Trigonometric Identities

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Questions and Answers

What is the formula for cos(A + B)?

  • sin A cos B + cos A sin B
  • sin A sin B - cos A cos B
  • cos A cos B + sin A sin B
  • cos A cos B - sin A sin B (correct)

The formula sin(π/2-A) = cos A is always true.

True (A)

What is the formula for tan(A+B)?

[(tan A + tan B)/(1 – tan A tan B)]

The formula for sin(A -B) is _______.

<p>sin A cos B – cos A sin B</p> Signup and view all the answers

Match the following trigonometric identities with their formulas:

<p>sin2A = 2sinA cosA cos2A = cos2A–sin2A tan2A = (2 tan A)/(1-tan2A)</p> Signup and view all the answers

What is the formula for cos3A?

<p>4cos3A – 3cosA (D)</p> Signup and view all the answers

What is the value of sin(π-A) in terms of sin A?

<p>sin A (D)</p> Signup and view all the answers

The formula tan(A+B) = [(tan A + tan B)/(1 + tan A tan B)] is always true.

<p>False (B)</p> Signup and view all the answers

What is the formula for cos2A in terms of sin A and cos A?

<p>cos2A = cos2A–sin2A = 1–2sin2A = 2cos2A–1</p> Signup and view all the answers

The formula sin(A+B) sin(A–B) = _______

<p>sin2A–sin2B=cos2B–cos2A</p> Signup and view all the answers

What is the value of tan 2A in terms of tan A?

<p>(2 tan A)/(1-tan2A) (D)</p> Signup and view all the answers

Match the following trigonometric identities with their formulas:

<p>cos(A+B) cos(A–B) = cos2A–sin2B=cos2B–sin2A sin(A+B) sin(A–B) = sin2A–sin2B=cos2B–cos2A sin3A = 3sinA – 4sin3A cos3A = 4cos3A – 3cosA</p> Signup and view all the answers

The formula sin(A+B) = sin A cos B + cos A sin B is always true.

<p>True (A)</p> Signup and view all the answers

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Study Notes

Trigonometric Identities: Sum and Difference Formulas

  • cos(A + B) = cos A cos B – sin A sin B
  • cos(A – B) = cos A cos B + sin A sin B
  • sin(A+B) = sin A cos B + cos A sin B
  • sin(A -B) = sin A cos B – cos A sin B

Complementary Angle Identities

  • sin(π/2-A) = cos A
  • cos(π/2-A) = sin A
  • sin(π-A) = sin A
  • cos(π-A) = -cos A
  • sin(π+A)=-sin A
  • cos(π+A)=-cos A
  • sin(2π-A) = -sin A
  • cos(2π-A) = cos A

Tangent and Cotangent Identities

  • tan(A+B) = [(tan A + tan B)/(1 – tan A tan B)] (if none of the angles A, B, and (A ± B) is an odd multiple of π/2)
  • tan(A-B) = [(tan A – tan B)/(1 + tan A tan B)] (if none of the angles A, B, and (A ± B) is an odd multiple of π/2)
  • cot(A+B) = [(cot A cot B − 1)/(cot B + cot A)] (if none of the angles A, B, and (A ± B) is a multiple of π)
  • cot(A-B) = [(cot A cot B + 1)/(cot B – cot A)] (if none of the angles A, B, and (A ± B) is a multiple of π)

Product Identities

  • cos(A+B) cos(A–B)=cos2A–sin2B=cos2B–sin2A
  • sin(A+B) sin(A–B) = sin2A–sin2B=cos2B–cos2A

Sum-to-Product Identities

  • sinA+sinB = 2 sin (A+B)/2 cos (A-B)/2

Double Angle Identities

  • sin2A = 2sinA cosA = [2tan A /(1+tan2A)]
  • cos2A = cos2A–sin2A = 1–2sin2A = 2cos2A–1= [(1-tan2A)/(1+tan2A)]
  • tan 2A = (2 tan A)/(1-tan2A)

Triple Angle Identities

  • sin3A = 3sinA – 4sin3A
  • cos3A = 4cos3A – 3cosA
  • tan3A = [3tanA–tan3A]/[1−3tan2A]

Trigonometric Identities: Sum and Difference Formulas

  • cos(A + B) = cos A cos B – sin A sin B
  • cos(A – B) = cos A cos B + sin A sin B
  • sin(A+B) = sin A cos B + cos A sin B
  • sin(A -B) = sin A cos B – cos A sin B

Complementary Angle Identities

  • sin(π/2-A) = cos A
  • cos(π/2-A) = sin A
  • sin(π-A) = sin A
  • cos(π-A) = -cos A
  • sin(π+A)=-sin A
  • cos(π+A)=-cos A
  • sin(2π-A) = -sin A
  • cos(2π-A) = cos A

Tangent and Cotangent Identities

  • tan(A+B) = [(tan A + tan B)/(1 – tan A tan B)] (if none of the angles A, B, and (A ± B) is an odd multiple of π/2)
  • tan(A-B) = [(tan A – tan B)/(1 + tan A tan B)] (if none of the angles A, B, and (A ± B) is an odd multiple of π/2)
  • cot(A+B) = [(cot A cot B − 1)/(cot B + cot A)] (if none of the angles A, B, and (A ± B) is a multiple of π)
  • cot(A-B) = [(cot A cot B + 1)/(cot B – cot A)] (if none of the angles A, B, and (A ± B) is a multiple of π)

Product Identities

  • cos(A+B) cos(A–B)=cos2A–sin2B=cos2B–sin2A
  • sin(A+B) sin(A–B) = sin2A–sin2B=cos2B–cos2A

Sum-to-Product Identities

  • sinA+sinB = 2 sin (A+B)/2 cos (A-B)/2

Double Angle Identities

  • sin2A = 2sinA cosA = [2tan A /(1+tan2A)]
  • cos2A = cos2A–sin2A = 1–2sin2A = 2cos2A–1= [(1-tan2A)/(1+tan2A)]
  • tan 2A = (2 tan A)/(1-tan2A)

Triple Angle Identities

  • sin3A = 3sinA – 4sin3A
  • cos3A = 4cos3A – 3cosA
  • tan3A = [3tanA–tan3A]/[1−3tan2A]

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