Reference angles: Worked examples — Maths deep dive

This page is the drill-ground for Reference Angles. The parent note built the rules; here we throw every kind of angle at those rules — big, negative, on-axis, radians, word problems — and grind each one to a number. Guess first, then check.

Before we start, one reminder in plain words:

Everything below is that same four-move dance: reduce → quadrant → reference → sign.

The scenario matrix

Every angle you can ever be handed falls into exactly one of these case classes. If we work one example from each row, you will never meet a situation you haven't seen.

#	Case class	What's tricky about it	Covered by
A	Plain Q1 angle	nothing — warm-up	Ex 1
B	Q2 angle (sign flip on cos)	subtract from $180°$	Ex 2
C	Q3 angle (sign flip on sin)	subtract $180°$	Ex 3
D	Q4 angle (sign flip on tan)	subtract from $360°$	Ex 4
E	Big angle $> 360°$	must reduce first	Ex 5
F	Negative angle	must add turns first	Ex 6
G	Radian angle	keep $π$ , don't convert	Ex 7
H	On-axis / degenerate ( $0°, 90°, 180°, 270°$ )	reference angle is $0$ or $90°$ ; one ratio may be undefined	Ex 8
I	Word problem (real-world)	translate a story into an angle	Ex 9
J	Exam twist (solve backwards)	given a value, find all angles	Ex 10

Rows A–D cover all four quadrants. Rows E–F cover all sizes and signs. Row G covers radians. Row H covers the degenerate boundaries where the quadrant is ambiguous. Rows I–J are the applications. That is genuinely every cell.

Example 1 — Case A (plain Q1)

Example 2 — Case B (Q2, cosine flips)

See the picture: the arrow at $135°$ and its little reference tilt back to the negative x-axis.

Example 3 — Case C (Q3, sine flips)

Example 4 — Case D (Q4, tangent flips)

Example 5 — Case E (angle bigger than one turn)

Example 6 — Case F (negative angle)

Example 7 — Case G (radians, keep $π$ )

Example 8 — Case H (on-axis / degenerate)

When the terminal side lands exactly on an axis, there is no "corner" and the reference angle is degenerate ( $0°$ or $90°$ ). Watch what happens.

Worked example Compute

$cos 180°$ , $sin 90°$ , and $tan 90°$ Forecast: these sit on the axes (green points in the figure), so their coordinates are $\pm 1$ or $0$ — clean numbers, but one ratio blows up.

(a) $cos 180°$ — terminal side on the negative x-axis.

Reference: the gap to the nearest x-axis direction is $0°$ — the side is already on the axis, so $θ^{'} = 0°$ .
Point on unit circle: $(- 1, 0)$ . Cosine = x-coordinate = $- 1$ . $cos 180° = - 1.$

(b) $sin 90°$ — terminal side on the positive y-axis.

Reference: nearest x-axis direction is $0°$ ; the gap is a full $90°$ , so $θ^{'} = 90°$ (the maximum a reference angle can be).
Point: $(0, 1)$ . Sine = y-coordinate = $1$ . $sin 90° = 1.$

(c) $tan 90°$ — the degenerate case.

$tan 90° = \frac{sin 90°}{cos 90°} = \frac{1}{0}$ . Dividing by zero.
Why does it blow up? At $90°$ the point is $(0, 1)$ : the horizontal shadow $x$ is zero, so the slope $y / x$ is vertical — undefined. $tan 90° is undefined.$

Verify: $cos 180° = - 1$ , $sin 90° = 1$ , and $cos 90° = 0$ makes $tan 90° = 1/0$ undefined. ✔

Example 9 — Case I (real-world word problem)

Worked example A radar dish rotates. Its beam direction is

$θ = 200°$ measured counter-clockwise from due East (the positive x-axis). How far east (signed) does a target at distance $1$ km lie, i.e. find its east-coordinate $cos 200°$ ? Forecast: $200°$ points down-and-left (south-west-ish), so the east component should be negative — the target is to the west.

Reduce: $200°$ already in range.
Quadrant: $180° < 200° \leq 270°$ → Q3. Why? Tells us east ( $x$ ) is negative.
Reference: Q3 rule $θ^{'} = 200° - 180° = 20°$ .
Helper: $cos 20° \approx 0.9397$ (calculator, non-standard angle).
Sign: Q3 → cosine negative. $cos 200° \approx - 0.9397 km east = 0.9397 km west .$

Verify: direct calculator $cos 200° \approx - 0.9397$ ; sign is negative as the forecast predicted (target lies west). Units: a coordinate on a $1$ km circle is measured in km. ✔

Example 10 — Case J (exam twist: solve backwards)

Active recall

Recall Answers

$1000° - 2 \times 360° = 280°$ → Q4. ::: reduce twice. $tan 270° = sin / cos = - 1/0$ undefined; $tan 180° = 0/ (- 1) = 0$ . ::: numerator zero is fine, denominator zero is not. Two solutions, Q1 ( $60°$ ) and Q4 ( $300°$ ), since cosine is positive there. ::: ASTC gives the corners.

Connections

Reference Angles — parent: the rules these examples drill.
Coterminal Angles — every "reduce" step (Ex 5, 6).
Signs of Trig Functions (ASTC) — every "sign" step.
Trigonometric Ratios of Standard Angles — the helper values.
Unit Circle — the coordinate checks in every "Verify".
Radian and Degree Measure — Ex 7's radian handling.