Trigonometric Identities and Equations

Outcome 4: Trigonometric Identities and Equations · Blueprint Pillar 3 · PGCC MAT-1350 (interim) · Download .docx

Objectives

Apply the Pythagorean identities to simplify trigonometric expressions.
Use sum, difference, and double angle formulas to find exact values.
Verify a trigonometric identity by transforming one side only.
Solve basic trigonometric equations on the interval [0, 2π).
Write the general solution to a trigonometric equation using period.

Key terms

Pythagorean identity: sin²θ + cos²θ = 1; equivalently 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ.
sum formula (sin): sin(A + B) = sinA cosB + cosA sinB.
difference formula (sin): sin(A − B) = sinA cosB − cosA sinB.
sum formula (cos): cos(A + B) = cosA cosB − sinA sinB.
difference formula (cos): cos(A − B) = cosA cosB + sinA sinB.
double angle (sin): sin(2θ) = 2 sinθ cosθ.
double angle (cos): cos(2θ) = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1.
verifying: Transforming one side of a potential identity using algebra and known identities until it matches the other side.
general solution: All solutions to a trig equation: for sin(θ) = k with reference angle α, θ = α + 2nπ or θ = π − α + 2nπ (n integer).

The concept

Trigonometric identities are equations that are true for every angle in their domain. They are not equations to solve — they are tools for transforming expressions. The three families of identities used most are: Pythagorean identities (connecting sin and cos), sum/difference formulas (for sin and cos of combined angles), and double angle formulas (special cases of sum formulas with A = B = θ).

The Pythagorean identity sin²θ + cos²θ = 1 is the foundation. Rearrange it: sin²θ = 1 − cos²θ or cos²θ = 1 − sin²θ. These substitutions allow you to write any expression purely in terms of one trig function. Divide the identity by cos²θ to get 1 + tan²θ = sec²θ; divide by sin²θ to get cot²θ + 1 = csc²θ.

Sum and difference formulas let you find exact values for angles not on the standard circle. sin(75°) = sin(45° + 30°) = sin45°cos30° + cos45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4. No calculator needed.

Verifying an identity requires working on one side only and transforming it into the other. The most common strategies: convert everything to sin and cos, factor, combine fractions, multiply by a conjugate, or apply a Pythagorean identity. Never move terms across the equals sign — that assumes what you are trying to prove.

Solving trig equations uses the same process as solving algebraic equations, with the extra step of converting from a trig value to an angle. sin(θ) = 1/2 has solutions θ = π/6 and θ = 5π/6 on [0, 2π) — two solutions because sine is positive in both Q1 and Q2. For more complex equations, factor if possible (2sin²θ − sinθ = 0 factors to sinθ(2sinθ − 1) = 0).

Blueprint Pillar 3 — Technology and Society: Trigonometric identities are the foundation of Fourier analysis — the mathematical tool that decomposes any signal into sine and cosine waves. Every MP3, JPEG, and digital phone call uses Fourier transforms. Signal processing, image compression, wireless communication, and medical imaging (MRI, CT scans) all rely on this mathematics. Identities are not abstract exercises — they are the algebra of waves.

Worked examples

Example 1: Verifying: sin²θ · sec²θ + 1 = sec²θ. Work left side only. sin²θ · (1/cos²θ) + 1 = sin²θ/cos²θ + 1 = tan²θ + 1. By Pythagorean identity: tan²θ + 1 = sec²θ. Left side = sec²θ = right side. ✓

Example 2: Solving 2cos²θ − 1 = 0 on [0, 2π): cos²θ = 1/2 → cosθ = ±1/√2 = ±√2/2. cos positive → Q1: π/4; Q4: 7π/4. cos negative → Q2: 3π/4; Q3: 5π/4. Four solutions: π/4, 3π/4, 5π/4, 7π/4.

Common mistakes

Working both sides simultaneously when verifying an identity. This assumes the conclusion before proving it. Always transform one side only.
Losing solutions when dividing by a trig function. If 2sinθcosθ = sinθ, do not divide both sides by sinθ — factor instead: sinθ(2cosθ − 1) = 0 → sinθ = 0 or cosθ = 1/2. Dividing drops the sinθ = 0 solutions.
Confusing cos(2θ) = cos²θ − sin²θ with cos²θ + sin²θ = 1. The double angle formula has a minus sign; the Pythagorean identity has a plus sign.

Self-check

Try each one before you look. A miss here costs nothing and tells you exactly what to reread.

1. Using sin²θ + cos²θ = 1, express sin²θ in terms of cosθ.

2. Evaluate sin(90° − θ) using the difference formula.

3. Solve: sinθ = √3/2 on [0, 2π). How many solutions are there?

4. Which strategy is ALWAYS correct when verifying a trig identity?

5. What is sin(2θ) according to the double angle formula?

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