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{min}=\tfrac{\pi}{6},\;x{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).