The Trig Toolbox — Functions, Symmetries, and Core Identities

0. Big picture

This lecture turns angle measure into functions. We define $\sin,\cos,\tan,\csc,\sec,\cot$ on the unit circle, learn how signs flip by quadrant, and assemble the identity toolkit you’ll use all semester. The flow:

1. Definitions & domains → 2) Periodicity & symmetry → 3) Core identities → 4) Angle-addition → 5) Double/half-angle → 6) Product–sum.

---

1. Defining the six trigonometric functions

1.1 Unit-circle definition (primary)

Let $\theta$ be in standard position and let its terminal side meet the unit circle $x^2+y^2=1$ at $P(x,y)$.

$$\begin{aligned} \sin\theta &= y,\\ \cos\theta &= x,\\ \tan\theta &= \frac{y}{x} \quad (x \neq 0). \end{aligned}$$

Reciprocals:

$$\begin{aligned} \csc\theta &= \frac{1}{\sin\theta},\\ \sec\theta &= \frac{1}{\cos\theta},\\ \cot\theta &= \frac{1}{\tan\theta} = \frac{x}{y} \quad (y \neq 0). \end{aligned}$$

Domains & ranges.

• $\sin\theta,\cos\theta\in[-1,1]$ for all $\theta\in\mathbb{R}$.
• $\tan\theta,\sec\theta$ undefined where $\cos\theta=0$ (odd multiples of $\tfrac{\pi}{2}$).
• $\cot\theta,\csc\theta$ undefined where $\sin\theta=0$ (multiples of $\pi$).

Right-triangle view (for acute $\theta$): in a right triangle with hypotenuse $r$, opposite $y$, adjacent $x$,

$$\begin{aligned} \sin\theta &= \frac{y}{r},\\ \cos\theta &= \frac{x}{r},\\ \tan\theta &= \frac{y}{x}. \end{aligned}$$

Extends to all $\theta$ via signs on the unit circle.

---

1.2 Signs by quadrant (ASTC rule)

Let QI, QII, QIII, QIV denote the four quadrants.

• QI $(0,\tfrac{\pi}{2})$: all positive.
• QII $(\tfrac{\pi}{2},\pi)$: $\sin,\csc>0$; $\cos,\sec,\tan,\cot<0$.
• QIII $(\pi,\tfrac{3\pi}{2})$: $\tan,\cot>0$; $\sin,\csc,\cos,\sec<0$.
• QIV $(\tfrac{3\pi}{2},2\pi)$: $\cos,\sec>0$; $\sin,\csc,\tan,\cot<0$.

Reference angle $\theta_{\text{ref}}\in(0,\tfrac{\pi}{2})$ aids evaluation; attach the correct sign from the quadrant.

---

2. Periodicity and symmetry (even–odd)

Periods.

$$\begin{aligned} \sin(\theta+2\pi) &= \sin\theta,\\ \cos(\theta+2\pi) &= \cos\theta,\\ \tan(\theta+\pi) &= \tan\theta. \end{aligned}$$

Thus: $\sin,\cos,\sec,\csc$ have period $2\pi$; $\tan,\cot$ have period $\pi$.

Even–odd symmetries.

$$\begin{aligned} \cos(-\theta) &= \cos\theta &&(\text{even}),\\ \sin(-\theta) &= -\sin\theta &&(\text{odd}),\\ \tan(-\theta) &= -\tan\theta &&(\text{odd}). \end{aligned}$$

Consequently, $\sec$ is even; $\csc,\cot$ are odd.

Cofunction (complement) symmetries.

$$\begin{aligned} \sin\!\left(\frac{\pi}{2}-\theta\right) &= \cos\theta,\\ \cos\!\left(\frac{\pi}{2}-\theta\right) &= \sin\theta,\\ \tan\!\left(\frac{\pi}{2}-\theta\right) &= \cot\theta, \end{aligned}$$

and reciprocals accordingly.

