Trig Notes – Angle Reduction, Identities & Graph Transformations

Section 2.3 – Exact Values of Trigonometric Functions (No Calculator)

Periodic Properties (a.k.a. Coterminal‐Angle Rules)

• For $\sin,\; \cos,\; \sec,\; \csc$ : $f(\theta)=f(\theta\pm360^\circ)=f(\theta\pm2\pi)$
• For $\tan,\; \cot$ : $f(\theta)=f(\theta\pm180^\circ)=f(\theta\pm\pi)$

➔ Strategy: keep adding or subtracting the appropriate period until the angle is placed in Quadrant I or II ( $0<\theta<180^\circ$ ) where most exact values are known.

Even / Odd Symmetry

• Even functions (unchanged by $\theta\to-\theta$)
$\cos(-\theta)=\cos\theta,\qquad \sec(-\theta)=\sec\theta$
• Odd functions (pick up a minus sign)
$\sin(-\theta)=-\sin\theta,\;\; \csc(-\theta)=-\csc\theta,\;\; \tan(-\theta)=-\tan\theta,\;\; \cot(-\theta)=-\cot\theta$

Pythagorean / Reciprocal Identities Re-used All Class

• $\sin^2\theta+\cos^2\theta=1$
• $\tan^2\theta+1=\sec^2\theta$
• $1+\cot^2\theta=\csc^2\theta$
• Reciprocals: $\sec\theta=\dfrac1{\cos\theta},\; \csc\theta=\dfrac1{\sin\theta},\; \cot\theta=\dfrac{\cos\theta}{\sin\theta}$

Walk-Through of the Five In-Class Examples

Example 1

Evaluate $\displaystyle\cot315^\circ-\cot(135^\circ)+\cot(-495^\circ)$

1. Reduce each angle with tangent/cotangent period $180^\circ$:
   • $315-180=135^\circ$ (still in Quadrant II)
   • $-495+360=-135^\circ\;(\text{still outside })\implies -135^\circ+180^\circ=45^\circ$

2. Use even/odd: $\cot(-135^\circ)=-\cot135^\circ$

3. All three terms become $\cot135^\circ$; two positive, one negative ⇒ total $0$.

Example 2

$\sec(-360^\circ)-\dfrac1{\cos(-360^\circ)}$
• Period of $\sec$ is $360^\circ$, so $\sec(-360)=\sec0=1$.
• Second term is also $1$.
⇒ Difference $0$.

Example 3

$\dfrac{\tan^2\bigl(14\pi/3\bigr)-\sin^2\bigl(14\pi/3\bigr)}{\cos^2\bigl(20\pi/3\bigr)}$

1. Convert to one revolution ($0\le\theta<2\pi$) via $\theta-2\pi k$ with $k\in\mathbb Z$.
   • $14\pi/3-2\pi \cdot 2=2\pi/3$
   • $20\pi/3-2\pi \cdot 3=2\pi/3$
   So every trig term now uses $2\pi/3$.

2. Recognize $\tan^2\theta-\dfrac{\sin^2\theta}{\cos^2\theta}$ simplifies to $0$ (since numerator becomes $\dfrac{\sin^2-\sin^2}{\cos^2}=0$).

Example 4

$\tan^2 330^\circ + \sec^2 870^\circ - \dfrac1{\sec^2 510^\circ}$

1. Angle reduction:
   • $330-180=150^\circ$
   • $870-2\cdot360=150^\circ$
   • $510-360=150^\circ$

2. Rewrite $\dfrac1{\sec^2}=\cos^2$. Expression becomes
   $\tan^2 150^\circ+\sec^2 150^\circ-\cos^2 150^\circ$

3. Use identities $\tan^2+1=\sec^2\;\Rightarrow\;(\tan^2+1)-\cos^2$. Since $\sec^2=\dfrac1{\cos^2}\Rightarrow\tan^2=\dfrac1{\cos^2}-1$, substitution eventually leaves $\sin^2\theta+\cos^2\theta=1$. Then $1+1=2$.

