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Trigonometric Identities

Description: This quiz is designed to assess your understanding of trigonometric identities. Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. These identities are useful in solving a variety of trigonometric problems.
Number of Questions: 15
Created by:
Tags: trigonometry trigonometric identities sine cosine tangent cosecant secant cotangent
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Which of the following is an identity?

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

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

  3. sin(x) - cos(x) = 1

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

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Correct Option: A
Explanation:

The identity sin^2(x) + cos^2(x) = 1 is known as the Pythagorean identity. It is true for all values of x.

What is the reciprocal identity for sine?

  1. sin(x) = 1/csc(x)

  2. sin(x) = 1/sec(x)

  3. sin(x) = 1/tan(x)

  4. sin(x) = 1/cot(x)

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Correct Option: A
Explanation:

The reciprocal identity for sine is sin(x) = 1/csc(x).

Which of the following is the quotient identity for tangent?

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

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

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

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

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Correct Option: A
Explanation:

The quotient identity for tangent is tan(x) = sin(x) / cos(x).

What is the double-angle identity for cosine?

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

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

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

  4. cos(2x) = 2cos(x) - 1

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Correct Option: A
Explanation:

The double-angle identity for cosine is cos(2x) = cos^2(x) - sin^2(x).

Which of the following is the half-angle identity for sine?

  1. sin(x/2) = sqrt((1 - cos(x))/2)

  2. sin(x/2) = sqrt((1 + cos(x))/2)

  3. sin(x/2) = (1 - cos(x))/2

  4. sin(x/2) = (1 + cos(x))/2

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Correct Option: A
Explanation:

The half-angle identity for sine is sin(x/2) = sqrt((1 - cos(x))/2).

What is the sum-to-product identity for sine and cosine?

  1. sin(x) + cos(x) = 2sin((x+y)/2)cos((x-y)/2)

  2. sin(x) + cos(x) = 2cos((x+y)/2)sin((x-y)/2)

  3. sin(x) + cos(x) = sin((x+y)/2) + cos((x-y)/2)

  4. sin(x) + cos(x) = sin((x+y)/2) - cos((x-y)/2)

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Correct Option: A
Explanation:

The sum-to-product identity for sine and cosine is sin(x) + cos(x) = 2sin((x+y)/2)cos((x-y)/2).

Which of the following is the difference-to-product identity for sine and cosine?

  1. sin(x) - cos(x) = 2sin((x-y)/2)cos((x+y)/2)

  2. sin(x) - cos(x) = 2cos((x-y)/2)sin((x+y)/2)

  3. sin(x) - cos(x) = sin((x-y)/2) + cos((x+y)/2)

  4. sin(x) - cos(x) = sin((x-y)/2) - cos((x+y)/2)

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Correct Option: A
Explanation:

The difference-to-product identity for sine and cosine is sin(x) - cos(x) = 2sin((x-y)/2)cos((x+y)/2).

What is the product-to-sum identity for sine and cosine?

  1. sin(x)cos(y) = (sin(x+y) + sin(x-y))/2

  2. sin(x)cos(y) = (sin(x+y) - sin(x-y))/2

  3. sin(x)cos(y) = (cos(x+y) + cos(x-y))/2

  4. sin(x)cos(y) = (cos(x+y) - cos(x-y))/2

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Correct Option: A
Explanation:

The product-to-sum identity for sine and cosine is sin(x)cos(y) = (sin(x+y) + sin(x-y))/2.

Which of the following is the product-to-sum identity for cosine and sine?

  1. cos(x)sin(y) = (sin(x+y) - sin(x-y))/2

  2. cos(x)sin(y) = (sin(x+y) + sin(x-y))/2

  3. cos(x)sin(y) = (cos(x+y) + cos(x-y))/2

  4. cos(x)sin(y) = (cos(x+y) - cos(x-y))/2

Show answer
Correct Option: A
Explanation:

The product-to-sum identity for cosine and sine is cos(x)sin(y) = (sin(x+y) - sin(x-y))/2.

What is the sum-to-product identity for tangent?

  1. tan(x) + tan(y) = (sin(x+y))/(cos(x)cos(y))

  2. tan(x) + tan(y) = (sin(x-y))/(cos(x)cos(y))

  3. tan(x) + tan(y) = (cos(x+y))/(sin(x)sin(y))

  4. tan(x) + tan(y) = (cos(x-y))/(sin(x)sin(y))

Show answer
Correct Option: A
Explanation:

The sum-to-product identity for tangent is tan(x) + tan(y) = (sin(x+y))/(cos(x)cos(y)).

Which of the following is the difference-to-product identity for tangent?

  1. tan(x) - tan(y) = (sin(x-y))/(cos(x)cos(y))

  2. tan(x) - tan(y) = (sin(x+y))/(cos(x)cos(y))

  3. tan(x) - tan(y) = (cos(x-y))/(sin(x)sin(y))

  4. tan(x) - tan(y) = (cos(x+y))/(sin(x)sin(y))

Show answer
Correct Option: A
Explanation:

The difference-to-product identity for tangent is tan(x) - tan(y) = (sin(x-y))/(cos(x)cos(y)).

What is the half-angle identity for tangent?

  1. tan(x/2) = (1 - cos(x))/(1 + cos(x))

  2. tan(x/2) = (1 + cos(x))/(1 - cos(x))

  3. tan(x/2) = (sin(x))/(cos(x))

  4. tan(x/2) = (cos(x))/(sin(x))

Show answer
Correct Option: A
Explanation:

The half-angle identity for tangent is tan(x/2) = (1 - cos(x))/(1 + cos(x)).

Which of the following is the double-angle identity for tangent?

  1. tan(2x) = (2tan(x))/(1 - tan^2(x))

  2. tan(2x) = (2tan(x))/(1 + tan^2(x))

  3. tan(2x) = (tan(x))/(1 - tan^2(x))

  4. tan(2x) = (tan(x))/(1 + tan^2(x))

Show answer
Correct Option: A
Explanation:

The double-angle identity for tangent is tan(2x) = (2tan(x))/(1 - tan^2(x)).

What is the power-reducing identity for sine?

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

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

  3. sin^2(x) = (1 - sin(2x))/2

  4. sin^2(x) = (1 + sin(2x))/2

Show answer
Correct Option: A
Explanation:

The power-reducing identity for sine is sin^2(x) = (1 - cos(2x))/2.

Which of the following is the power-reducing identity for cosine?

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

  2. cos^2(x) = (1 - cos(2x))/2

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

  4. cos^2(x) = (1 - sin(2x))/2

Show answer
Correct Option: A
Explanation:

The power-reducing identity for cosine is cos^2(x) = (1 + cos(2x))/2.

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