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Trigonometric Identities and Their Significance

Description: Trigonometric Identities and Their Significance Quiz
Number of Questions: 15
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Tags: trigonometry trigonometric identities applications of trigonometry
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What is the Pythagorean identity?

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

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

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

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

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Correct Option: A
Explanation:

The Pythagorean identity is a fundamental trigonometric identity that states that the square of the sine of an angle plus the square of the cosine of the same angle is equal to 1.

What is the double-angle identity for sine?

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

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

  3. sin(2x) = sin(x) - sin(x)

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

Show answer
Correct Option: A
Explanation:

The double-angle identity for sine states that the sine of twice an angle is equal to twice the sine of the angle multiplied by the cosine of the angle.

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

Show answer
Correct Option: A
Explanation:

The double-angle identity for cosine states that the cosine of twice an angle is equal to the square of the cosine of the angle minus the square of the sine of the angle.

What 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

Show answer
Correct Option: A
Explanation:

The half-angle identity for sine states that the sine of half an angle is equal to the square root of one minus the cosine of the angle divided by two.

What is the half-angle identity for cosine?

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

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

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

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

Show answer
Correct Option: A
Explanation:

The half-angle identity for cosine states that the cosine of half an angle is equal to the square root of one plus the cosine of the angle divided by two.

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

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

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

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

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

Show answer
Correct Option: A
Explanation:

The sum-to-product identity for sine states that the sum of two sines is equal to twice the sine of half the sum of the angles multiplied by the cosine of half the difference of the angles.

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

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

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

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

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

Show answer
Correct Option: A
Explanation:

The sum-to-product identity for cosine states that the sum of two cosines is equal to twice the cosine of half the sum of the angles multiplied by the cosine of half the difference of the angles.

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

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

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

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

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

Show answer
Correct Option: A
Explanation:

The product-to-sum identity for sine states that the product of two sines is equal to half the difference of the cosines of the sum and difference of the angles.

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

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

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

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

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

Show answer
Correct Option: A
Explanation:

The product-to-sum identity for cosine states that the product of two cosines is equal to half the sum of the cosines of the sum and difference of the angles.

What is the addition formula for sine?

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

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

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

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

Show answer
Correct Option: A
Explanation:

The addition formula for sine states that the sine of the sum of two angles is equal to the sine of the first angle multiplied by the cosine of the second angle plus the cosine of the first angle multiplied by the sine of the second angle.

What is the addition formula for cosine?

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

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

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

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

Show answer
Correct Option: A
Explanation:

The addition formula for cosine states that the cosine of the sum of two angles is equal to the cosine of the first angle multiplied by the cosine of the second angle minus the sine of the first angle multiplied by the sine of the second angle.

What is the subtraction formula for sine?

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

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

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

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

Show answer
Correct Option: A
Explanation:

The subtraction formula for sine states that the sine of the difference of two angles is equal to the sine of the first angle multiplied by the cosine of the second angle minus the cosine of the first angle multiplied by the sine of the second angle.

What is the subtraction formula for cosine?

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

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

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

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

Show answer
Correct Option: A
Explanation:

The subtraction formula for cosine states that the cosine of the difference of two angles is equal to the cosine of the first angle multiplied by the cosine of the second angle plus the sine of the first angle multiplied by the sine of the second angle.

What is the double-angle formula for sine?

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

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

  3. sin(2x) = sin(x) - sin(x)

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

Show answer
Correct Option: A
Explanation:

The double-angle formula for sine states that the sine of twice an angle is equal to twice the sine of the angle multiplied by the cosine of the angle.

What is the double-angle formula 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

Show answer
Correct Option: A
Explanation:

The double-angle formula for cosine states that the cosine of twice an angle is equal to the square of the cosine of the angle minus the square of the sine of the angle.

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