Trig Graphs, Reciprocal & Inverse Trig Functions – Detailed Notes

Sine Graph Example – $y = \sin\bigl(\tfrac{\pi}{2}x-\pi\bigr)+2$

• Factor out the coefficient of $x$ to see all shifts clearly.
• $y = \sin\Bigl[\tfrac{\pi}{2}\bigl(x-2\bigr)\Bigr]+2$ (because $\pi \div \tfrac{\pi}{2}=2$).
• Amplitude = $|1| = 1$ (positive so the curve is not inverted).
• Period = $\tfrac{2\pi}{\tfrac{\pi}{2}} = 4$.
• Phase (horizontal) shift = $+2$ units (to the right).
• Vertical shift = $+2$ units (up) – anything added outside the trig term is a vertical shift.
• Key points (quarter–period apart):
• Starts at $(2,2)$ → mid-line raised.
• Max at $(3,3)$, crosses mid-line again at $(4,2)$, min at $(5,1)$, ends one period later at $(6,2)$.
• Sketch tip: first draw the usual sine in light pencil, then slide it right 2 and up 2.

Cosine Graph Example – $y=-3\cos(2x-\tfrac{\pi}{3})$

• Write in standard form by factoring the $2$: $y=-3\cos\bigl[2\bigl(x-\tfrac{\pi}{6}\bigr)\bigr]$.
• Amplitude = $|{-3}| = 3$; minus sign flips (inverts) the curve.
• Period = $\tfrac{2\pi}{2}=\pi$.
• Phase shift = $+\tfrac{\pi}{6}$ (right).
• Endpoints of one fundamental cycle: start $x=\tfrac{\pi}{6}$, finish $x=\tfrac{\pi}{6}+\pi=\tfrac{7\pi}{6}$.
• Quarter-points (do fraction arithmetic):
• $x=\tfrac{\pi}{6},\;\tfrac{5\pi}{12},\;\tfrac{2\pi}{3},\;\tfrac{11\pi}{12},\;\tfrac{7\pi}{6}$.
• Corresponding $y$ values: $-3,0,3,0,-3$.
• Calculator check: set window $x<em>{min}=\tfrac{\pi}{6},\;x</em>{max}=\tfrac{7\pi}{6}$ in radian mode – only the fundamental period appears.

General Calculator / Desmos Tips

• Always confirm RADIAN vs DEGREE mode.
• Use a custom window so exactly one period (fundamental cycle) fills the screen – useful for verifying algebra.

Tangent Function Basics

• Undefined where cosine $=0$ → vertical asymptotes at $x=\tfrac{\pi}{2}+n\pi$.
• Domain: all $x\neq \tfrac{\pi}{2}+n\pi$.
• Range: $(-\infty,\infty)$.
• Fundamental period length = $\pi$ (not $2\pi$).
• Crosses $x$-axis at integer multiples of $\pi$.

Example: $y=2\tan\bigl(x-\tfrac{\pi}{4}\bigr)$
• Period still $\pi$.
• Phase shift $+\tfrac{\pi}{4}$.
• New asymptotes at $x=-\tfrac{\pi}{4},\;x=\tfrac{3\pi}{4}$.
• Choice of $y$-scale (±4, ±10, …) just stretches vertically; asymptotes stay put.

Cotangent Function

• Undefined where sine $=0$ → vertical asymptotes at $x=n\pi$.
• Domain: $x\neq n\pi$ ; Range: $(-\infty,\infty)$.
• Always decreasing on each period (contrast tangent).
• Fundamental portion usually drawn from $0$ to $\pi$.

Secant Function (reciprocal of cosine)

• Same points of undefinedness as tangent → asymptotes $x=\tfrac{\pi}{2}+n\pi$.
• Domain: $x\neq \tfrac{\pi}{2}+n\pi$.
• Graph made of upward and downward branches (“U” and “∩”) sitting outside $y=\pm1$.
• Range: $(-\infty,-1]\cup[1,\infty)$.
• Fundamental interval convenient to use: $(-\tfrac{\pi}{2},\tfrac{3\pi}{2})$.

Cosecant Function (reciprocal of sine)

• Undefined where sine $=0$ → asymptotes $x=n\pi$.
• Graph identical to secant but shifted $\tfrac{\pi}{2}$ left/right.
• Range again $(-\infty,-1]\cup[1,\infty)$.

