Trigonometric Identities

Introduction

Trigonometric identities are equations involving trigonometric functions that hold for all values where both sides are defined. They are not mere formulas to memorize—they are tools that reveal the deep structure connecting sine, cosine, and the other trigonometric functions.

These identities enable us to simplify complex expressions, solve equations, evaluate integrals, and prove new relationships. In calculus, physics, and engineering, facility with trigonometric identities is essential for manipulating expressions involving periodic phenomena, waves, and rotations.

Most trigonometric identities ultimately derive from the definitions of sine and cosine on the unit circle and the Pythagorean theorem. Understanding the geometric origins makes the identities intuitive rather than arbitrary.

Pythagorean Identities

From the unit circle, a point at angle θ has coordinates (cos(θ),sin(θ)) Since this point lies on the unit circle x2+y2=1

sin2(θ)+cos2(θ)=1

This fundamental identity generates two others by division:

1+tan2(θ)=sec2(θ) divide by cos2(θ)

1+cot2(θ)=csc2(θ) divide by sin2(θ)

These identities are invaluable for converting between functions and simplifying expressions.

Sum and Difference Formulas

These formulas express the sine and cosine of a sum or difference in terms of sines and cosines of the individual angles:

sin(α±β)=sin(α)*cos(β)±cos(α)*sin(β)

cos(α±β)=cos(α)*cos(β)∓sin(α)*sin(β)

tan(α±β)=(tan(α)±tan(β))/(1∓tan(α)*tan(β))

The sum formulas can be proved geometrically using rotations or algebraically using complex exponentials. They are the foundation for many other identities.

Double Angle Formulas

Setting α=β=θ in the sum formulas yields:

sin(2*θ)=2*sin(θ)*cos(θ)

cos(2*θ)=cos2(θ)−sin2(θ)=2*cos2(θ)−1=1−2*sin2(θ)

tan(2*θ)=(2*tan(θ))/(1−tan2(θ))

The three forms for cos(2*θ) are interchangeable via sin2(θ)+cos2(θ)=1 Choose the form most convenient for the problem at hand.

Half Angle Formulas

Solving the double angle formulas for sin(θ) and cos(θ) gives:

sin2(θ)=(1−cos(2*θ))/2,cos2(θ)=(1+cos(2*θ))/2

Replacing θ with θ/2

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

The sign depends on the quadrant of θ/2 These formulas are useful in integration (converting powers of sine and cosine to first powers).

Product-to-Sum Formulas

These identities convert products of sines and cosines into sums, which is particularly useful in integration and signal processing:

sin(α)*cos(β)=1/2*[sin(α+β)+sin(α−β)]

cos(α)*cos(β)=1/2*[cos(α−β)+cos(α+β)]

sin(α)*sin(β)=1/2*[cos(α−β)−cos(α+β)]

These follow directly from adding or subtracting the sum and difference formulas. They transform multiplication into addition, which is often simpler to handle.

Sum-to-Product Formulas

The reverse transformation converts sums into products:

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

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

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

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

These are derived by substituting α=(A+B)/2 and β=(A−B)/2 into the product-to-sum formulas.

Worked Examples

Example 1: Simplifying an Expression

Simplify (1−cos(2*θ))/sin(2*θ)

Using 1-cos(2*θ)=2*sin2(θ) and sin(2*θ)=2*sin(θ*cos(θ))

(1−cos(2*θ))/sin(2*θ)=(2*sin2(θ))/(2*sin(θ)*cos(θ))=sin(θ)/cos(θ)=tan(θ)

Example 2: Evaluating Without a Calculator

Find the exact value of cos(75).

Write 75=45+30and apply this formula:

cos(75)=cos(45)*cos(30)-sin(45)*sin(30)=√(,2)/2⋅√(,3)/2−√(,2)/2⋅1/2=(√(,6)−√(,2))/4

Example 3: Proving an Identity

Prove that sin(2*θ)/(1+cos(2*θ))=tan(θ)

Using double angle formulas: sin(2*θ)=2*sin(θ)*cos(θ) and 1+cos(2*θ)=1+2*cos2(θ)-1=2*cos2(θ)

sin(2*θ)/(1+cos(2*θ))=(2*sin(θ)*cos(θ))/(2*cos2(θ))=sin(θ)/cos(θ)=tan(θ)

Example 4: Integration Application

Evaluate (∫_^)(sin2(x)*d(x))

Using the half-angle formula sin2(x)=(1−cos(2*x))/2

(∫_^)(sin2(x)*d(x))=(∫_^)((1−cos(2*x))/2*d(x))=1/2*(x−sin(2*x)/2)+C=x/2−sin(2*x)/4+C

Connection to Complex Exponentials

Euler's formula e(i*θ)=cos(θ)+i*sin(θ) provides a powerful framework for deriving trigonometric identities. From this:

cos(θ)=(e(i*θ)+e(−i*θ))/2,sin(θ)=(e(i*θ)−e(−i*θ))/(2*i)

Many identities become simple algebraic manipulations. For example, the product e(i*α)⋅e(i*β)=e(i*(α+β)) immediately yields the sum formulas when we expand both sides using Euler's formula.

De Moivre's theorem (cos(θ)+i*sin(θ))n=cos(n*θ)+i*sin(n*θ) generates formulas for cos(n*θ) and sin(n*θ) in terms of powers of cos(θ) and sin(θ)

Summary

Trigonometric identities are equations valid for all angles. The Pythagorean identity sin2(θ)+cos2(θ)=1 is fundamental. Sum and difference formulas express sin(α±β) and cos(α±β) in terms of the individual angles.

Double angle formulas as sin(2*θ)=2*sin(θ)*cos(θ), cos(2*θ)=cos2(θ)-sin2(θ) and half-angle formulas convert between single and double angles. Product-to-sum and sum-to-product formulas interchange products and sums.

Complex exponentials via Euler's formula unify all these identities elegantly. Mastery of trigonometric identities is essential for calculus, physics, signal processing, and any field involving periodic phenomena.