12. Trigonometry
High SchoolTrigonometry connects angles to lengths. Born from right triangles, it grows into the language of circles, waves and periodic motion — the mathematics of anything that repeats.
The ratios, the unit circle and identities
In a right triangle, the three primary ratios relate an angle θ to its sides (SOH-CAH-TOA):
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent = sin θ / cos θ
On the unit circle (radius 1), a point at angle θ has coordinates (cos θ, sin θ). This extends the ratios to all angles and makes them periodic with period 360° (or 2π radians). The master identity, straight from Pythagoras on that circle:
For any triangle (not just right ones), the sine and cosine rules apply:
c² = a² + b² − 2ab·cos C
From triangles to waves
The genius move in trigonometry is the unit circle. A right-triangle ratio only makes sense for angles between 0° and 90°, but a point travelling around a circle has a sine and cosine for every angle, including negative and beyond 360°. As the point goes round and round, its height oscillates smoothly up and down — that is the sine wave, and it is why trig describes sound, light, tides and alternating current.
Radians feel strange at first but are the natural unit: one radian is the angle that wraps an arc equal to the radius, so a full circle is 2π radians. Measuring angles this way makes calculus of trig functions clean (as you will see in Chapter 15). The identity sin²θ + cos²θ = 1 is just Pythagoras applied to the circle's radius, restated in trig language.
Worked example: height by angle of elevation
Standing 50 m from a tower, you measure the angle of elevation to its top as 32°. How tall is the tower (ignore eye height)?
- The height is opposite the angle; the 50 m is adjacent. Use tangent: tan 32° = height / 50.
- So height = 50 × tan 32°.
- tan 32° ≈ 0.6249, giving height ≈ 50 × 0.6249 ≈ 31.2 m.
In a triangle with a = 7, b = 8, C = 60°, the third side: c² = 49 + 64 − 2(7)(8)cos60° = 113 − 112(0.5) = 57, so c = √57 ≈ 7.55.
Test your understanding
What is sin 30°, and why is it exactly 1/2?
sin 30° = 1/2 exactly. Take an equilateral triangle of side 2 and drop an altitude: it splits into two right triangles with hypotenuse 2, base 1, and a 30° angle at the apex. The side opposite the 30° is 1, so sin 30° = 1/2. The same triangle gives cos 30° = √3/2.
Convert 135° to radians.
Multiply by π/180: 135 × π/180 = 3π/4 radians. As a check, 180° is π, and 135° is three-quarters of the way from 90° to 180°, landing at 3π/4 ≈ 2.36.
Why does the graph of y = sin x repeat forever?
Because sine is defined by a point going round the unit circle. After a full turn of 2π, the point returns to exactly where it started with the same height, so the values recur identically. This periodicity — period 2π, range [−1, 1] — is what makes sine the model for every repeating phenomenon.
Use an identity to simplify (sin θ)/(cos θ) × cos θ.
(sin θ / cos θ) × cos θ = sin θ, since the cos θ cancels. Equivalently, tan θ × cos θ = sin θ because tan θ = sin θ / cos θ by definition. Recognising tan as a ratio of sin and cos simplifies a great deal.
How the ideas connect
Every key idea in this chapter, branching from the core concept — use it to see the whole picture at a glance.
The key facts, visualised
Worked problems, step by step
Follow each solution line by line, then try to reproduce it on paper before moving on.
Example 1A tree casts a shadow, and the angle of elevation to its top is 30 degrees from 20 m away. Find the height.
- tan(30) = height / 20
- height = 20 x tan(30) = 20 x 0.577
- height is about 11.5 m
Example 2Find sin(30) + cos(60).
- sin(30) = 1/2
- cos(60) = 1/2
- Sum = 1/2 + 1/2 = 1
Now you try
Work each one out first, then tap to reveal the worked answer.