Trigonometric identities

 Trigonometric identities are mathematical relationships involving trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) that hold true for all angles. These identities are fundamental in trigonometry and are essential in solving various problems in mathematics, physics, engineering, and other fields.

Here are some of the most commonly used trigonometric identities:

1. Pythagorean Identities:

  • sin²θ + cos²θ = 1
  • 1 + tan²θ = sec²θ
  • 1 + cot²θ = csc²θ

2. Reciprocal Identities:

cscθ = 1/sinθ
secθ = 1/cosθ
cotθ = 1/tanθ

3. Quotient Identities:

tanθ = sinθ/cosθ
cotθ = cosθ/sinθ

4. Co-function Identities:

  • sin(π/2 - θ) = cosθ
  • cos(π/2 - θ) = sinθ
  • tan(π/2 - θ) = cotθ
  • cot(π/2 - θ) = tanθ
  • sec(π/2 - θ) = cscθ
  • csc(π/2 - θ) = secθ

5. Even-Odd Identities:
  • sin(-θ) = -sinθ
  • cos(-θ) = cosθ
  • tan(-θ) = -tanθ
  • csc(-θ) = -cscθ
  • sec(-θ) = secθ
  • cot(-θ) = -cotθ

6. Double Angle Identities:

sin(2θ) = 2sinθcosθ
cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
tan(2θ) = (2tanθ) / (1 - tan²θ)

7. Half Angle Identities:

sin(θ/2) = ±√((1 - cosθ)/2)
cos(θ/2) = ±√((1 + cosθ)/2)
tan(θ/2) = ±√((1 - cosθ)/(1 + cosθ))

8. Sum and Difference Identities:

sin(A ± B) = sinAcosB ± cosAsinB
cos(A ± B) = cosAcosB ∓ sinAsinB
tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA*tanB)

Product-to-Sum and Sum-to-Product Identities:

sinA*sinB = (1/2)[cos(A - B) - cos(A + B)]
cosA*cosB = (1/2)[cos(A - B) + cos(A + B)]
sinA*cosB = (1/2)[sin(A + B) + sin(A - B)]
cosA*sinB = (1/2)[sin(A + B) - sin(A - B)]


These are just some of the many trigonometric identities that exist. Understanding and utilizing these identities can greatly simplify trigonometric calculations and help solve complex problems involving angles and triangles.






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