Trigonometric Identities: Master the Art of Trig with Online Quizzes

Question 1

Which of the following expresses the Pythagorean identity?

Accepted Answer

sin²θ + cos²θ = 1

Answer

tanθ = sinθ / cosθ

Answer

cotθ = cosθ / sinθ

Answer

sin(θ + φ) = sinθcosφ + cosθsinφ

Question 2

Which formula represents the double-angle formula for sine?

Accepted Answer

sin2θ = 2sinθcosθ

Answer

cos2θ = 2cos²θ - 1

Answer

tan2θ = 2tanθ / (1 - tan²θ)

Answer

sin(θ - φ) = sinθcosφ - cosθsinφ

Question 3

Identify the sum-to-product formula for sine.

Accepted Answer

sin(θ + φ) = sinθcosφ + cosθsinφ

Answer

cos(θ + φ) = cosθcosφ - sinθsinφ

Answer

tan(θ + φ) = (tanθ + tanφ) / (1 - tanθtanφ)

Answer

sin(a - b) = sinacosb - cosasinb

Question 4

Which identity aids in solving for the unknown angle in a right triangle?

Accepted Answer

Pythagorean identity

Answer

Double-angle formulas

Answer

Sum-to-product formulas

Answer

Half-angle formulas

Question 5

In a triangle with sides a, b, and c, which trigonometric identity facilitates the calculation of the area?

Accepted Answer

Area = (1/2)absinC

Answer

Area = (1/2)abcosC

Answer

Area = (1/2)bccosA

Answer

Area = (1/2)cassinB

Question 6

Explain how double-angle trigonometric formulas are used to simplify trigonometric expressions and derive new identities.

Accepted Answer

Double-angle formulas can be used to derive other identities, such as sum-to-product and product-to-sum formulas.

Answer

Double-angle formulas can only be used with expressions involving a single trigonometric function.

Answer

Double-angle formulas can only be used to simplify equations, not identities.

Answer

Double-angle formulas are only used to expand trigonometric expressions, not simplify them.

Question 7

How does the unit circle aid in deriving the Pythagorean identity, sin² θ + cos² θ = 1?

Accepted Answer

It offers a visual representation of trigonometric ratios, aiding in understanding their relationships.

Answer

It defines the domain and range of trigonometric functions.

Answer

It simplifies calculations of trigonometric values directly.

Answer

It is unnecessary for deriving trigonometric identities.

Question 8

Which of the following is the Pythagorean identity?

Accepted Answer

sin²θ + cos²θ = 1

Answer

sinθ + cosθ = 1

Answer

tan²θ + 1 = sec²θ

Answer

cot²θ - 1 = csc²θ

Question 9

Which of the following is the sum-to-product formula for sine?

Accepted Answer

sin(A + B) = sinA cosB + cosA sinB

Answer

sin(A - B) = sinA cosB - cosA sinB

Answer

cos(A + B) = cosA cosB - sinA sinB

Answer

cos(A - B) = cosA cosB + sinA sinB

Question 10

Which of the following identities is used to find the sine of a sum of two angles?

Accepted Answer

sin(A + B) = sinA cosB + cosA sinB

Answer

sin(A - B) = sinA cosB - cosA sinB

Answer

cos(A + B) = cosA cosB - sinA sinB

Answer

tan(A + B) = tanA + tanB

Question 11

Identify the trigonometric identity for (sin x + cos x)²:

Accepted Answer

1 + sin 2x

Answer

1 - sin 2x

Answer

2 sin x cos x

Answer

None of the options match the result

Question 12

Which of the following is the Pythagorean identity?

Accepted Answer

sin² x + cos² x = 1

Answer

tan x = sin x / cos x

Answer

cot x = 1 / tan x

Answer

sec x = 1 / cos x

Question 13

Which of the following is the sum-to-product formula for sine?

Accepted Answer

sin (a + b) = sin a cos b + cos a sin b

Answer

sin (a - b) = sin a cos b - cos a sin b

Answer

cos (a + b) = cos a cos b - sin a sin b

Answer

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

Question 14

Which of the following is a common engineering application of trigonometric identities?

Accepted Answer

All of the options listed

Answer

Calculating the torque on a shaft

Answer

Designing bridges and structures

Answer

Determining the angle of repose of a material

Question 15

Which of the following is the correct trigonometric identity for the expression sin (a - b)?

Accepted Answer

sin a cos b - cos a sin b

Answer

sin a cos b + cos a sin b

Answer

cos a cos b - sin a sin b

Answer

sin a sin b - cos a cos b

Question 16

Prove the trigonometric identity: (1 + tan x)² = sec² x

Accepted Answer

Use the Pythagorean identity

Answer

Use the double-angle formula for tangent

Answer

Use the sum-to-product formula for tangent

Answer

Use the half-angle formula for tangent

Question 17

Which of the following is the Pythagorean identity?

Accepted Answer

sin²θ + cos²θ = 1

Answer

tan²θ + cot²θ = 1

Answer

sinθ + cosθ = 1

Answer

tanθ + cotθ = 1

Question 18

Select the double-angle formula for sine

Accepted Answer

sin(2θ) = 2sinθcosθ

Answer

sin(2θ) = sin²θ + cos²θ

Answer

sin(2θ) = 1 - 2sin²θ

Answer

sin(2θ) = tanθ + cotθ

Question 19

To prove the identity: sin(π/2 - θ) = cosθ, which method should be used?

Accepted Answer

Employ the unit circle and the definitions of sine and cosine

Answer

Combine the Pythagorean identity and the double-angle formula for sine

Answer

Utilize the sum-to-product formula for cosine

Question 20

Which of the following is NOT a trigonometric identity?

Accepted Answer

sinθ = tanθ

Answer

cotθ = 1/tanθ

Answer

tanθ = sinθ/cosθ

Answer

cos²θ + sin²θ = 1

Question 21

Which of the following identities is used to find the sine of half an angle?

Accepted Answer

sin(θ/2) = ±√((1 - cosθ)/2)

Answer

tan(θ/2) = ±√((1 - sinθ)/2)

Answer

cos(θ/2) = ±√((1 + cosθ)/2)

Answer

cot(θ/2) = ±√((1 + sinθ)/2)

Question 22

Prove the identity: tan(α - β) = (tanα - tanβ)/(1 + tanαtanβ)

Accepted Answer

Use the definitions of tangent and the difference formula for tangent

Answer

Combine the Pythagorean identity and the double-angle formula for sine

Answer

Utilize the sum-to-product formula for cosine