigonometric identity can be used to find the sine of an angle in a right triangle if the cosine is known?

Accepted Answer

sin(x) = √(1 - cos²(x))

Answer

sin²(x) + cos²(x) = 1

Answer

cos(x) = √(1 - sin²(x))

Answer

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

Question 5

Which trigonometric identity correctly defines the sine of a double angle?

Accepted Answer

sin(2x) = 2sin(x)cos(x)

Answer

sin(2x) = 2sin(x)

Answer

sin(2x) = sin(x) + sin(x)

Answer

sin(2x) = sin^2(x)

Question 6

In a right triangle, the length of one leg is 3 cm, and the hypotenuse measures 5 cm. What is the length of the other leg?

Accepted Answer

4 cm

Answer

6 cm

Answer

8 cm

Answer

10 cm

Question 7

Solve the trigonometric equation: 2sin(x)cos(x) - sin(x) = 0

Accepted Answer

x = 0°, x = 180°

Answer

x = 45°, x = 135°

Answer

x = 60°, x = 120°

Answer

The equation has no solutions.

Question 8

In a right triangle, one angle measures 30 degrees and the opposite side is 6 units long. What is the length of the hypotenuse?

Accepted Answer

6√2

Answer

3

Answer

6√3

Answer

12

Question 9

When graphing trigonometric functions, what is the effect of a vertical shift on the graph?

Accepted Answer

Moves the graph up or down along the y-axis, without affecting the period or phase shift.

Answer

Changes the amplitude of the function, making it narrower or wider.

Answer

Alters the period of the function, compressing or stretching the graph.

Question 10

Which trigonometric property is illustrated by the multiple solutions to the equation `sin(x) = 0.5`, where `x = π/6 + 2πn` and `x = 5π/6 + 2πn` and `n` is an integer?

Accepted Answer

Periodicity

Answer

Amplitude

Answer

Phase Shift

Answer

Vertical Shift

Question 11

Identify the option that is NOT a trigonometric ratio.

Accepted Answer

Radius

Answer

Sine

Answer

Cosine

Answer

Tangent

Question 12

What is the period of the sine function?

Accepted Answer

2π

Answer

π

Answer

π/2

Answer

1

Question 13

Identify the graph of the function y = 2cos(x).

Accepted Answer

A cosine wave with amplitude 2 and period π

Answer

A sine wave with amplitude 2 and period π

Answer

A cosine wave with amplitude 2π and period 1

Answer

A sine wave with amplitude 2π and period 1

Question 14

Determine the inverse function of cosine.

Accepted Answer

Arcsine

Answer

Sine

Answer

Cosine

Answer

Tangent

Question 15

How does a vertical shift affect the graph of a trigonometric function? Provide a real-world example to illustrate your explanation.

Accepted Answer

Moves the graph up or down without changing its shape or other characteristics.

Answer

Changes the amplitude of the function.

Answer

Changes the period of the function.

Answer

Changes the phase shift of the function.

Question 16

Prove the trigonometric identity `tan(a+b) = (tan(a)+tan(b))/(1−tan(a)tan(b))` using the sum-to-product formula for tangent.

Accepted Answer

`tan(a+b) = (tan(a)+tan(b))/(1−tan(a)tan(b))`

Answer

`tan(a+b) = tan(a) + tan(b)`

Answer

`tan(a+b) = tan(a) - tan(b)`

Answer

`tan(a+b) = (tan(a)−tan(b))/(1+tan(a)tan(b))`

Question 17

Find the exact value of `cos(π/12) + cos(5π/12) + cos(7π/12) + cos(11π/12)`.

Accepted Answer

0

Answer

1

Answer

-1

Answer

√3

Question 18

In trigonometry, the unit circle is defined as a circle with radius:

Accepted Answer

1

Answer

0

Answer

2

Answer

π

Question 19

The period of the trigonometric function y = sin x is:

Accepted Answer

2π

Answer

π

Answer

360°

Answer

720°

Question 20

The unit circle is a circle with a radius of:

Accepted Answer

1

Answer

0

Answer

2

Answer

π

Question 21

Determine the magnitude of the vector represented by the complex number in trigonometric form: 2(cos 30° + i sin 30°)

Accepted Answer

2

Answer

1

Answer

4

Answer

-2

Question 22

How does increasing the frequency of a trigonometric function impact the graph's oscillations within a fixed interval?

