Trig Identities & Equation-Solving – Comprehensive Lecture Notes

Administrative Announcements

  • Exam 3 Date & Location

    • Exactly one week from the lecture day (Tuesday): takes place in room SE-141.

    • Students who must test at Leonardtown or Prince Frederick: email the instructor with your chosen Thursday time.

  • Coverage: Chapters 4 & 5.

  • Practice: Extra problems posted in MyMathLab.

  • Cheat-Sheet Rules

    • One sheet of paper, front & back, any content the student chooses.

    • Recommended: list of identities already studied (reciprocal, Pythagorean, cofunction, negative-angle, sum/difference, double-angle, half-angle).

    • Do NOT waste space on Product-to-Sum or Sum-to-Product; no direct exam questions on them.

  • Content taught after this lecture (Thursday’s material) appears only on the final, not on Exam 3.

Identities Already Mastered (Quick Recap)

  • Reciprocal identities: \sin\theta=\dfrac{1}{\csc\theta},\;\cos\theta=\dfrac{1}{\sec\theta},\;\tan\theta=\dfrac{1}{\cot\theta} and the inverses.

  • Pythagorean set: \sin^2\theta+\cos^2\theta=1, \;1+\tan^2\theta=\sec^2\theta,\;1+\cot^2\theta=\csc^2\theta.

  • Cofunction: \sin(\tfrac{\pi}{2}-\theta)=\cos\theta and cyclic variants.

  • Negative-angle: \sin(-\theta)=-\sin\theta,\;\cos(-\theta)=\cos\theta,\;\tan(-\theta)=-\tan\theta.

  • Sum & Difference, Double-Angle, Half-Angle formulas (all previously derived and recommended for the cheat sheet).

Product-to-Sum (P→S) Identities

Purpose: Convert a product of two trig functions (different angles) into a sum/difference, simplifying integration or algebraic manipulation (heavily used in Calculus II for integrals).

  1. \cos A\,\cos B=\tfrac12\big[\cos(A-B)+\cos(A+B)\big]

  2. \sin A\,\sin B=\tfrac12\big[\cos(A-B)-\cos(A+B)\big]

  3. \cos A\,\sin B=\tfrac12\big[\sin(A+B)-\sin(A-B)\big]

Key remarks

  • Rarely memorised verbatim; instead, re-derive using sum–difference formulas if needed.

  • Verify numerically when unsure (calculator check).

Example

  • \cos56^\circ\cos18^\circ=\tfrac12\big[\cos(38^\circ)+\cos(74^\circ)\big]\approx0.53182.

  • Symbolic example: \sin(5\theta)\sin(2\theta)=\tfrac12\big[\cos(3\theta)-\cos(7\theta)\big].

    • Helpful because a sum of cosines is easier to integrate than a product of sines.

Sum-to-Product (S→P) Identities

Purpose: Condense a sum/difference into a product — occasionally useful for factoring, cancellation, or integration.

  1. \cos A+\cos B=2\cos\Bigl(\dfrac{A+B}{2}\Bigr)\cos\Bigl(\dfrac{A-B}{2}\Bigr)

  2. \cos A-\cos B=-2\sin\Bigl(\dfrac{A+B}{2}\Bigr)\sin\Bigl(\dfrac{A-B}{2}\Bigr)

  3. \sin A+\sin B=2\sin\Bigl(\dfrac{A+B}{2}\Bigr)\cos\Bigl(\dfrac{A-B}{2}\Bigr)

  4. \sin A-\sin B=2\sin\Bigl(\dfrac{A-B}{2}\Bigr)\cos\Bigl(\dfrac{A+B}{2}\Bigr)

Instructor’s advice

  • Lowest practical value for day-to-day problems; keep as reference, not memorisation priority.

Example

  • \sin52^\circ-\sin6^\circ=2\sin23^\circ\cos29^\circ.

  • \cos78^\circ+\cos25^\circ=2\cos51.5^\circ\cos26.5^\circ.

Proof sketch (for \dfrac{\sin7\theta-\sin3\theta}{\cos7\theta-\cos3\theta}=\tan2\theta):

  1. Apply S→P separately to numerator & denominator.

  2. Factors cancel; remaining \frac{\sin2\theta}{\cos2\theta}=\tan2\theta.

Memorisation Strategy

  1. Must-know: Pythagorean, Reciprocal, Sum/Difference, Double-Angle, Half-Angle.

  2. Nice-to-know reference: Product-to-Sum and Sum-to-Product.

  3. Use a logically ordered cheat-sheet; avoid clogging it with low-yield identities.

Solving Trigonometric Equations

General principles

  • Identity vs. Equation: identities hold for all inputs; equations are true only for specific solutions.

  • Transform until you achieve
    \text{trig-function(expression)} = \text{constant}.

  • Respect algebra: never divide by a variable/trig expression that may be zero (avoids lost solutions).

  • Specify solution set on a given interval; instructor notation:

    • [0,2\pi) → give answers in radians.

    • [0,360^\circ) → give answers in degrees.