---

3. Fundamental identity chain

3.1 Pythagorean identities

From $x^2+y^2=1\Rightarrow \cos^2\theta+\sin^2\theta=1$:

$$\begin{aligned} \sin^2\theta + \cos^2\theta &= 1,\\ 1 + \tan^2\theta &= \sec^2\theta,\\ 1 + \cot^2\theta &= \csc^2\theta. \end{aligned}$$

3.2 Quotient & reciprocal identities

$$\begin{aligned} \tan\theta &= \frac{\sin\theta}{\cos\theta},\\ \cot\theta &= \frac{\cos\theta}{\sin\theta},\\ \sec\theta &= \frac{1}{\cos\theta},\\ \csc\theta &= \frac{1}{\sin\theta}. \end{aligned}$$

---

4. Angle addition & subtraction formulas (derivations + use)

Core set (to be memorized and applied fluently):

$$\begin{aligned} \sin(\alpha\pm\beta)&=\sin\alpha\cos\beta\ \pm\ \cos\alpha\sin\beta,\\ \cos(\alpha\pm\beta)&=\cos\alpha\cos\beta\ \mp\ \sin\alpha\sin\beta,\\ \tan(\alpha\pm\beta)&=\dfrac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta}. \end{aligned}$$

Sketch of derivation idea. View $(\cos\theta,\sin\theta)$ as a rotation on the plane. The rotation matrix

$$R(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\ \sin\theta&\cos\theta\end{pmatrix}$$

satisfies $R(\alpha)R(\beta)=R(\alpha+\beta)$. Multiply the matrices to read off the $\cos(\alpha+\beta)$ and $\sin(\alpha+\beta)$ formulas.

Exact-value classic.
$\displaystyle \sin 15^\circ=\sin(45^\circ-30^\circ)=\frac{\sqrt6-\sqrt2}{4}$,
$\displaystyle \cos 75^\circ=\cos(45^\circ+30^\circ)=\frac{\sqrt6-\sqrt2}{4}$.

---

5. Double-angle, power-reduction, and half-angle

5.1 Double-angle

$$\begin{aligned} \sin(2\alpha) &= 2\sin\alpha\cos\alpha,\\ \cos(2\alpha) &= \cos^2\alpha - \sin^2\alpha = 1 - 2\sin^2\alpha = 2\cos^2\alpha - 1,\\ \tan(2\alpha) &= \dfrac{2\tan\alpha}{1-\tan^2\alpha} \quad (1-\tan^2\alpha \neq 0). \end{aligned}$$

5.2 Power-reduction (from $\cos 2\alpha$)

$$\begin{aligned} \sin^2\alpha &= \frac{1-\cos 2\alpha}{2},\\ \cos^2\alpha &= \frac{1+\cos 2\alpha}{2}. \end{aligned}$$

5.3 Half-angle

$$\begin{aligned} \sin^2\frac{\alpha}{2} &= \frac{1-\cos\alpha}{2},\\ \cos^2\frac{\alpha}{2} &= \frac{1+\cos\alpha}{2}. \end{aligned}$$

With quadrant-aware signs:

$$\begin{aligned} \sin\frac{\alpha}{2} &= \pm\sqrt{\frac{1-\cos\alpha}{2}},\\ \cos\frac{\alpha}{2} &= \pm\sqrt{\frac{1+\cos\alpha}{2}},\\ \tan\frac{\alpha}{2} &= \frac{\sin\alpha}{1+\cos\alpha} = \frac{1-\cos\alpha}{\sin\alpha}. \end{aligned}$$

---

6. Product–to–sum and sum–to–product (linearizing products)

Useful to linearize products or factor sums:

$$\begin{aligned} \sin A\sin B &= \tfrac12\big[\cos(A-B)-\cos(A+B)\big],\\ \cos A\cos B &= \tfrac12\big[\cos(A-B)+\cos(A+B)\big],\\ \sin A\cos B &= \tfrac12\big[\sin(A+B)+\sin(A-B)\big]. \end{aligned}$$

Converses (sum–to–product), e.g.

$$\begin{aligned} \cos A-\cos B &= -2\sin\frac{A+B}{2}\,\sin\frac{A-B}{2},\\ \sin A+\sin B &= 2\sin\frac{A+B}{2}\,\cos\frac{A-B}{2}. \end{aligned}$$

---

7. Worked examples (step-by-step)

Example 1 — Signs & reference angle

Evaluate $\sin!\left(-\dfrac{5\pi}{6}\right)$ and $\cos!\left(-\dfrac{5\pi}{6}\right)$.

