Trigonometry – Inverse Trig Functions (Section 3.1)

Administrative Announcements

• Test #1
  • Opens tonight at 10 PM
  • 15 questions, 90-minute limit
  • Some items are multiple-choice, but work must be shown for every exercise; no credit for answers without process
• Section being covered today: 3.1 – Inverse of Sine, Cosine & Tangent (Textbook pp. 189–193)
• Next meetings
  • Finish §3.1 on Thursday
  • Begin §3.2 afterward (chapter 3 is harder than chapter 2 → practice!)
• Practice worksheet for today’s material will be posted in Blackboard (B 12) later tonight or tomorrow

Review – One-to-One Functions & Inverses

• To possess an inverse, a relation must be one-to-one (horizontal-line test)
• Examples
  • $f(x)=\sqrt{x}$
  • Passes horizontal-line test → inverse exists on its natural domain
  • $f(x)=x^2$
  • Fails; each $x$ has two $y$’s unless the domain is restricted (e.g.
    $x\ge 0$)
• For trig functions, the natural graphs are periodic and not one-to-one, so we impose domain restrictions (windows) to create invertible pieces
• After restriction, the inverse graph is obtained by reflecting across $y=x$; domain & range swap

Standard Trig Graphs, Restrictions, Domains & Ranges

Sine (normal)

• Periodic curve passing through $(0,0)$, peak $1$, trough $-1$
• Domain: $(-\infty,\infty)$
• Range: $[-1,1]$
• Not one-to-one → restrict to $[-\pi/2,\;\pi/2]$ (central hump)

Inverse Sine $y=\sin^{-1}x$ (a.k.a. arcsin)

• Domain (swapped): $[-1,1]$
• Range: $[-\pi/2,\;\pi/2]$ (quadrants IV & I)

Cosine (normal)

• Starts at $1$ when $x=0$, crosses $0$ at $\pi/2$, minimum $-1$ at $\pi$
• Domain: $(-\infty,\infty)$
• Range: $[-1,1]$
• Restrict to $[0,\pi]$ to make one-to-one

Inverse Cosine $y=\cos^{-1}x$ (arccos)

• Domain: $[-1,1]$
• Range: $[0,\pi]$ (quadrants I & II)

Tangent (normal)

• Period $\pi$, vertical asymptotes where $\cos x = 0$ → $x=\pm\pi/2,\, \pm3\pi/2,\ldots$
• Domain: all reals except odd multiples of $\pi/2$
• Range: $(-\infty,\infty)$
• Restrict to $(-\pi/2,\;\pi/2)$ for invertibility

Inverse Tangent $y=\tan^{-1}x$ (arctan)

• Domain: $(-\infty,\infty)$
• Range: $(-\pi/2,\;\pi/2)$

Quadrant Map for Inverse Values (Based on Ranges)

• Quadrant I (0 < \theta < \pi/2): All inverses take positive values here
  • $\sin^{-1},\;\cos^{-1},\;\tan^{-1}$ positive
  • Related reciprocals adopt same sign rule (\csc, \sec, \cot via their base functions)
• Quadrant II (\pi/2 < \theta < \pi): used only by arccos (and thus $\sec^{-1}$ when negative)
• Quadrant III: Never appears in principal value sets of inverses
• Quadrant IV (-\pi/2<\theta<0 written clockwise): hosts negative values of $\sin^{-1}$, $\tan^{-1}$, $\csc^{-1}$

Recap table (principal value location):

• $\sin^{-1}x,\;\csc^{-1}x$: Q I if x>0, Q IV if x<0
• $\cos^{-1}x,\;\sec^{-1}x$: Q I if x>0, Q II if x<0
• $\tan^{-1}x,\;\cot^{-1}x$ (cot defined here so that positive values go to Q I, negatives Q II)

