Trigonometry Class – Quiz Review, Identities, and Graph Transformations

Administrative & Course Logistics

• Upcoming assessments
  • Test #1: Thursday, Feb 18
    • Review packet handed out on Thursday, Feb 11
    • Covers Sections 2.3 – 2.6 (exact values, identities, graphing)
  • Quizzes: students may use their own handwritten notes, but must show complete work; no collaborating or sharing answers.
• Homework
  • Section 2.3 due Thu Feb 11
  • Section 2.4 due Tue Feb 16
• Calculator policy
  • Allowed only to check numeric answers/graphs, not to replace algebraic or geometric work.
  • Graphs submitted without supporting work earn $0$.
• Academic honesty
  • Working together on graded work ⇒ grade of $0$ if detected.
  • Instructor stresses integrity and effort over shortcuts.

Study & Success Tips (Instructor’s Personal Story)

• Feeling “weak” in math is normal; improvement comes from hours of deliberate practice.
• Example: instructor spent 3-4 hrs/day in the library solving 40–60 algebra problems to turn an initial struggle into consistent A-performance.
• Key takeaway: practice + process > raw talent.
• Memorising common radian angles and identities speeds exams, even if reference sheets are allowed.

Quiz #1 – Frequently Asked Questions

Q5 – Evaluate $\cos^2!\left(\tfrac{2\pi}{3}\right) - \tan^2!\left(-\tfrac{5\pi}{3}\right)$

1. Use even–odd ideas or coterminal angles. Instructor rewrote $-\tfrac{5\pi}{3}\;\to\; \tfrac{\pi}{3}$ because tangent values repeat every $\pi$.
2. Compute individual pieces:
   $\sin\left(\tfrac{2\pi}{3}\right)=\tfrac{\sqrt3}{2},\qquad\sin\left(\tfrac{\pi}{3}\right)=\tfrac{\sqrt3}{2},\qquad \cos\left(\tfrac{\pi}{3}\right)=\tfrac12$
   $\cos^2!\left(\tfrac{2\pi}{3}\right)=\Bigl(\tfrac{\sqrt3}{2}\Bigr)^2=\tfrac34$
   $\tan\left(\tfrac{\pi}{3}\right)=\tfrac{\sqrt3}{1}$ and so $\tan^2\bigl(\cdot\bigr)=\tfrac34$
3. Subtract: $\tfrac34-\tfrac34=0$ (Instructor’s board arithmetic later simplified to $-\tfrac53$ after alternative arrangement—main idea: show every step for credit.)

Q7 – Given $\csc\theta=-\tfrac53$ and \tan\theta>0, find all six trig functions

1. $\csc\theta=1/\sin\theta \Rightarrow \sin\theta=-\tfrac35$.
2. Tangent positive while sine negative ⟹ cosine negative ⇒ Quadrant III.
3. Build right‐triangle:
   $\text{opposite}=-3,\; \text{hypotenuse}=5 \Rightarrow \text{adjacent}=-4$ via $a^2+b^2=c^2$.
4. Functions
   $\sin\theta=-\tfrac35\,,\qquad \cos\theta=-\tfrac45\,,\qquad \tan\theta=\tfrac34$
   $\csc\theta=-\tfrac53\,,\qquad \sec\theta=-\tfrac54\,,\qquad \cot\theta=\tfrac43$
5. Reminder: full triangle + Pythagorean work must be shown to earn points.

Triangle Reconstruction from Fractions (Homework Pattern)

• Exercises often give matching denominators, e.g.
  $\cos\theta=-\tfrac{\sqrt3}{2}, \; \sin\theta=-\tfrac12$
• Recognise signs ⇒ Quadrant III.
• Triangle sides are directly readable: adjacent $=-\sqrt3$, opposite $=-1$, hypotenuse $=2$ (no need to re-use $a^2+b^2=c^2$ when two legs + hypo clearly appear).
• Derive remaining four functions quickly from those side lengths.

Even & Odd Properties

• Will be explicitly requested: “Use even/odd properties to evaluate …”
• Quick recall:
  $\sin(-x)=-\sin x,\; \tan(-x)=-\tan x,\; \csc(-x)=-\csc x$
  $\cos(-x)=\cos x,\; \sec(-x)=\sec x,\; \cot(-x)=-\cot x$

Graphing Sine & Cosine Transformations (Section 2.4)

General form
$y=A\,\sin(Bx-C)+D \quad\text{or}\quad y=A\,\cos(Bx-C)+D$

• $|A|$ → amplitude (vertical stretch/compression)
• \text{sgn}(A)<0 → reflection over $x$-axis
• $B$ modifies period: $\text{period}=\dfrac{2\pi}{|B|}$
• Horizontal shift (phase): $C/B$
• Vertical shift: $D$ moves entire graph up/down.

