The Essential Trigonometric Identities and Their Derivations

The Essential Trigonometric Identities and Their Derivations

Trigonometric identities describe exact relationships between the trigonometric functions. These identities hold for all valid angles and can be derived using the unit circle, right-triangle geometry, and rotation principles.

1. The Unit Circle and the Fundamental Identity

On the unit circle, a point has coordinates (cos θ, sin θ). Since the radius is 1:

(cos θ)² + (sin θ)² = 1

sin² θ + cos² θ = 1

2. Quotient and Reciprocal Identities

2.1 Basic Ratios

  • sin θ = opposite / hypotenuse
  • cos θ = adjacent / hypotenuse
  • tan θ = opposite / adjacent

2.2 Quotient Identity

tan θ = (opposite / adjacent)
= (sin θ / cos θ)

tan θ = sin θ / cos θ

2.3 Reciprocal Functions

  • csc θ = 1 / sin θ
  • sec θ = 1 / cos θ
  • cot θ = cos θ / sin θ

3. Pythagorean Extensions

From sin² θ + cos² θ = 1:

Divide by cos² θ:

tan² θ + 1 = sec² θ

Divide by sin² θ:

1 + cot² θ = csc² θ

4. Sum and Difference Identities

Rotating by α then β is equivalent to a single rotation by α + β. This yields:

  • cos(α + β) = cos α cos β − sin α sin β
  • sin(α + β) = sin α cos β + cos α sin β

Replacing β with −β:

  • cos(α − β) = cos α cos β + sin α sin β
  • sin(α − β) = sin α cos β − cos α sin β

5. Double-Angle Identities

5.1 Cosine

cos(2α) = cos² α − sin² α

  • cos(2α) = 2cos² α − 1
  • cos(2α) = 1 − 2sin² α

5.2 Sine

sin(2α) = 2 sin α cos α

5.3 Tangent

tan(2α) = (2 tan α) / (1 − tan² α)

6. Half-Angle Identities

sin² θ = (1 − cos(2θ)) / 2

cos² θ = (1 + cos(2θ)) / 2

  • sin(θ / 2) = ± √((1 − cos θ) / 2)
  • cos(θ / 2) = ± √((1 + cos θ) / 2)

7. Product-to-Sum Example

cos(α + β) + cos(α − β) = 2 cos α cos β

cos α cos β = (cos(α + β) + cos(α − β)) / 2

8. Summary of Core Identities

  • sin² θ + cos² θ = 1
  • tan θ = sin θ / cos θ
  • cot θ = cos θ / sin θ
  • sec θ = 1 / cos θ
  • csc θ = 1 / sin θ
  • tan² θ + 1 = sec² θ
  • 1 + cot² θ = csc² θ
  • sin(α ± β) = sin α cos β ± cos α sin β
  • cos(α ± β) = cos α cos β ∓ sin α sin β
  • sin(2α) = 2 sin α cos α
  • cos(2α) = cos² α − sin² α
  • tan(2α) = (2 tan α) / (1 − tan² α)
  • sin² θ = (1 − cos(2θ)) / 2
  • cos² θ = (1 + cos(2θ)) / 2
  • cos α cos β = (cos(α + β) + cos(α − β)) / 2

© mathematics.proofs

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