“Problem-child” Function Types to Watch

• Rational expressions (division by 0).
• Even-index roots (negative radicands).
• Logarithms (non-positive arguments).
• Trig reciprocals & tangents (division by 0 at specific angles).

Inverse-Function Review

• To find an inverse, interchange $x$ and $y$ and solve for $y$.
• Eg. $y=3x+2\Rightarrow x=3y+2\Rightarrow y=\tfrac{x-2}{3}$.
• For exponentials $y=2^x\Rightarrow x=2^y\Rightarrow y=\log_2x$.

Inverse Trig Concept & Notation

• $\sin^{-1}x,\;\cos^{-1}x,\;\tan^{-1}x$ or $\operatorname{arcsin}x,\;\operatorname{arccos}x,\;\operatorname{arctan}x$.
• Important: $\sin^{-1}x \neq \dfrac{1}{\sin x}$.
• An inverse trig output IS AN ANGLE whose original trig ratio equals the given number.
• Calculator: the 2nd-key of SIN/COS/TAN provides the inverse functions.

Graph & Range Restrictions of Inverse Trig

Inverse Sine (arcsin)

• To make $\sin x$ one-to-one, restrict domain to $[-\tfrac{\pi}{2},\tfrac{\pi}{2}]$.
• Domain $[-1,1]$; Range $[-\tfrac{\pi}{2},\tfrac{\pi}{2}]$.

Inverse Tangent (arctan)

• Horizontal asymptotes at $y=\pm\tfrac{\pi}{2}$.
• Domain $(-\infty,\infty)$; Range $(-\tfrac{\pi}{2},\tfrac{\pi}{2})$ (open).
• Most frequently used inverse in Calculus.

Inverse Cosine (arccos)

• Restrict $\cos x$ to domain $[0,\pi]$ so it is one-to-one.
• Domain $[-1,1]$; Range $[0,\pi]$.
• Outputs are always between $0^{\circ}$ and $180^{\circ}$.

Evaluating Inverse Trig – Examples and Rules

• $\sin^{-1}\bigl(\tfrac12\bigr)=30^{\circ}=\tfrac{\pi}{6}$.
• $\cos^{-1}\bigl(\tfrac{\sqrt2}{2}\bigr)=45^{\circ}=\tfrac{\pi}{4}$.
• $\tan^{-1}(1)=45^{\circ}=\tfrac{\pi}{4}$.
• Negative inputs:
• $\sin^{-1}\bigl(-\tfrac{\sqrt3}{2}\bigr)=-60^{\circ}=-\tfrac{\pi}{3}$ (must stay in $[-90^{\circ},90^{\circ}]$).
• $\tan^{-1}(-\sqrt3)=-60^{\circ}$.
• $\cos^{-1}\bigl(-\tfrac{1}{2}\bigr)=120^{\circ}$ (must stay in $[0^{\circ},180^{\circ}]$).
• Non-special numbers (use calculator): $\tan^{-1}(3.2)\approx72.65^{\circ}$, so $\tan^{-1}(-3.2)\approx-72.65^{\circ}$.
• Reciprocal trick when the calculator lacks a key:
• $\sec^{-1}(3)=\cos^{-1}\bigl(\tfrac{1}{3}\bigr)\approx70.53^{\circ}$.
• $\cot^{-1}(0.5)=\tan^{-1}(2)\approx63.43^{\circ}$.

Using Inverse Trig to Solve Right Triangles

Example 3-4-5 triangle (right angle at $C$):
• Let $\beta$ opposite the 4 side, $\alpha$ opposite the 3 side.
• Biggest angle $\beta$ first:
• $\sin\beta=\tfrac45 \Rightarrow \beta=\sin^{-1}\bigl(\tfrac45\bigr)\approx53.13^{\circ}$.
• (Checks: $\cos^{-1}\bigl(\tfrac35\bigr)$ or $\tan^{-1}\bigl(\tfrac43\bigr)$ give the same.)
• $\alpha=90^{\circ}-\beta\approx36.87^{\circ}$, or directly $\alpha=\cos^{-1}\bigl(\tfrac45\bigr)$, etc.

Miscellaneous Reminders

• Always note whether homework requests degrees or radians.
• Incorrect mode is the #1 graphing/solving error (degrees for graphs, radians for triangle solutions, etc.).
• Upcoming topics: remainder of inverse-trig details and beginning of Chapter 5 (trig identities).