9.2 Sum and Difference Identities

Sum and difference identities are powerful tools for manipulating trigonometric expressions. They allow us to break down complex angle combinations into simpler terms, making calculations easier and revealing hidden relationships between angles.

These identities are crucial for solving advanced trigonometric equations and simplifying complex expressions. They're the building blocks for understanding more complex trigonometric concepts and are widely used in fields like physics and engineering.

Sum and Difference Identities

Sum and difference formulas

• Sum formula for sine expresses $\sin(A + B)$ in terms of $\sin A$, $\cos A$, $\sin B$, and $\cos B$: $\sin(A + B) = \sin A \cos B + \cos A \sin B$
• Difference formula for sine expresses $\sin(A - B)$ in terms of $\sin A$, $\cos A$, $\sin B$, and $\cos B$: $\sin(A - B) = \sin A \cos B - \cos A \sin B$
• Sum formula for cosine expresses $\cos(A + B)$ in terms of $\cos A$, $\cos B$, $\sin A$, and $\sin B$: $\cos(A + B) = \cos A \cos B - \sin A \sin B$
• Difference formula for cosine expresses $\cos(A - B)$ in terms of $\cos A$, $\cos B$, $\sin A$, and $\sin B$: $\cos(A - B) = \cos A \cos B + \sin A \sin B$
• Sum formula for tangent expresses $\tan(A + B)$ in terms of $\tan A$ and $\tan B$: $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$
• Difference formula for tangent expresses $\tan(A - B)$ in terms of $\tan A$ and $\tan B$: $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$
• These formulas are essential for angle addition and angle subtraction in trigonometric functions

Cofunction identities in formulas

• Cofunction identities relate trigonometric functions of complementary angles ($\frac{\pi}{2} - \theta$)
  • Sine and cosine are cofunctions: $\sin(\frac{\pi}{2} - \theta) = \cos \theta$ and $\cos(\frac{\pi}{2} - \theta) = \sin \theta$
  • Tangent and cotangent are cofunctions: $\tan(\frac{\pi}{2} - \theta) = \cot \theta$ and $\cot(\frac{\pi}{2} - \theta) = \tan \theta$
  • Secant and cosecant are cofunctions: $\sec(\frac{\pi}{2} - \theta) = \csc \theta$ and $\csc(\frac{\pi}{2} - \theta) = \sec \theta$
• Substitute cofunction identities into sum and difference formulas when angles are in the form $\frac{\pi}{2} - \theta$ to simplify expressions

Verification of trigonometric identities

• Simplify the left-hand side (LHS) and right-hand side (RHS) of the identity separately by applying sum and difference formulas and utilizing cofunction identities when necessary
• Compare the simplified LHS and RHS to verify their equality
• If the simplified LHS and RHS are identical, the identity is verified

Solving equations with identities

• Rewrite the trigonometric equation using sum and difference formulas and substitute cofunction identities if required
• Solve the resulting equation for the unknown variable using algebraic techniques (factoring, quadratic formula, or linear equation solving)
• Determine the solutions within the given domain, considering the period of the trigonometric functions involved ($2\pi$ for sine and cosine, $\pi$ for tangent and cotangent)

Simplification of complex expressions

• Identify opportunities to apply sum and difference formulas within the trigonometric expression, particularly when sums or differences of angles appear within trigonometric functions
• Rewrite the expression using the appropriate sum and difference formulas and utilize cofunction identities when necessary
• Simplify the resulting expression using algebraic techniques (combining like terms, factoring, or expanding)
• Evaluate the simplified expression for given angle values, if required

Relationship to periodic functions and the unit circle

• Sum and difference identities are derived from the properties of periodic functions and their behavior on the unit circle
• These identities help in understanding how trigonometric functions combine and interact, which is crucial for analyzing complex periodic phenomena
• The unit circle provides a geometric interpretation of these identities, allowing for visual representation of angle addition and subtraction

Applying Sum and Difference Identities

Solving equations with identities

• Example: Solve $\sin(2x) + \sin x = 1$ for $0 \leq x \leq 2\pi$
  1. Rewrite $\sin(2x)$ using the double angle formula: $\sin(2x) = 2\sin x \cos x$
  2. Substitute: $2\sin x \cos x + \sin x = 1$
  3. Factor out $\sin x$: $\sin x(2\cos x + 1) = 1$
  4. Solve $\sin x = 1$ and $2\cos x + 1 = 1$ separately
     • $\sin x = 1$ yields $x = \frac{\pi}{2}$
     • $2\cos x + 1 = 1$ yields $\cos x = 0$, so $x = \frac{\pi}{2}, \frac{3\pi}{2}$
  5. Solutions within the given domain: $x = \frac{\pi}{2}$

Simplification of complex expressions

• Example: Simplify $\tan(x + \frac{\pi}{4}) - \tan(x - \frac{\pi}{4})$
  1. Apply the sum and difference formulas for tangent:
     • $\tan(x + \frac{\pi}{4}) = \frac{\tan x + \tan \frac{\pi}{4}}{1 - \tan x \tan \frac{\pi}{4}}$
     • $\tan(x - \frac{\pi}{4}) = \frac{\tan x - \tan \frac{\pi}{4}}{1 + \tan x \tan \frac{\pi}{4}}$
  2. Substitute $\tan \frac{\pi}{4} = 1$:
     • $\tan(x + \frac{\pi}{4}) = \frac{\tan x + 1}{1 - \tan x}$
     • $\tan(x - \frac{\pi}{4}) = \frac{\tan x - 1}{1 + \tan x}$
  3. Subtract the expressions and simplify:
     • $\frac{\tan x + 1}{1 - \tan x} - \frac{\tan x - 1}{1 + \tan x}$
     • $= \frac{(\tan x + 1)(1 + \tan x) - (\tan x - 1)(1 - \tan x)}{(1 - \tan x)(1 + \tan x)}$
     • $= \frac{2}{1 - \tan^2 x}$
     • $= 2\sec^2 x$