In Grade 10, trig was about right-angled triangles and acute angles. In Grade 11, we extend trig to any angle — including angles bigger than $90°$, bigger than $360°$, and even negative angles. The tool that makes this work is the CAST diagram.
The CAST Diagram#
The Cartesian plane is divided into 4 quadrants. In each quadrant, only certain trig ratios are positive:
| Quadrant | Angle range | Positive ratios | Memory aid |
|---|---|---|---|
| Q1 | $0° < \theta < 90°$ | All (sin, cos, tan) | A |
| Q2 | $90° < \theta < 180°$ | Sin only | S |
| Q3 | $180° < \theta < 270°$ | Tan only | T |
| Q4 | $270° < \theta < 360°$ | Cos only | C |
Read anti-clockwise from Q4: C-A-S-T.
Why?#
In each quadrant, the signs of $x$ and $y$ change:
- $\sin\theta = \frac{y}{r}$ — positive when $y > 0$ (Q1, Q2)
- $\cos\theta = \frac{x}{r}$ — positive when $x > 0$ (Q1, Q4)
- $\tan\theta = \frac{y}{x}$ — positive when $x$ and $y$ have the same sign (Q1, Q3)
$r$ is always positive.
The Complete Reduction Formula Table#
The goal: reduce any angle to an acute angle ($0° – 90°$) by finding which quadrant the angle is in and applying the correct sign.
$180°$ formulas (Q2 and Q3)#
| Formula | Sign | Reason |
|---|---|---|
| $\sin(180° - \theta) = +\sin\theta$ | + | Sin positive in Q2 |
| $\cos(180° - \theta) = -\cos\theta$ | − | Cos negative in Q2 |
| $\tan(180° - \theta) = -\tan\theta$ | − | Tan negative in Q2 |
| $\sin(180° + \theta) = -\sin\theta$ | − | Sin negative in Q3 |
| $\cos(180° + \theta) = -\cos\theta$ | − | Cos negative in Q3 |
| $\tan(180° + \theta) = +\tan\theta$ | + | Tan positive in Q3 |
$360°$ formulas (Q4 and full rotation)#
| Formula | Sign | Reason |
|---|---|---|
| $\sin(360° - \theta) = -\sin\theta$ | − | Sin negative in Q4 |
| $\cos(360° - \theta) = +\cos\theta$ | + | Cos positive in Q4 |
| $\tan(360° - \theta) = -\tan\theta$ | − | Tan negative in Q4 |
Negative angles#
A negative angle rotates clockwise:
$\sin(-\theta) = -\sin\theta$
$\cos(-\theta) = +\cos\theta$
$\tan(-\theta) = -\tan\theta$
This is the same as $360° - \theta$ (Q4).
Co-functions: The $90°$ “Switch”#
When the angle involves $90°$, the function changes type (sin ↔ cos):
| Formula | Result | Why |
|---|---|---|
| $\sin(90° - \theta) = \cos\theta$ | Switch, Q1 → positive | |
| $\cos(90° - \theta) = \sin\theta$ | Switch, Q1 → positive | |
| $\sin(90° + \theta) = \cos\theta$ | Switch, Q2 → sin still positive | |
| $\cos(90° + \theta) = -\sin\theta$ | Switch, Q2 → cos is negative |
The logic#
$90°$ is on the boundary between quadrants, so it “swaps” the role of $x$ and $y$ coordinates — which swaps sin and cos.
Memory trick: $90°$ makes sin ↔ cos. $180°$ and $360°$ keep the function the same.
Worked Example 1: Basic Reduction#
Simplify $\sin(180° + 30°)$
$180° + 30° = 210°$ (Q3 — sin is negative)
$\sin(210°) = -\sin(30°) = -\frac{1}{2}$
Worked Example 2: Multi-Step#
Simplify: $\frac{\cos(360° - \theta) \cdot \sin(90° + \theta)}{\sin(180° + \theta)}$
Step 1 — Reduce each piece:
$\cos(360° - \theta) = \cos\theta$ (Q4, cos positive)
$\sin(90° + \theta) = \cos\theta$ (co-function switch, Q2 sin positive)
$\sin(180° + \theta) = -\sin\theta$ (Q3, sin negative)
Step 2 — Substitute:
$= \frac{\cos\theta \cdot \cos\theta}{-\sin\theta} = \frac{\cos^2\theta}{-\sin\theta} = -\frac{\cos^2\theta}{\sin\theta}$
Worked Example 3: With Negative Angle#
Simplify: $\tan(180° + \theta) \cdot \cos(-\theta) \cdot \sin(90° - \theta)$
$\tan(180° + \theta) = \tan\theta$
$\cos(-\theta) = \cos\theta$
$\sin(90° - \theta) = \cos\theta$
$= \tan\theta \cdot \cos\theta \cdot \cos\theta = \frac{\sin\theta}{\cos\theta} \cdot \cos^2\theta = \sin\theta \cdot \cos\theta$
Worked Example 4: Finding an Exact Value#
Find the value of $\cos 150°$ without a calculator.
$150° = 180° - 30°$ (Q2)
$\cos(180° - 30°) = -\cos 30° = -\frac{\sqrt{3}}{2}$
Worked Example 5: Given Information#
If $\sin 23° = p$, express $\cos 293°$ in terms of $p$.
$293° = 360° - 67°$ (Q4)
$\cos(360° - 67°) = \cos 67°$
$67° = 90° - 23°$
$\cos(90° - 23°) = \sin 23° = p$
Answer: $\cos 293° = p$
The Strategy: How to Reduce ANY Angle#
- Is it > 360°? Subtract 360° until it’s between 0° and 360°.
- Is it negative? Add 360° to make it positive.
- Which quadrant? Use CAST to determine the sign.
- Which reference angle? Strip away the $180°$, $360°$, or $90°$ part.
- Does it involve 90°? If yes, switch sin ↔ cos.
🚨 Common Mistakes#
- Forgetting the sign after the switch: Students remember to switch $\cos(90° + \theta)$ to $\sin\theta$ but forget the NEGATIVE sign. The co-function switch gives the new function, but the QUADRANT determines the sign.
- Not reducing all the way: $\sin(330°) = \sin(360° - 30°) = -\sin 30°$. Don’t stop at $\sin(330°)$ — always reduce to a special angle.
- Negative angle confusion: $\sin(-\theta) = -\sin\theta$ but $\cos(-\theta) = +\cos\theta$. Cos doesn’t change sign for negative angles!
- Angles > 360°: $\sin(420°) = \sin(420° - 360°) = \sin(60°)$. Just subtract 360° first.
- Not using the given information: When a question says “if $\sin\alpha = \frac{3}{5}$…”, they want your answer in terms of $\frac{3}{5}$. Use Pythagoras to find the other ratios, then apply reduction.
💡 Pro Tip: The Two-Question Checklist#
For every reduction, ask yourself TWO questions:
- Does the function stay the same or switch? (Only switches for $90°$)
- What sign does it get? (Use CAST with the ORIGINAL angle’s quadrant)
If you answer these two questions correctly every time, you’ll never get a reduction wrong.
🔗 Related Grade 11 topics:
- Trig Identities & Equations — reduction is used in almost every identity proof
- Sine, Cosine & Area Rules — applies trig to non-right-angled triangles
- The Parabola — trig graphs use the same domain/range language
📌 Grade 10 foundation: Trig Ratios & Special Angles