Worked Evaluation Examples

Direct inverse evaluations

1. $\sin^{-1}(1)=\frac{\pi}{2}$ (lies at top of range)
2. $\tan^{-1}!\left(-\sqrt{3}\right)=-\frac{\pi}{3}$ (negative → Q IV)
3. $\cot^{-1}(-1)=\frac{3\pi}{4}$ (cot negative → Q II)
4. $\csc^{-1}!\left(-\dfrac{2\sqrt3}{3}\right)=-\frac{\pi}{3}$ (convert to sine: $-\sqrt3/2$)
5. $\tan^{-1}!\left(\dfrac{\sqrt3}{3}\right)=\frac{\pi}{6}$
6. $\sec^{-1}(-2)=\frac{2\pi}{3}$ (sec negative → Q II)

Practice pair solved in class

• $\cos^{-1}!\left(-\dfrac{\sqrt3}{2}\right)=\frac{5\pi}{6}$
• $\cot^{-1}!\left(\dfrac{\sqrt3}{3}\right)=-\frac{\pi}{6}$

Composite Expressions $f(f^{-1}(x))$ & $f^{-1}(f(x))$

Key idea

• If inverse is outside, we are constrained by the range of the inverse
• If inverse is inside, we must check the domain of the inverse (= range of original function after restriction)

Domain-check samples (inverse inside)

• $\sin\bigl(\sin^{-1}(-0.8)\bigr)=-0.8$ because $-0.8\in[-1,1]$
• $\sin\bigl(\sin^{-1}(\pi)\bigr)$ undefined; $\pi\notin[-1,1]$
• $\cos\bigl(\cos^{-1}(4/3)\bigr)$ undefined; 4/3>1

Range-check samples (inverse outside)

• $\cos^{-1}\bigl(\cos(5\pi/6)\bigr)=5\pi/6$ (angle already within $[0,\pi]$)
• $\tan^{-1}\bigl(\tan(-\pi/4)\bigr)=-\pi/4$ because $-\pi/4\in(-\pi/2,\pi/2)$
• $\sin^{-1}\bigl(\sin(7\pi/6)\bigr)$
  • $\sin(7\pi/6)=-1/2$
  • Need an angle in $[-\pi/2,\pi/2]$ with same sine ⇒ $-\pi/6$
• $\tan^{-1}\bigl(\tan(4\pi/3)\bigr)$
  • $\tan(4\pi/3)=\sqrt3$ (positive)
  • Choose principal angle in Q I with tan $\sqrt3$ ⇒ $\pi/3$

Additional tricky composite

• $\csc^{-1}\bigl(\csc(7\pi/4)\bigr)$
  • $\csc(7\pi/4)=\; -\sqrt2$ (Q IV, sine negative)
  • Principal csc inverse range $[-\pi/2,\pi/2]\setminus{0}$
  • Same reciprocal value occurs at $-\pi/4$ → answer $-\pi/4$

Summary of Evaluation Strategy

1. Convert reciprocal inputs to base functions when helpful
2. Identify sign (+/–) → locate correct quadrant using principal-value map
3. Recall special-angle sine/cosine/tangent values
4. If composite, first compute inner function, then project result into inverse’s restricted range

Key Special-Angle Values (reminder)

Practice for Self-Study (from lecture)

1. $\sin^{-1}\bigl(\sin(5\pi/6)\bigr)=\pi/6$
2. Compute $\tan^{-1}(2)$ (not special → numeric)
3. Evaluate $\sec^{-1}!\left(-\dfrac{5}{4}\right)$ → convert to cos $=-4/5$, answer in Q II
4. Verify domain before computing $\sin(\sin^{-1}(1.2))$ (undefined)

Ethical & Practical Notes Mentioned

• Academic honesty: guessing multiple-choice without showing work earns zero
• Study recommendations: practice immediately after Test #1; chapter 3 concepts require repetition
• Instructor’s closing: encouragement, prayer for students, reminder to keep “Jesus 1st 3-in-1 in your heart”