📐 Trig Functions & Graphs
Master all six trig functions, unit circle values, graph transformations, and more — with worked examples and practice problems.
📚 Table of Contents
1. The Six Trig Functions
📘 Key Concept
For a right triangle with angle \(\theta\), opposite side \(o\), adjacent side \(a\), and hypotenuse \(h\):
| Function | Definition | Reciprocal | Reciprocal Definition |
|---|---|---|---|
| Sine | \(\sin\theta = \dfrac{o}{h}\) | Cosecant | \(\csc\theta = \dfrac{h}{o}\) |
| Cosine | \(\cos\theta = \dfrac{a}{h}\) | Secant | \(\sec\theta = \dfrac{h}{a}\) |
| Tangent | \(\tan\theta = \dfrac{o}{a}\) | Cotangent | \(\cot\theta = \dfrac{a}{o}\) |
🟢 ASTC Rule — Signs by Quadrant
| Quadrant | Positive Functions |
|---|---|
| Q I | All (sin, cos, tan) |
| Q II | Sin only |
| Q III | Tan only |
| Q IV | Cos only |
Memory tip: All Students Take Calculus
✏️ Worked Example
Problem: A right triangle has opposite = 5, hypotenuse = 13. Find all six trig values.
Step 1: Find adjacent: \(\sqrt{13^2 – 5^2} = \sqrt{169 – 25} = \sqrt{144} = 12\)
Step 2: Apply definitions:
- \(\sin\theta = \dfrac{5}{13}\) \(\csc\theta = \dfrac{13}{5}\)
- \(\cos\theta = \dfrac{12}{13}\) \(\sec\theta = \dfrac{13}{12}\)
- \(\tan\theta = \dfrac{5}{12}\) \(\cot\theta = \dfrac{12}{5}\)
2. Unit Circle Exact Values
📘 Key Concept
The unit circle gives exact values for sine and cosine at key angles. Every point on the unit circle is \((\cos\theta,\ \sin\theta)\).
| Angle (°) | Angle (rad) | \(\sin\theta\) | \(\cos\theta\) | \(\tan\theta\) |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | \(\frac{\pi}{6}\) | \(\frac{1}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{\sqrt{3}}\) |
| 45° | \(\frac{\pi}{4}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{2}}{2}\) | 1 |
| 60° | \(\frac{\pi}{3}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{2}\) | \(\sqrt{3}\) |
| 90° | \(\frac{\pi}{2}\) | 1 | 0 | undefined |
| 180° | \(\pi\) | 0 | −1 | 0 |
| 270° | \(\frac{3\pi}{2}\) | −1 | 0 | undefined |
| 360° | \(2\pi\) | 0 | 1 | 0 |
🟢 Reference Angle Method
- Find the reference angle (acute angle to the x-axis)
- Look up the value for that reference angle
- Apply the correct sign using ASTC
✏️ Worked Example
Problem: Find the exact value of \(\sin(240°)\)
Step 1: \(240°\) is in Q III → reference angle = \(240° – 180° = 60°\)
Step 2: \(\sin(60°) = \dfrac{\sqrt{3}}{2}\)
Step 3: Sine is negative in Q III
$$\sin(240°) = -\frac{\sqrt{3}}{2}$$
3. Sine & Cosine Graphs
📘 Key Properties
| Property | \(y = \sin(x)\) | \(y = \cos(x)\) |
|---|---|---|
| Domain | \((-\infty, \infty)\) | \((-\infty, \infty)\) |
| Range | \([-1, 1]\) | \([-1, 1]\) |
| Period | \(2\pi\) | \(2\pi\) |
| Amplitude | 1 | 1 |
| Midline | \(y = 0\) | \(y = 0\) |
| Starts at (0, ?) | (0, 0) | (0, 1) |
| Symmetry | Odd: \(\sin(-x) = -\sin(x)\) | Even: \(\cos(-x) = \cos(x)\) |
✏️ Worked Example
Problem: What is the range of \(y = 3\sin(x) – 2\)?
Amplitude: 3 Midline: \(y = -2\)
Maximum: \(-2 + 3 = 1\) Minimum: \(-2 – 3 = -5\)
$$\text{Range} = [-5,\ 1]$$
4. Transformations
📘 General Form
$$y = A\sin(Bx – C) + D \qquad \text{or} \qquad y = A\cos(Bx – C) + D$$
| Parameter | What It Controls | Formula |
|---|---|---|
| \(A\) | Amplitude & reflection | Amplitude \(= |A|\); if \(A < 0\), graph reflects |
| \(B\) | Period | Period \(= \dfrac{2\pi}{|B|}\) |
| \(C\) | Phase shift | Phase shift \(= \dfrac{C}{B}\) (right if positive) |
| \(D\) | Vertical shift / midline | Midline: \(y = D\) |
🟢 Finding Equation from a Graph
$$A = \frac{\text{max} – \text{min}}{2} \qquad D = \frac{\text{max} + \text{min}}{2} \qquad B = \frac{2\pi}{\text{period}}$$
$$\text{Range} = [D – |A|,\ D + |A|]$$
✏️ Worked Example 1 — Identify Transformations
Problem: Find the amplitude, period, phase shift, and vertical shift of \(y = 4\cos(2x – \pi) + 3\)
- Amplitude: \(|4| = 4\)
- Period: \(\dfrac{2\pi}{2} = \pi\)
- Phase shift: \(\dfrac{\pi}{2}\) to the right
- Vertical shift: up 3 (midline \(y = 3\))
✏️ Worked Example 2 — Write the Equation
Problem: A sine graph has maximum 7, minimum −1, and period \(4\pi\). Write its equation.