Example 5

Huge “monster” problem (fourth powers). Skeleton of instructor’s solution:
• Repeatedly add/subtract $360^\circ$ to place angles at $\pm30^\circ$.
• Exploit even/odd so negatives disappear from $\cos$ but not $\sin$.
• Convert $\dfrac1{\cos^2}$ into $\sec^2$.
• Group as $\tan^2\theta-\sec^2\theta$ and recall identity $\tan^2-\sec^2=-1$.
• Final numeric result $1$ (after adding $+2$ and raising to $4^{\text{th}}$ power).

Overarching Tips From Instructor

• DO NOT turn angles into decimal form; full credit requires symbolic work.
• Calculator allowed only to verify, never to generate steps.
• If an angle is still outside $0^\circ!\le!\theta<360^\circ$ after one subtraction, keep sub-tracting.

Section 2.4 – Graphs of $\sin$ & $\cos$ With Transformations

(Page 147)

Two Transformations Focused On

1. Amplitude change (vertical stretch/shrink)
   • For $y=A\,\sin(Bx)$ or $y=A\,\cos(Bx)$, amplitude $=|A|$.

2. Period change (horizontal stretch/shrink)
   • Period $=\displaystyle\frac{2\pi}{|B|}$ for sine/cosine.

3. A leading negative sign creates a reflection across the $x$-axis.

No phase shifts or vertical shifts were covered in this lecture.

Example 1 : $y=3\sin x$

• Start with “unit” sine wave (peaks at $\pm1$).
• Multiply every $y$-value by $3$ ⇒ new peaks $\pm3$.
• Period remains $2\pi$. Sketch shown with key points $\bigl(0,0\bigr),(\tfrac\pi2,3),(\pi,0),(\tfrac{3\pi}2,-3),(2\pi,0)$.

Example 2 : $y=-2\sin(4x)$

Step-by-step instructor logic:

1. Parent function $\sin x$ sketched.

2. Amplitude factor $2$ ⇒ peaks to $\pm2$.

3. Negative sign ⇒ reflect across $x$-axis (max becomes min, etc.).

4. Inside factor $4$ ⇒ period $=\dfrac{2\pi}{4}=\dfrac{\pi}{2}$. Achieved by dividing all standard $x$ intercepts $\bigl(0,\pi/2,\pi,3\pi/2,2\pi\bigr)$ by $4$: $0,\tfrac\pi8,\tfrac\pi4,\tfrac{3\pi}8,\tfrac\pi2$.

5. Plot reflected/stretched points accordingly and connect with smooth sine curve.

Vocabulary Recap

• Amplitude – maximum vertical displacement, $|A|$.
• Period – horizontal length of one complete cycle, $\dfrac{2\pi}{|B|}$.
• Reflection – produced by a leading negative sign.
• Shrinking vs. stretching:
– $|A|>1$ stretches vertically; $0<|A|<1$ shrinks. – $|B|>1$ compresses horizontally; $0<|B|<1$ stretches.

Instructor’s Meta-Advice

• Graphical effects can be applied in any order, but mentally the sequence “amplitude → reflection → period” tends to be fastest.
• Every tick mark should be labelled with its
exact radian measure (e.g.
$\pi/8,\;3\pi/8$).
• Practise redrawing standard waves repeatedly; graphs appear on quizzes/tests.

Administrative / Logistic Notes

• Quiz 1 answer key available in room B-12 during the week.
• Questions about grading: email instructor.
• Calculator use policy reiterated (process must be manual).
• Homework for §2.4 not due until next Tuesday (after lecture is finished).
• Instructor’s mantra: “Practice, practise, practise – re-work each in-class example 3–4 times.”
• Section 2.4 will continue next class; upcoming content will include additional graph features.

Ethical & Practical Takeaways

• Building skill = sustained repetition; there is no shortcut.
• Class video re-watch strongly encouraged; class problems resemble exam problems more than textbook homework does.
• Students juggling jobs, families, multiple courses must still budget “hours, hours, hours” of trig practice for mastery.