• Reduce: $-\dfrac{5\pi}{6}$ is in QIII after adding $\pi$ (or think clockwise $150^\circ$ lands in QIII).
• Reference angle: $\theta_{\text{ref}}=\dfrac{\pi}{6}$.
• In QIII: $\sin<0,\ \cos<0$.
  Thus:

$$\sin!\left(-\frac{5\pi}{6}\right)=-\sin\frac{\pi}{6}=-\frac{1}{2},\quad
\cos!\left(-\frac{5\pi}{6}\right)=-\cos\frac{\pi}{6}=-\frac{\sqrt3}{2}.$$

---

Example 2 — Prove an identity

Show $\displaystyle \frac{1-\cos 2x}{\sin 2x}=\tan x$ when $\sin 2x\neq 0$.

Left-hand side:

$$\frac{1-(1-2\sin^2 x)}{2\sin x\cos x}
=\frac{2\sin^2 x}{2\sin x\cos x}
=\frac{\sin x}{\cos x}
=\tan x.$$

---

Example 3 — Exact value with addition formula

Find $\cos 75^\circ$.

$$\cos(45^\circ+30^\circ)
=\cos45^\circ\cos30^\circ-\sin45^\circ\sin30^\circ
=\frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2}-\frac{\sqrt2}{2}\cdot\frac{1}{2}
=\frac{\sqrt6-\sqrt2}{4}.$$

---

Example 4 — Double-angle to linearize a power

Compute $\sin^2 20^\circ+\sin^2 40^\circ+\sin^2 60^\circ+\sin^2 80^\circ$.

Use $\sin^2 t=\tfrac{1-\cos 2t}{2}$ and denote the total by $S$:

$$S=\tfrac12\big[4-\big(\cos40^\circ+\cos80^\circ+\cos120^\circ+\cos160^\circ\big)\big].$$

Pair with sum–to–product:

$$\cos40^\circ+\cos160^\circ=2\cos100^\circ\cos60^\circ=\cos100^\circ,$$

$$\cos80^\circ+\cos120^\circ=2\cos100^\circ\cos20^\circ.$$

Also $\cos100^\circ=-\cos80^\circ$. Continuing carefully (or using symmetry around $90^\circ$) yields $S=2$.
(Neater route: group as conjugate pairs around $90^\circ$; the cosines cancel.)

---

Example 5 — Product–to–sum to solve an equation

Solve $\cos 2x+2\cos 3x+\cos 4x=0$ for all real $x$.

Group as $(\cos 4x+\cos 2x)+2\cos 3x=0$. Use sum–to–product:

$$\cos 4x+\cos 2x=2\cos 3x\cos x.$$

Then

$$2\cos 3x\cos x+2\cos 3x=2\cos 3x(\cos x+1)=0.$$

Hence

$$\cos 3x=0\quad \text{or}\quad \cos x=-1.$$

Solutions:

$$3x=\frac{\pi}{2}+\pi k\ \Rightarrow\ x=\frac{\pi}{6}+\frac{\pi}{3}k,\quad
\text{or}\quad x=\pi+2\pi n,\qquad k,n\in\mathbb Z.$$

---

Example 6 — Triangle identity classic

In $\triangle ABC$ (angles $A,B,C$ sum to $\pi$), show

$$\tan A+\tan B+\tan C=\tan A,\tan B,\tan C.$$

Since $C=\pi-(A+B)$, $\tan C=-\tan(A+B)=\dfrac{-(\tan A+\tan B)}{1-\tan A\tan B}$.
Multiply both sides by $1-\tan A\tan B$:

$$\tan C-\tan A\tan B\tan C=-(\tan A+\tan B).$$

Rearrange:

$$\tan A+\tan B+\tan C=\tan A\tan B\tan C.$$

---

8. Common pitfalls (and fixes)

1. Forgetting domain holes: dividing by $\cos\theta$ or $\sin\theta$ assumes they’re nonzero—state restrictions.
2. Wrong period: $\tan$ repeats every $\pi$, not $2\pi$.
3. Addition-formula signs: $\cos(\alpha\pm\beta)$ uses opposite internal sign.
4. Half-angle signs: choose $+$ or $−$ by the quadrant of $\tfrac{\alpha}{2}$.
5. Mixing degrees/radians: keep units consistent when substituting numeric angles.

---

9. Short practice (with answers)

Q1. Determine the period and parity (even/odd) of each:
(a) $\sin\theta$ (b) $\tan\theta$ (c) $\sec\theta$.

