Pre-Calculus - Week 11

Trigonometric Identities and Equations

A trigonometric identity is an equation that is true for every value in its domain. Learning to prove and use identities builds the algebraic fluency that calculus demands — you will use them to integrate trig functions, simplify derivatives, and analyze Fourier series. This week you master the fundamental identities, sum and difference formulas, and double-angle formulas, then solve trig equations.

Learning Objectives

- Recognize and apply the Pythagorean, reciprocal, and quotient identities. - Simplify trigonometric expressions using identities. - Prove trigonometric identities by transforming one side into the other. - Apply the sum and difference formulas for sine, cosine, and tangent. - Apply the double-angle formulas for sine and cosine. - Solve basic trigonometric equations on [0, 2π). - Find all solutions to trigonometric equations over all reals.

1. Fundamental Identities

Reciprocal identities: csc(x) = 1/sin(x), sec(x) = 1/cos(x), cot(x) = 1/tan(x). Quotient identities: tan(x) = sin(x)/cos(x), cot(x) = cos(x)/sin(x). Pythagorean identities: sin^2(x) + cos^2(x) = 1. 1 + tan^2(x) = sec^2(x). 1 + cot^2(x) = csc^2(x). Even/Odd: cos(-x) = cos(x), sin(-x) = -sin(x), tan(-x) = -tan(x). Memorise all of these. They are used in every calculus derivation involving trig.

2. Simplifying Trig Expressions

Example 1: Simplify sin(x)*cot(x). sin(x) * cos(x)/sin(x) = cos(x). Example 2: Simplify (1 - cos^2(x))/sin(x). = sin^2(x)/sin(x) = sin(x). Example 3: Simplify sec^2(x) - tan^2(x). = 1 (Pythagorean identity: 1 + tan^2 = sec^2, so sec^2 - tan^2 = 1). Example 4: Simplify cos(x) + sin(x)*tan(x). = cos(x) + sin(x)*(sin(x)/cos(x)) = cos(x) + sin^2(x)/cos(x) = (cos^2(x) + sin^2(x))/cos(x) = 1/cos(x) = sec(x).

3. Proving Identities

Rules for proving identities: 1. Work on ONE SIDE only (usually the more complex side). 2. Convert everything to sines and cosines if stuck. 3. Try multiplying by a conjugate or factoring. 4. Never move terms across the equals sign. Example: Prove tan(x) + cot(x) = sec(x)*csc(x). LHS = sin/cos + cos/sin = (sin^2 + cos^2)/(sin*cos) = 1/(sin*cos) = (1/cos)*(1/sin) = sec(x)*csc(x). ✓

4. Sum and Difference Formulas for Cosine

cos(A + B) = cos(A)cos(B) - sin(A)sin(B) cos(A - B) = cos(A)cos(B) + sin(A)sin(B) Note: the cosine addition formula has a MINUS sign. Example 1: Find cos(75°) = cos(45° + 30°). = cos(45°)cos(30°) - sin(45°)sin(30°) = (sqrt(2)/2)(sqrt(3)/2) - (sqrt(2)/2)(1/2) = sqrt(6)/4 - sqrt(2)/4 = (sqrt(6) - sqrt(2))/4. Example 2: Find cos(15°) = cos(45° - 30°). = cos(45°)cos(30°) + sin(45°)sin(30°) = sqrt(6)/4 + sqrt(2)/4 = (sqrt(6) + sqrt(2))/4.

5. Sum and Difference Formulas for Sine and Tangent

sin(A + B) = sin(A)cos(B) + cos(A)sin(B) sin(A - B) = sin(A)cos(B) - cos(A)sin(B) tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B)) tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B)) Example: Find sin(105°) = sin(60° + 45°). = sin(60°)cos(45°) + cos(60°)sin(45°) = (sqrt(3)/2)(sqrt(2)/2) + (1/2)(sqrt(2)/2) = sqrt(6)/4 + sqrt(2)/4 = (sqrt(6)+sqrt(2))/4.

6. Double-Angle Formulas

Derived from sum formulas with A = B: sin(2A) = 2*sin(A)*cos(A). cos(2A) = cos^2(A) - sin^2(A) = 2*cos^2(A) - 1 = 1 - 2*sin^2(A). tan(2A) = 2*tan(A) / (1 - tan^2(A)). Example: If sin(A) = 3/5 and A is in QI, find sin(2A) and cos(2A). cos(A) = 4/5. sin(2A) = 2(3/5)(4/5) = 24/25. cos(2A) = 1 - 2*(9/25) = 1 - 18/25 = 7/25.

7. Half-Angle Formulas

From the double-angle formulas: sin(A/2) = ±sqrt((1 - cos(A))/2) cos(A/2) = ±sqrt((1 + cos(A))/2) tan(A/2) = sin(A)/(1 + cos(A)) = (1 - cos(A))/sin(A) The sign depends on the quadrant of A/2. Example: Find sin(π/8) = sin(π/4 / 2). cos(π/4) = sqrt(2)/2. sin(π/8) = sqrt((1 - sqrt(2)/2)/2) = sqrt((2 - sqrt(2))/4) = sqrt(2-sqrt(2))/2.

8. Solving Trig Equations on [0, 2π)

Example 1: Solve 2*sin(x) - 1 = 0 on [0, 2π). sin(x) = 1/2. Reference angle: π/6. sin is positive in QI and QII. x = π/6 and x = π - π/6 = 5π/6. Example 2: Solve cos^2(x) = 3/4 on [0, 2π). cos(x) = ±sqrt(3)/2. For cos(x)=sqrt(3)/2: x=π/6, 11π/6. For cos(x)=-sqrt(3)/2: x=5π/6, 7π/6. All four solutions.

9. Solving Trig Equations: All Solutions

To give all real solutions, add the period: For sin(x) = c: general solutions are x = arcsin(c) + 2πk and x = π - arcsin(c) + 2πk, for all integers k. For cos(x) = c: x = arccos(c) + 2πk and x = -arccos(c) + 2πk. For tan(x) = c: x = arctan(c) + πk (period is π, only one family). Example: Solve tan(x) = 1 for all x. Reference: π/4. tan positive in QI and QIII. General solution: x = π/4 + nπ for all integers n.

10. Equations Requiring Identities

Sometimes you must apply an identity before solving. Example 1: Solve 2*sin^2(x) - 1 = 0 on [0, 2π). sin^2(x) = 1/2. sin(x) = ±sqrt(2)/2. x = π/4, 3π/4, 5π/4, 7π/4. Example 2: Solve sin(2x) = cos(x) on [0, 2π). Use double-angle: 2*sin(x)*cos(x) = cos(x). 2*sin(x)*cos(x) - cos(x) = 0. cos(x)(2*sin(x) - 1) = 0. cos(x) = 0: x = π/2, 3π/2. sin(x) = 1/2: x = π/6, 5π/6. All solutions: π/6, π/2, 5π/6, 3π/2.

11. Common Mistakes

- Dividing both sides of an equation by sin(x) or cos(x) — you LOSE solutions where that function equals 0. Factor instead. - Using sin(A+B) = sin(A) + sin(B) (WRONG). Always use the correct formula. - Forgetting to find solutions in ALL relevant quadrants. - Mixing up the double-angle formula for cosine (three versions — use the one that's most helpful). - Forgetting that proving an identity requires working one side only, not moving terms across.