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.