Accepted Answer

Increases oscillations, decreases period

Answer

Decreases oscillations, increases period

Answer

Increases both oscillations and period

Answer

Decreases both oscillations and period

Question 23

Determine which of the following transformations will result in a horizontal shift of 2 units to the left and a vertical stretch by a factor of 3 for the function f(x)=cos(x)?

Accepted Answer

g(x) = 3cos(x + 2)

Answer

g(x) = 3cos(x - 2)

Answer

g(x) = cos(3x + 2)

Answer

g(x) = cos(x + 2) / 3

Question 24

In a right triangle, the hypotenuse is 10 units and one leg is 6 units. What is the measure of the angle opposite the 6-unit leg, rounded to the nearest degree?

Accepted Answer

37°

Answer

53°

Answer

60°

Answer

45°

Question 25

In trigonometry, what is the sine of an angle equal to π/2 radians?

Accepted Answer

1

Answer

0

Answer

-1

Answer

Undefined

Question 26

Which of the following equations represents the unit circle?

Accepted Answer

x^2 + y^2 = 1

Answer

x^2 + y^2 = 2

Answer

x^2 - y^2 = 1

Answer

x^2 - y^2 = 2

Question 27

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

Accepted Answer

2

Answer

1

Answer

3

Answer

6

Question 28

Which of the following is the trigonometric identity that expresses the Pythagorean relationship?

Accepted Answer

sin²θ + cos²θ = 1

Answer

secθ + cscθ = 1

Answer

sinθ + cosθ = 1

Answer

tanθ + cotθ = 1

Question 29

In a right triangle, the ratio of the side opposite an angle to the side adjacent to the angle is known as the:

Accepted Answer

Sine

Answer

Cosine

Answer

Tangent

Answer

Cotangent

Question 30

Which trigonometric identity relates sine and cosine?

Accepted Answer

sin²θ + cos²θ = 1

Answer

cotθ = 1/tanθ

Answer

tanθ = sinθ/cosθ

Answer

cscθ = 1/sinθ

Question 31

Find the period of the function y = 2sin(3x).

Accepted Answer

2π/3

Answer

2π

Answer

3π/2

Answer

π/3

Question 32

What is the inverse function of cosine?

Accepted Answer

Arccosine

Answer

Arcsine

Answer

Arctangent

Answer

Arccotangent

Question 33

Find the amplitude of the function y = -2cos(x + π/3).

Accepted Answer

2

Answer

-1

Answer

1

Answer

-2

Question 34

In a right triangle, the ratio of the length of the opposite side to the length of the hypotenuse is termed as

Accepted Answer

Sine

Answer

Cotangent

Answer

Tangent

Answer

Cosine

Question 35

Given that sin(A) = 0.5, ascertain the measure of angle A in degrees:

Accepted Answer

30°

Answer

60°

Answer

90°

Answer

45°

Question 36

The amplitude of the trigonometric function f(x) = 3 cos(x) is:

Accepted Answer

3

Answer

1

Answer

-3

Answer

0

Question 37

Specify the domain of the trigonometric function f(x) = tan(x):

Accepted Answer

All real numbers except multiples of π/2

Answer

All positive real numbers

Answer

All real numbers

Answer

All negative real numbers

Question 38

In a triangle with sides of length a, b, and c, the Law of Cosines is defined as:

Accepted Answer

c² = a² + b² - 2ab cos(∠C)

Answer

a² + b² = c² + 2ab cos(∠C)

Answer

c² + a² = b² - 2ab cos(∠C)

Answer

a² + b² = c² - 2ab cos(∠C)

Question 39

What is the sine of an angle measuring 30°?

Accepted Answer

1/2

Answer

1

Answer

2

Answer

1/4

Question 40

What is the cosine of an angle measuring 45°?

Accepted Answer

√2/2

Answer

√3/2

Answer

0

Answer

1

Question 41

What is the tangent of an angle measuring 60°?