Q2. Prove $\displaystyle \frac{1+\cos x}{\sin x}=\cot\frac{x}{2}$ for $x\not\equiv 0\pmod{2\pi}$.

Q3. Find the exact value of $\sin 75^\circ$.

Q4. Solve for all real $\theta$: $\sin\theta+\sin 3\theta=0$.

Q5. Simplify to a single trig function: $\displaystyle \frac{\sin x,\cos x}{\cos^2 x-\sin^2 x}$.

Answers.

• A1. (a) period $2\pi$, odd; (b) period $\pi$, odd; (c) period $2\pi$, even.
• A2. Use $\cot\frac{x}{2}=\frac{1+\cos x}{\sin x}$ (half-angle identity) or divide $\sin x$ by $\sin x$ after writing $\cos x=1-2\sin^2\frac{x}{2}$.
• A3. $\sin(45^\circ+30^\circ)=\tfrac{\sqrt6+\sqrt2}{4}$.
• A4. Product–to–sum: $\sin\theta+\sin3\theta=2\sin2\theta\cos\theta=0$. So $\sin2\theta=0\Rightarrow 2\theta=\pi k\Rightarrow \theta=\tfrac{\pi k}{2}$ or $\cos\theta=0\Rightarrow \theta=\tfrac{\pi}{2}+\pi n$. Combined set is $\theta=\tfrac{\pi k}{2}$ (covers both).
• A5. $\dfrac{\sin x\cos x}{\cos2x}=\dfrac{\tfrac12\sin2x}{\cos2x}=\tfrac12\tan2x$.

---

10. Micro-checkpoints (self-check while solving)

• C1. Did I pick the correct period ($2\pi$ vs $\pi$)?
• C2. Are there domain restrictions before dividing by $\sin$ or $\cos$?
• C3. Did I determine the quadrant before assigning a sign to a root or half-angle?
• C4. For products/sums, would product–sum or sum–product simplify things?
• C5. Can I rewrite powers using double-angle or power-reduction?

---

11. One-page summary (carry sheet)

$$\begin{aligned} &\textbf{Unit circle: } \sin\theta=y,\ \cos\theta=x,\ \tan\theta=\frac{y}{x}.\\ &\textbf{Pythagorean: } \sin^2\theta+\cos^2\theta=1;\ 1+\tan^2\theta=\sec^2\theta;\ 1+\cot^2\theta=\csc^2\theta.\\ &\textbf{Periods: } \sin,\cos,(\sec,\csc)\ 2\pi;\ \tan,\cot\ \pi.\\ &\textbf{Parity: } \cos,\sec\ \text{even};\ \sin,\csc,\tan,\cot\ \text{odd}.\\ &\textbf{Cofunction: } \sin(\tfrac{\pi}{2}-\theta)=\cos\theta,\ \tan(\tfrac{\pi}{2}-\theta)=\cot\theta.\\ &\textbf{Addition: } \sin(\alpha\!\pm\!\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta;\\ &\qquad\qquad \cos(\alpha\!\pm\!\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta;\\ &\qquad\qquad \tan(\alpha\!\pm\!\beta)=\dfrac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta}.\\ &\textbf{Double-angle: } \sin2\alpha=2\sin\alpha\cos\alpha;\ \cos2\alpha=1-2\sin^2\alpha=2\cos^2\alpha-1;\\ &\qquad\qquad \tan2\alpha=\dfrac{2\tan\alpha}{1-\tan^2\alpha}.\\ &\textbf{Power/half-angle: } \sin^2\alpha=\tfrac{1-\cos2\alpha}{2},\ \cos^2\alpha=\tfrac{1+\cos2\alpha}{2};\\ &\qquad\qquad \sin\frac{\alpha}{2}=\pm\sqrt{\tfrac{1-\cos\alpha}{2}},\ \cos\frac{\alpha}{2}=\pm\sqrt{\tfrac{1+\cos\alpha}{2}}.\\ &\textbf{Prod–sum: } \sin A\sin B=\tfrac12[\cos(A-B)-\cos(A+B)],\ \dots \end{aligned}$$

---

12. What’s next

We’ll use these identities to solve trigonometric equations systematically: principal values of inverse trig, general-solution templates, and multi-angle factoring—leading into the third part of the chapter.