Common Tactics

  1. Isolate the trig function.

  2. Factor when multiple trig factors appear; set each equal to zero (Zero-Product Property).

  3. Pythagorean substitution: eliminate mixed trig types (e.g., replace \sin^2x with 1-\cos^2x).

  4. Quadratic-in-trig: substitute u=\sin x or u=\cos x, then solve via factoring or quadratic formula, remembering -1\le u\le1.

  5. Multiple-angle k\,x:

    • Solve for k\,x first, then divide.

    • Go around the unit circle k times to capture all distinct solutions in the stated interval.

  6. Squaring both sides (or any power) can introduce extraneous solutions ⇒ always check.

Illustrative Examples

  1. 3\tan x = \sqrt3\quad(0< x<2\pi)

    • \tan x = \sqrt3/3 ⇒ x = \pi/6, 7\pi/6.

  2. \sin x + \sqrt2 = -\sin x

    • Collect terms: 2\sin x = -\sqrt2 ⇒ \sin x=-\dfrac{\sqrt2}{2} ⇒ x=5\pi/4, 7\pi/4.

  3. 2\sin x\cos x - \sin x =0

    • Factor: \sin x(2\cos x -1)=0.

    • Solutions: \sin x=0 \Rightarrow x=0,\pi; \; 2\cos x=1 \Rightarrow \cos x=\tfrac12 \Rightarrow x=\frac{\pi}{3},\frac{5\pi}{3}.

  4. Quadratic pattern
    2\tan^2 x -1 =0

    • \tan^2x=\tfrac12\Rightarrow \tan x=\pm\dfrac{1}{\sqrt2}=\pm\dfrac{\sqrt2}{2}.

    • Gives four solutions in [0,2\pi).

  5. Mixed angles requiring identity
    \cos2x = \cos x with choice \cos2x=2\cos^2x-1.

    • Rearrange: 2\cos^2x-1-\cos x=0.

    • Factor \bigl(2\cos x+1\bigr)(\cos x-1)=0.

    • \cos x=-\tfrac12 \Rightarrow x=120^\circ,240^\circ; \cos x=1 \Rightarrow x=0^\circ.

  6. Division danger

    • Attempting \dfrac{\sin x\tan x}{\sin x}=\tan x could drop solutions where \sin x=0 (division by zero). Always factor instead of cancel.

  7. Multiple-angle
    \tan 3x = -1 \quad (0< x<360^\circ)

    • \tan 3x = -1 at 3x=135^\circ,315^\circ,495^\circ,675^\circ,855^\circ,1035^\circ.

    • Divide by 3 ⇒ x=45^\circ,105^\circ,165^\circ,225^\circ,285^\circ,345^\circ.

  8. Half-angle
    \sin\tfrac{x}{2}=\dfrac{\sqrt2}{2},\;0< x<360^\circ

    • Half-angle solutions: \tfrac{x}{2}=45^\circ,135^\circ → x=90^\circ,270^\circ.

  9. Squaring both sides example
    \cos x+1 = \sin x (degrees)

    1. Square → \cos^2x+2\cos x+1 = \sin^2x.

    2. Substitute \sin^2x=1-\cos^2x → 2\cos^2x+2\cos x=0.

    3. Factor & solve → x=90^\circ,270^\circ,0^\circ.

    4. Check original equation: only 90^\circ and 0^\circ valid; 270^\circ extraneous.

  10. Quadratic formula with non-factorable coefficients
    \sin^2x-3\sin x-2=0 ⇒ u=\sin x → u=\dfrac{3\pm\sqrt{17}}{2}.

    • u=\dfrac{3+\sqrt{17}}{2}>1 (reject).

    • u=\dfrac{3-\sqrt{17}}{2}\approx-0.5590.

    • Primary inverse-sine gives x\approx-34.16^\circ; adjust to required interval ⇒ x\approx325.84^\circ,\;214.16^\circ.

Calculator & Domain Safety

  • Always set DEG vs RAD correctly.

  • Remember \sin^{-1},\cos^{-1},\tan^{-1} have restricted ranges; supplement missing quadrants manually.

  • If RHS constant falls outside [-1,1] for sine or cosine, or any real number for tangent, declare no solution for that branch.

Ethical / Practical Implications

  • Rigorous algebra prevents overlooked or phantom solutions; dividing by zero or erroneous cancellation yields incomplete or incorrect answer sets.

  • Cross-checking after operations that can enlarge the solution set (e.g.
    squaring) is mandatory.

  • Systematic methods improve success in Calculus II integrals (Product-to-Sum) and in engineering applications involving signal decomposition.

Quick Reference: Recommended Cheat-Sheet Essentials

  • Pythagorean identities.

  • Reciprocal identities.

  • Sum/Difference formulas.

  • Double-Angle & Half-Angle formulas.

  • Basic cofunction & negative-angle relations.

  • Special angle values (unit-circle chart) and quadrant signs (ASTC).

Avoid: lengthy Product-to-Sum or Sum-to-Product tables unless space remains.