$$A = \frac{7-(-1)}{2} = 4 \qquad D = \frac{7+(-1)}{2} = 3 \qquad B = \frac{2\pi}{4\pi} = \frac{1}{2}$$
$$y = 4\sin\!\left(\frac{x}{2}\right) + 3$$
✏️ Worked Example 3 — Range of Transformed Function
Problem: What is the range of \(y = -6\cos(x) + 1\)?
Amplitude: \(|-6| = 6\) Midline: \(y = 1\)
Maximum: \(1 + 6 = 7\) Minimum: \(1 – 6 = -5\)
$$\text{Range} = [-5,\ 7]$$
Note: the negative sign reflects the graph but does NOT change the range.
5. Tangent & Cotangent Graphs
📘 Key Properties
| Property | \(y = \tan(x)\) | \(y = \cot(x)\) |
|---|---|---|
| Period | \(\pi\) | \(\pi\) |
| Range | \((-\infty, \infty)\) | \((-\infty, \infty)\) |
| Amplitude | None | None |
| Asymptotes | \(x = \frac{\pi}{2} + n\pi\) | \(x = n\pi\) |
| x-intercepts | \(x = n\pi\) | \(x = \frac{\pi}{2} + n\pi\) |
| Period formula | \(\dfrac{\pi}{|B|}\) | |
✏️ Worked Example
Problem: Find the period and asymptotes of \(y = \tan(3x)\)
Period: \(\dfrac{\pi}{3}\)
Asymptotes: where \(3x = \dfrac{\pi}{2} + n\pi\), so \(x = \dfrac{\pi}{6} + \dfrac{n\pi}{3}\)
6. Secant & Cosecant Graphs
📘 Key Properties
| Property | \(y = \sec(x)\) | \(y = \csc(x)\) |
|---|---|---|
| Defined as | \(\frac{1}{\cos(x)}\) | \(\frac{1}{\sin(x)}\) |
| Period | \(2\pi\) | \(2\pi\) |
| Range | \((-\infty,-1]\cup[1,\infty)\) | \((-\infty,-1]\cup[1,\infty)\) |
| Asymptotes | \(x = \frac{\pi}{2} + n\pi\) | \(x = n\pi\) |
| x-intercepts | None | None |
🟢 Range of Transformed Sec & Csc
For \(y = A\sec(Bx) + D\) or \(y = A\csc(Bx) + D\):
$$\text{Range} = (-\infty,\ D – |A|] \cup [D + |A|,\ \infty)$$
✏️ Worked Example
Problem: Find the range of \(y = 3\sec(x) – 1\)
\(|A| = 3,\ D = -1\)
Lower bound: \(-1 – 3 = -4\) Upper bound: \(-1 + 3 = 2\)
$$\text{Range} = (-\infty,\ -4] \cup [2,\ \infty)$$
7. Practice Problems
📘 Try These!
1. A right triangle has opposite = 7, hypotenuse = 25. Find all six trig values.
👁️ Show Solution
Adjacent: \(\sqrt{625 – 49} = 24\)
- \(\sin\theta = \frac{7}{25}\) \(\csc\theta = \frac{25}{7}\)
- \(\cos\theta = \frac{24}{25}\) \(\sec\theta = \frac{25}{24}\)
- \(\tan\theta = \frac{7}{24}\) \(\cot\theta = \frac{24}{7}\)
2. Find the exact value of \(\cos(300°)\)
👁️ Show Solution
Q IV, reference angle = 60°, cosine is positive in Q IV
$$\cos(300°) = \frac{1}{2}$$
3. Find the amplitude, period, phase shift, and vertical shift of \(y = -3\sin(4x – \pi) + 2\)
👁️ Show Solution
- Amplitude: 3
- Period: \(\frac{2\pi}{4} = \frac{\pi}{2}\)
- Phase shift: \(\frac{\pi}{4}\) to the right
- Vertical shift: up 2 (midline \(y = 2\))
- Reflection: yes (negative A)
4. What is the range of \(y = -4\cos(x) + 3\)?
👁️ Show Solution
Amplitude = 4, Midline = 3
Maximum: \(3 + 4 = 7\) Minimum: \(3 – 4 = -1\)
$$\text{Range} = [-1,\ 7]$$
5. Find the period and asymptotes of \(y = \cot\!\left(\frac{x}{2}\right)\)
👁️ Show Solution
Period: \(\frac{\pi}{1/2} = 2\pi\)
Asymptotes: where \(\frac{x}{2} = n\pi\), so \(x = 2n\pi\)
6. Find the range of \(y = 2\csc(x) + 4\)
👁️ Show Solution
\(|A| = 2,\ D = 4\)
$$\text{Range} = (-\infty,\ 2] \cup [6,\ \infty)$$
