Mini-Lesson

Trigonometry

This mini-lesson covers radians (arc length and sector area), exact values, the sine and cosine rules, the core identities, compound and double angle formulae, the R sin(θ + α) harmonic form, small-angle approximations and how to solve trig equations without losing solutions.

Where this sits: Unit AS 1: Pure Mathematics — trigonometry, radian measure, sine/cosine rules, graphs and simple identities. Unit A2 1 — reciprocal and inverse functions, compound and double angle formulae, R-form and harder equations.

Work through each screen, answer the questions as you go (some are reasoning, some are calculations) and collect ⭐ stars. Press Start when you are ready.

Radians

Radian measure, arc length and sector area

s = rθ · A = ½r²θθ MUST be in radians · π radians = 180° · 1 rad ≈ 57.3°

Worked example — r = 8 cm, θ = 0.75 rad

Arc length s = rθ = 8 × 0.75 = 6 cm

Sector area A = ½r²θ = ½ × 64 × 0.75 = 32 × 0.75 = 24 cm²

Convert: degrees → radians, multiply by π/180. So 60° = π/3, 45° = π/4, 30° = π/6, 90° = π/2. Calculator in the wrong mode is the single biggest source of lost marks in this topic.

Calculate

Arc length

1A sector of a circle has radius 8 cm and angle 0.75 radians. Find the arc length.

cm

Hint: s = rθ = 8 × 0.75.

Calculate

Sector area

2The same sector has radius 8 cm and angle 0.75 radians. Find its area.

cm²

Hint: A = ½r²θ = 0.5 × 8² × 0.75 = 0.5 × 64 × 0.75.

Triangles

Sine rule, cosine rule and area

a/sin A = b/sin B = c/sin C · a² = b² + c² − 2bc cos A · Area = ½ab sin CUse the cosine rule when you have two sides and the included angle, or all three sides

Worked example — cosine rule

Triangle with a = 7, b = 9, included angle C = 60°. Find c.

c² = 7² + 9² − 2(7)(9)cos 60° = 49 + 81 − 126(0.5) = 130 − 63 = 67

c = √67 = 8.19 (3 s.f.)

The ambiguous case: when the sine rule gives sin θ = k, there are usually two angles in 0° < θ < 180°: θ and 180° − θ. Check whether both fit the triangle.

Calculate

Cosine rule

3A triangle has sides 7 cm and 9 cm with an included angle of 60°. Find the third side to 3 significant figures.

cm

Hint: c² = 7² + 9² − 2 × 7 × 9 × cos 60° = 49 + 81 − 126 × 0.5 = 67, so c = √67 = 8.185…

Identities

The identities you must know cold

sin²θ + cos²θ ≡ 1 · tan θ ≡ sin θ / cos θdivide the first by cos²θ: 1 + tan²θ ≡ sec²θ · divide by sin²θ: 1 + cot²θ ≡ cosec²θ

Reciprocal functions (A2): sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = 1/tan θ.

Compound angles:

sin(A ± B) = sin A cos B ± cos A sin Bcos(A ± B) = cos A cos B ∓ sin A sin B — note the SWAPPED sign

Double angles (put B = A): sin 2θ = 2 sin θ cos θ, and cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.

Sort it

Exact values

Tap an exact value, then tap the number it equals.

½

√3/2

1/√2

A2 skill

The R-harmonic form

Any expression a sin θ + b cos θ can be written as a single sine (or cosine) wave — which instantly gives its maximum and minimum.

a sin θ + b cos θ = R sin(θ + α)R = √(a² + b²) and tan α = b/a, with R > 0 and α acute

Worked example — 3 sin θ + 4 cos θ

R = √(3² + 4²) = √25 = 5

tan α = 4/3 ⇒ α = arctan(4/3) = 0.927 rad (53.1°)

So 3 sin θ + 4 cos θ = 5 sin(θ + 0.927), with maximum 5 and minimum −5.

Calculate

Find R

4Write 3 sin θ + 4 cos θ in the form R sin(θ + α) with R > 0. Find R.

Hint: R = √(a² + b²) = √(3² + 4²) = √25.

Calculate

Find α

5For 3 sin θ + 4 cos θ = R sin(θ + α), find α in radians to 3 decimal places.

rad

Hint: tan α = b/a = 4/3, so α = arctan(4/3) = 0.9273 rad (make sure your calculator is in radian mode).

Quick check

Small-angle approximations

?For small θ (in radians), which set of approximations is correct?

Quick check

Solving without losing roots

?Solve sin θ = 0.5 for 0° ≤ θ ≤ 360°. How many solutions are there and what are they?

Match it

Match the identity

Tap an expression on the left, then the identity it equals.

Statement

Answer

Quick check

Which cos 2θ form?

?You need to solve 3cos 2θ + 5 sin θ = 1. Which form of cos 2θ should you substitute?

Quick check

Radians or degrees?

?A sector has radius 5 cm and angle 40°. Which is the correct area?

Examiner traps

Trig pitfalls

Calculator mode. s = rθ and A = ½r²θ need radians; small-angle results need radians too.
Losing solutions. sin θ = k has two solutions per revolution (θ and 180° − θ); dividing an equation by cos θ throws away the roots where cos θ = 0.
Wrong cos 2θ form. Choose the form that leaves a single trig function.
The ambiguous case of the sine rule — check whether the obtuse angle also fits.
R-form: R is always positive, and α comes from tan α = b/a (with a and b from the ORIGINAL expression).

Recap

The big ideas to know

Radians: s = rθ, A = ½r²θ — radians only; π rad = 180°

Triangles: sine rule (watch the ambiguous case), cosine rule, area = ½ab sin C

Identities: sin²θ + cos²θ ≡ 1 · 1 + tan²θ ≡ sec²θ · tan θ ≡ sin θ/cos θ

Compound/double angle: sin 2θ = 2 sin θ cos θ · cos 2θ has three forms — pick the useful one

R-form: a sin θ + b cos θ = R sin(θ + α), R = √(a² + b²), tan α = b/a → gives max/min instantly

Small angles (radians): sin θ ≈ θ, tan θ ≈ θ, cos θ ≈ 1 − θ²/2

Trig identities are the engine room of A2 calculus — you will meet them again when differentiating and integrating. Press Finish to see your score.

🏆

Mini-lesson complete!

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