Comprehensive Trigonometry Review – Inverses, Identities & Complementary Angles

Quiz Logistics & Expectations

  • Quizzes are versioned; always state the question number and version when asking for help.
  • Show every algebraic step; calculator-only answers earn no credit.
  • For trigonometric simplifications, write down the identity you are using (e.g. 1+\tan^2\theta=\sec^2\theta) before substituting.
  • On the upcoming test you may skip exactly one problem. Instructor recommends skipping the longest problem to maximize time.
  • Participation matters: active students receive small grade boosts (e.g. 68→70). Silent attendance does not.

Inverse Trig Functions & Domain-Range Swapping

General Workflow

  1. Swap variables x \leftrightarrow y.
  2. Isolate y algebraically (add/subtract constants, divide coefficients, apply inverse trig).
  3. Identify the inner expression of the inverse trig; its natural domain controls the new domain.
  4. Solve the resulting compound inequality for x.

Example 1 (y=4-\cos(3x-12))

  1. Swap: x = 4-\cos(3y-12).
  2. Isolate:
    • x-4 = -\cos(3y-12)
    • -(x-4) = \cos(3y-12)
    • 3y-12 = \cos^{-1}!\bigl(-(x-4)\bigr)
    • 3y = \cos^{-1}!\bigl(-(x-4)\bigr)+12
    • y = -\tfrac13\cos^{-1}(x-4)+4
  3. Since \cos^{-1} requires -1\le (x-4)\le1:
    • -1\le x-4 \le 1\;\;\Rightarrow\;\;3\le x\le5 ← domain of the inverse.

Example 2 (y=7+5\sin(2x+9))

  1. Swap: x = 7+5\sin(2y+9)
  2. Isolate:
    \sin(2y+9)=\dfrac{x-7}{5}
    2y+9 = \sin^{-1}!\left(\dfrac{x-7}{5}\right)
    y = -\tfrac92+\tfrac12\sin^{-1}!\left(\dfrac{x-7}{5}\right)
  3. \sin^{-1} demands -1\le\dfrac{x-7}{5}\le1 ⇨ 2\le x\le12.

Key Reminders

  • Natural domains:
    • \sin^{-1},\;\cos^{-1}:\;[-1,1]
    • \tan^{-1}:\;(-\infty,\infty) (unchanged when taking inverses).
  • Do not shortcut by guessing range→domain; coefficients in front of trig terms can invalidate the guess.

Fundamental Trig Identities (used repeatedly)

  • Pythagorean set
    \sin^2\theta+\cos^2\theta=1
    1+\tan^2\theta=\sec^2\theta
    1+\cot^2\theta=\csc^2\theta
  • Cofunction (complementary-angle) set
    \sin(90^\circ-\theta)=\cos\theta\;; \cos(90^\circ-\theta)=\sin\theta
    \tan(90^\circ-\theta)=\cot\theta\;; \cot(90^\circ-\theta)=\tan\theta
    \sec(90^\circ-\theta)=\csc\theta\;; \csc(90^\circ-\theta)=\sec\theta
  • Angle-sum/difference (needed for calculator-free exact values)
    \tan(A-B)=\dfrac{\tan A-\tan B}{1+\tan A\tan B}
    \cos(A-B)=\cos A\cos B+\sin A\sin B (sign flips for +)

Complementary-Angle Theorem in Practice

Given two acute angles \alpha,\beta in a right triangle: \alpha+\beta=90^\circ.
Therefore any cofunction pair shares a value:

  • \sin 37^\circ = \cos 53^\circ,
  • \tan^2 23^\circ = \cot^2 67^\circ, etc.

Fast Evaluations

  • \sin^2 18^\circ+\cos^2 72^\circ = 1 (same angle after complement)
  • \csc^2 10^\circ-\cot^2 80^\circ = 1
  • Mixed powers example
    \sin^4 70^\circ-\cot^2 3^\circ+\sec^2 77^\circ
    • First two terms form 1
    • Remaining pair by identity ⇒ 1
    • Combined sign ⇒ final 1.

Radian Variant

  • Complement of 3\pi/8 is \pi/2-3\pi/8=\pi/8.
  • Example simplification
    \sin^2!\frac{3\pi}{8}+\tan^2!\frac{3\pi}{8}+\cos^2!\frac{\pi}{8}-\sec^2!\frac{3\pi}{8}
    → First and third form 1; last pair gives -1; result 0.

Exact-Value Problems Using Reference Triangles

Setup Guidelines

  1. Interpret interval to place the angle in the correct quadrant.
  2. Draw right triangle; assign signs (+/–) to legs.
  3. Use given trig ratio (e.g. \tan\alpha=-36/77) to label opposite/adjacent.
  4. Apply Pythagorean theorem to find hypotenuse.
  5. Extract all six trig values for that angle.

Example

Given \tan\alpha=-\tfrac{36}{77} with \alpha\in(\pi,3\pi/2) (Q III) and \cos\beta=\tfrac{\sqrt6}{3} with \beta\in(0,\tfrac\pi2) (Q I), find \cos(\alpha-\beta).

  • Triangle for \alpha: opposite 36, adjacent -77 ⇒ hyp 85.
    \cos\alpha=-77/85,\;\sin\alpha=36/85
  • Triangle for \beta: adjacent \sqrt6, hyp 3 ⇒ opposite \sqrt3.
    \cos\beta=\sqrt6/3,\;\sin\beta=\sqrt3/3
  • Apply formula:
    \cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta
    =(-\frac{77}{85})(\frac{\sqrt6}{3})+(\frac{36}{85})(\frac{\sqrt3}{3})
    =\frac{-77\sqrt6+36\sqrt3}{255}.

Identity Simplification Walk-Through (Instructor’s Board Example)

Expression: \sec^2\theta-\tan^2\theta\;\;\bigl(=1\bigr)

  1. Recognize direct Pythagorean identity → equals 1 immediately.
  2. Alternative: rewrite sec & tan in \sin,\cos, find common denominator, cancel.

Takeaway: multiple valid paths; credit awarded if algebra is sound.

Instructor Tips & Warnings

  • Never submit a raw calculator decimal (e.g. entering \tan(342^\circ) and writing -0.532\dots). Write angle-sum identity first, then exact radical.
  • Typical red flag: student writes 173 where only \sqrt3/2 is correct.
  • Time management: familiar topics → fast, unfamiliar → stall. Master the entire review sheet.
  • Allowed resources in online setting: notes, book, videos, friends; nevertheless, without practice time will expire.

Miscellaneous Reminders

  • Degree ↔ Radian: 90^\circ = \pi/2.
  • Always keep 2k\pi terms when solving trig equations; you can clear denominators later so they share the same base.
  • When multiplying an entire equation by a common denominator, distribute to every term including 2k\pi.

Ethical / Philosophical Nuggets

  • “Show the process or receive no credit” – mathematics values transparency.
  • Quizzes emulate real-life problem solving where intermediate reasoning is as important as answers.
  • Instructor’s closing note: personal encouragement and faith reference (invitation to “let Jesus live in your heart”).