Procedure used in class

1. Identify amplitude & reflection.
2. Find five key $x$ points of one cycle: $0,\;\dfrac{\text{period}}{4},\;\dfrac{\text{period}}{2},\;\dfrac{3\,\text{period}}{4},\;\text{period}$.
3. Apply vertical shift $D$ to all $y$–coordinates.
4. Plot and smoothly connect points.
5. Extend left/right if multiple cycles required.

Example 1

$f(x)=-3\cos\left(\tfrac\pi3 x\right)$

• Amplitude $|A|=3$, reflection (because A<0).
• Period $=\dfrac{2\pi}{\tfrac\pi3}=6$. Key $x$’s: $0,1.5,3,4.5,6$(all in units).
• Points after reflection: $(-3,0),(0),3,0,-3$
• No vertical shift.

Example 2

$g(x)=2\sin\left(\tfrac{3\pi}{4}x\right)+1$

• Amplitude $2$; no reflection.
• Period $=\dfrac{2\pi}{\tfrac{3\pi}{4}}=\dfrac{8}{3}$.
• Key $x$’s: $0,\tfrac23,\tfrac43,2,\tfrac83$.
• Compute sine template ${0,1,0,-1,0}$ then scale by $2$ → ${0,2,0,-2,0}$.
• Add vertical shift $+1$: ${1,3,1,-1,1}$.

Example 3

$h(x)=-\tfrac52\cos\left(\tfrac{5\pi}{6}x\right)-2$

• Amplitude $2.5$, reflection.
• Period $=\dfrac{2\pi}{\tfrac{5\pi}{6}}=\dfrac{12}{5}=2.4$.
• Five $x$ points: $0,0.6,1.2,1.8,2.4$ (or common‐fraction forms $0,\tfrac{3}{5},\tfrac65,\tfrac95,\tfrac{12}{5}$).
• Base cosine ${1,0,-1,0,1}$ → reflect & stretch ${-2.5,0,2.5,0,-2.5}$.
• Shift down 2 → ${-4.5,-2,0.5,-2,-4.5}$.

Practice Problem Solved in Class

$p(x)=-\tfrac34\sin\left(\tfrac{7\pi}{2}x\right)$

• Amplitude $0.75$ (compression), reflection.
• Period $=\dfrac{2\pi}{\tfrac{7\pi}{2}}=\dfrac{4\pi}{7}$.
• Key $x$’s: $0,-\tfrac{7\pi}{8},-\tfrac{7\pi}{4},-\tfrac{21\pi}{8},-\tfrac{7\pi}{2}$ (to the left because negative direction was chosen).
• Base sine pattern propagates, then scale/reflection → ${0,-0.75,0,0.75,0}$.

Common Numerical & Algebra Reminders

• Simplify radicals and fractions: e.g. $\bigl(\sqrt3\bigr)^2=3$, $\dfrac{\sqrt3}{1}=\sqrt3$.
• Rationalise denominators only when required.
• Period adjustment shortcut: if graph is $\sin(Bx)$ or $\cos(Bx)$, immediately write $\text{period}=\dfrac{2\pi}{|B|}$.
• Difference of squares identity: $\tan^2\theta+1=\sec^2\theta$, etc.—keep on a quick-access list for quizzes.

Expectation for Full Credit

1. Write down the identity or theorem being applied (e.g. Pythagorean identity, definition of trig ratios).
2. Show algebraic manipulation line-by-line.
3. For triangles: sketch, label sides, state $a^2+b^2=c^2$ computation, then list functions.
4. For graphs:
   • Table of transformed points
   • Indication of amplitude/period/phase/shift
   • Smooth curve, labelled axes/scale.

Failing to show process = no credit, even with a correct final number.

Key Formulas & Identities to Master

• Reciprocal pairs: $\csc\theta=\dfrac1{\sin\theta},\; \sec\theta=\dfrac1{\cos\theta},\; \cot\theta=\dfrac{\cos\theta}{\sin\theta}$
• Pythagorean set:
  $\sin^2\theta+\cos^2\theta=1$
  $1+\tan^2\theta=\sec^2\theta$
  $1+\cot^2\theta=\csc^2\theta$
• Period relations:
  $\text{period}<em>{\sin,\cos}=\dfrac{2\pi}{|B|},\quad \text{period}</em>{\tan,\cot}=\dfrac{\pi}{|B|}$
• Even/Odd: see earlier bullet.

Final Instructor Advice

• Re-watch lecture videos and re-draw every graph on fresh paper at least twice.
• Memorise unit-circle radian angles for speed.
• Show every intermediate step on quizzes/tests to secure partial credit.
• Practise daily; mathematics “muscle” grows with repetition, not last-minute cramming.