Mathematics.

identities

Trigonometric Identities

Trigonometry35 minDifficulty4 out of 10

You should know: trigonometric functions

Overview

Trigonometric identities are equations relating trig functions that hold for every value of the angle involved. They let you rewrite one trigonometric expression as another — simplifying integrals, solving equations, and proving other identities — much as algebraic identities like (a+b)² = a²+2ab+b² let you rewrite polynomial expressions.

Intuition

Because sine and cosine are just coordinates on the unit circle, many identities are really geometric facts in disguise. The Pythagorean identity sin²θ+cos²θ=1 is nothing but the equation of the unit circle itself; the angle-sum formulas come from rotating one point by a second angle and tracking how its coordinates transform.

Interactive Graph

sin(2x) — a double-angle identity in action

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Formal Definition

Definition

The core families of identities:

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
Pythagorean
sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A\cos B \pm \cos A \sin B
Angle sum/difference (sine)
cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A\cos B \mp \sin A \sin B
Angle sum/difference (cosine)
sin⁡(2θ)=2sin⁡θcos⁡θ,cos⁡(2θ)=cos⁡2θ−sin⁡2θ\sin(2\theta) = 2\sin\theta\cos\theta, \quad \cos(2\theta) = \cos^2\theta - \sin^2\theta
Double angle
sin⁡2θ=1−cos⁡(2θ)2,cos⁡2θ=1+cos⁡(2θ)2\sin^2\theta = \frac{1-\cos(2\theta)}{2}, \quad \cos^2\theta = \frac{1+\cos(2\theta)}{2}
Power-reduction / half-angle

Derivation

The cosine difference formula cos(A - B) = cos A cos B + sin A sin B can be derived using the distance between two points on the unit circle at angles A and B, computed two ways:

d2=(cos⁡A−cos⁡B)2+(sin⁡A−sin⁡B)2d^2 = (\cos A - \cos B)^2 + (\sin A - \sin B)^2

Distance formula between the two points

=2−2(cos⁡Acos⁡B+sin⁡Asin⁡B)= 2 - 2(\cos A \cos B + \sin A \sin B)

Expand, using sin²+cos²=1 for each point

d2=2−2cos⁡(A−B)d^2 = 2 - 2\cos(A-B)

Same distance, computed via the angle between the points directly

⇒cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\Rightarrow \cos(A-B) = \cos A \cos B + \sin A \sin B

Equate the two expressions

Properties

Tangent identity

tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}

Reciprocal Pythagorean forms

1+tan⁡2θ=sec⁡2θ,1+cot⁡2θ=csc⁡2θ1 + \tan^2\theta = \sec^2\theta, \quad 1 + \cot^2\theta = \csc^2\theta

Applications

Product-to-sum identities are used to analyze amplitude-modulated signals and beat frequencies.

Worked Examples

  1. Use the power-reduction identity.

    sin⁡2θ=1−0.62=0.2\sin^2\theta = \frac{1 - 0.6}{2} = 0.2

Answer: 0.2

Practice Problems

Difficulty 4/10

Verify that (1 - cos 2θ)/sin 2θ = tan θ.

Difficulty 6/10

In AC electrical power, instantaneous power involves sin²(ωt). Use a trig identity to rewrite sin²(ωt) in a form that makes its average value obvious.

Common Mistakes

Common Mistake

Assuming sin(A+B) = sin A + sin B.

Trig functions are NOT linear/additive. sin(A+B) = sin A cos B + cos A sin B — a common test case is A=B=45°: sin(90°)=1, but sin45°+sin45°=√2≈1.41≠1.

Quiz

The Pythagorean identity states that for any angle θ:
Trig identities are practically important because they let engineers:

Summary

  • Trig identities hold for all angles and let you rewrite expressions in equivalent forms.
  • Pythagorean: sin²θ+cos²θ=1 is just the unit circle equation.
  • Angle-sum/difference formulas let you compute trig values at non-standard angles.
  • Double-angle and power-reduction identities interconvert θ and 2θ expressions.

References

  1. WebsiteWikipedia — List of trigonometric identities