AP Precalculus: Trigonometric Identities

Master the fundamental identities that simplify and transform trig expressions

📐 Pythagorean 🔄 Reciprocal ➕ Sum/Difference 🔁 Cofunction

📚 Understanding Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variable where both sides are defined. They're essential tools for simplifying expressions, solving equations, and proving other identities. Memorize these foundational identities!

1 Pythagorean Identities

Derived from the Pythagorean theorem applied to the unit circle. These are the most frequently used identities.

Fundamental Identity

\(\sin^2 x + \cos^2 x = 1\)

Tangent Form

\(1 + \tan^2 x = \sec^2 x\)

Cotangent Form

\(1 + \cot^2 x = \csc^2 x\)

Derived Forms (Solve for Each Function)

\(\sin^2 x = 1 - \cos^2 x\)

\(\cos^2 x = 1 - \sin^2 x\)

\(\tan^2 x = \sec^2 x - 1\)

\(\cot^2 x = \csc^2 x - 1\)

💡 Memory Trick

The three Pythagorean identities all follow the pattern: (function)² + 1 = (co-reciprocal)² or 1 + (function)² = (reciprocal)²

2 Reciprocal Identities

Reciprocal identities express each trig function as the reciprocal of another function.

\(\sin x = \frac{1}{\csc x}\)

\(\cos x = \frac{1}{\sec x}\)

\(\tan x = \frac{1}{\cot x}\)

\(\csc x = \frac{1}{\sin x}\)

\(\sec x = \frac{1}{\cos x}\)

\(\cot x = \frac{1}{\tan x}\)

3 Quotient Identities

Quotient identities express tangent and cotangent in terms of sine and cosine.

Tangent

\(\tan x = \frac{\sin x}{\cos x}\)

Cotangent

\(\cot x = \frac{\cos x}{\sin x}\)

📌 Using Quotient Identities

Simplify: \(\frac{\sin x}{\cos x} \cdot \frac{1}{\sin x}\)

Solution: \(\frac{\sin x}{\cos x} \cdot \frac{1}{\sin x} = \frac{1}{\cos x} = \sec x\)

4 Cofunction (Complementary) Identities

Cofunction identities relate trig functions of complementary angles (angles that add to 90° or \(\frac{\pi}{2}\)).

\(\sin(90° - x) = \cos x\)

\(\cos(90° - x) = \sin x\)

\(\tan(90° - x) = \cot x\)

\(\cot(90° - x) = \tan x\)

\(\sec(90° - x) = \csc x\)

\(\csc(90° - x) = \sec x\)

💡 The "Co" Connection

Notice the pairing: sin ↔ cosine, tan ↔ cotangent, sec ↔ cosecant. The "co" in the name means "complement"!

5 Even & Odd Function Identities

Even functions satisfy \(f(-x) = f(x)\). Odd functions satisfy \(f(-x) = -f(x)\).

✅ Even Functions

\(\cos(-x) = \cos x\)
\(\sec(-x) = \sec x\)

🔄 Odd Functions

\(\sin(-x) = -\sin x\)
\(\tan(-x) = -\tan x\)
\(\cot(-x) = -\cot x\)
\(\csc(-x) = -\csc x\)

📌 Application

Simplify: \(\sin(-30°)\)

Solution: Since sine is odd: \(\sin(-30°) = -\sin(30°) = -\frac{1}{2}\)

6 Periodicity Identities

Trig functions repeat their values after a specific interval called the period.

Period = \(2\pi\)

\(\sin(x + 2\pi) = \sin x\)
\(\cos(x + 2\pi) = \cos x\)
\(\sec(x + 2\pi) = \sec x\)
\(\csc(x + 2\pi) = \csc x\)

Period = \(\pi\)

\(\tan(x + \pi) = \tan x\)
\(\cot(x + \pi) = \cot x\)

7 Sum & Difference Identities

These identities find the trig values of sums or differences of angles. Essential for computing exact values of non-standard angles.

Sine Sum/Difference

\(\sin(A + B) = \sin A \cos B + \cos A \sin B\)

\(\sin(A - B) = \sin A \cos B - \cos A \sin B\)

Pattern: Sin-Cos-Cos-Sin (same sign)

Cosine Sum/Difference

\(\cos(A + B) = \cos A \cos B - \sin A \sin B\)

\(\cos(A - B) = \cos A \cos B + \sin A \sin B\)

Pattern: Cos-Cos-Sin-Sin (opposite sign)

Tangent Sum/Difference \(\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}\)

📌 Example: Finding Exact Value

Find: \(\sin(75°)\)

Rewrite: \(75° = 45° + 30°\)

Apply sum formula:

\(\sin(75°) = \sin(45°)\cos(30°) + \cos(45°)\sin(30°)\)

\(= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}\)

\(= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}\)

⚠️ Sign Pattern

For cosine, the sign in the formula is OPPOSITE the sign in the angle. For sine and tangent, the signs match.

8 Strategies for Using Identities

Simplifying expressions and proving identities requires recognizing which identity to apply.

Common Strategies

Convert to sine and cosine: Replace tan, cot, sec, csc with their definitions
Look for Pythagorean patterns: \(\sin^2 + \cos^2\), \(1 + \tan^2\), \(1 + \cot^2\)
Factor expressions: Look for difference of squares, common factors
Combine fractions: Get common denominators
Work both sides: Sometimes simplify both sides to meet in the middle

📌 Example: Simplify

Simplify: \(\sec^2 x - 1\)

Recognize: This is the Pythagorean identity \(1 + \tan^2 x = \sec^2 x\)

Rearrange: \(\sec^2 x - 1 = \tan^2 x\)

📋 Quick Reference

Pythagorean

\(\sin^2 + \cos^2 = 1\)

Tangent Form

\(1 + \tan^2 = \sec^2\)

Quotient

\(\tan = \frac{\sin}{\cos}\)

Reciprocal

\(\csc = \frac{1}{\sin}\), etc.

Even/Odd

cos, sec: even; others: odd

Cofunction

\(\sin(90°-x) = \cos x\)

📚 Understanding Trigonometric Identities

1 Pythagorean Identities

Derived Forms (Solve for Each Function)

2 Reciprocal Identities

3 Quotient Identities

4 Cofunction (Complementary) Identities

5 Even & Odd Function Identities

6 Periodicity Identities

Period = \(2\pi\)

Period = \(\pi\)

7 Sum & Difference Identities

8 Strategies for Using Identities

Common Strategies

📋 Quick Reference

Pythagorean

Tangent Form

Quotient

Reciprocal

Even/Odd

Cofunction

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