Accepted Answer

√3

Answer

2

Answer

1

Answer

0

Question 42

Which of the following is the Pythagorean identity?

Accepted Answer

sin² θ + cos² θ = 1

Answer

tan² θ + cot² θ = 1

Answer

1 + tan² θ = sec² θ

Question 43

What is the inverse function of cosine?

Accepted Answer

arccos

Answer

arcsin

Answer

arctan

Answer

arccot

Question 44

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

Accepted Answer

3

Answer

6

Answer

1

Answer

-3

Question 45

In a right triangle with legs a and b and hypotenuse c, which trigonometric ratio represents the ratio of the length of the opposite side to the length of the hypotenuse?

Accepted Answer

Sine (a/c)

Answer

Cotangent (b/a)

Answer

Cosine (b/c)

Answer

Tangent (a/b)

Question 46

Find the value of cos(30 degrees).

Accepted Answer

√3/2

Answer

1/2

Answer

1

Answer

0

Question 47

Which trigonometric identity can be used to convert an expression involving cot(θ) to one involving tan(θ)?

Accepted Answer

cot(θ) = 1/tan(θ)

Answer

sin(θ)^2 + cos(θ)^2 = 1

Answer

sin(2θ) = 2sin(θ)cos(θ)

Answer

cos(θ - φ) = cos(θ)cos(φ) + sin(θ)sin(φ)

Question 48

A tree casts a shadow that is 40 meters long when the angle of elevation to the top of the tree is 35 degrees. What is the height of the tree?

Accepted Answer

40 tan(35 degrees) meters

Answer

40 cot(35 degrees) meters

Answer

40 sin(35 degrees) meters

Answer

40 cos(35 degrees) meters

Question 49

Which of the following is a trigonometric function?

Accepted Answer

Sine

Answer

Quadratic

Answer

Logarithm

Answer

Exponential

Question 50

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

Accepted Answer

sine

Answer

cotangent

Answer

tangent

Answer

cosine

Question 51

Which of the following is the reciprocal of the cosine function?

Accepted Answer

secant

Answer

sine

Answer

tangent

Question 52

The period of the sine function is:

Accepted Answer

2π

Answer

4π

Answer

π

Question 53

In a triangle, the law of cosines can be used to find:

Accepted Answer

The length of any side, given the lengths of the other two sides and the measure of the included angle

Answer

The measure of any angle, given the lengths of the other two sides

Question 54

If the sine of an angle is 0.5, what is the measure of the angle?

Accepted Answer

30°

Answer

60°

Answer

45°

Answer

90°

Question 55

Find the value of cosine 150° using the unit circle.

Accepted Answer

-½

Answer

0

Answer

1

Answer

½

Question 56

Determine the period of the function y = 2sin(3x + π/6).

Accepted Answer

2π/3

Answer

π

Answer

4π

Answer

2π

Question 57

What is the inverse function of y = tanx?

Accepted Answer

arctanx

Answer

cotx

Answer

sin⁻¹x

Question 58

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

Accepted Answer

Opposite leg to the hypotenuse

Answer

Opposite leg to the adjacent leg

Answer

Hypotenuse to the opposite leg

Answer

Adjacent leg to the hypotenuse

Question 59

What is the value of sin 30°?

Accepted Answer

1/2

Answer

√2/2

Answer

√3/2

Answer

1/√2

Question 60

What is the period of the function y = cos x?

Accepted Answer

2π

Answer

6π

Answer

4π

Answer

π

Question 61

What is the amplitude of the function y = 2 sin 3x?

Accepted Answer

2

Answer

6

Answer

1

Answer

3

Question 62

Which of the following is a Pythagorean identity?

Accepted Answer

sin² x + cos² x = 1

Answer

sin x = cos x

Answer

sin x + cos x = 1

Answer

tan x = sin x / cos x

Question 63

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

Accepted Answer

All real numbers except multiples of π/2

Answer

The interval [0, 2π]

Answer

The interval [-π/2, π/2]

Answer

All real numbers

Question 64

Which of the following is an example of a real-world application of trigonometry?

Accepted Answer

Calculating the height of a building using the shadow method

Answer

Solving a word problem

Answer

Measuring the distance to a star