Mini-Lesson

Trigonometry

OCR section 1.05 is where trigonometry grows up: radians replace degrees, small angle approximations appear, the reciprocal functions sec, cosec, cot arrive with their own identities, and you meet the compound and double angle formulae and the R-harmonic form.

Radians are not optional — calculus of trig functions only works in radians. Work through each screen, answer the questions as you go and collect ⭐ stars. Press Start when you're ready.

Trigonometry · radians

Radians, arcs and sectors

π radians = 180° · s = rθ · A = ½r²θθ MUST be in radians for these two formulae — that is the whole point of radians

Worked example — r = 8 cm, θ = 0.6 rad

Arc length s = rθ = 8 × 0.6 = 4.8 cm.

Sector area A = ½r²θ = ½ × 8² × 0.6 = ½ × 64 × 0.6 = 19.2 cm².

Converting: degrees → radians, multiply by π/180. So 135° = 135 × π/180 = 3π/4.

Check the whole circle: put θ = 2π into the formulae. s = 2πr (the circumference ✓) and A = ½r²(2π) = πr² (the area ✓). If a formula survives that test, you have remembered it right.

Calculate

Your turn — arc length

1A sector has radius 8 cm and angle 0.6 radians. Find the arc length.

cm

Hint: s = rθ = 8 × 0.6.

Calculate

Your turn — sector area

2The same sector has radius 8 cm and angle 0.6 radians. Find its area.

cm²

Hint: A = ½r²θ = ½ × 64 × 0.6.

Trigonometry · small angles

Small angle approximations

When θ is small and in radians, the trig functions collapse into polynomials:

sin θ ≈ θ · tan θ ≈ θ · cos θ ≈ 1 − θ²/2these are the first terms of the series — and they are only true in radians

Worked example — simplify (2θ²)/(1 − cos θ) for small θ

1 − cos θ ≈ 1 − (1 − θ²/2) = θ²/2.

∴ (2θ²)/(1 − cos θ) ≈ 2θ² ÷ (θ²/2) = 2θ² × (2/θ²) = 4.

Numerical check with θ = 0.01: cos(0.01) = 0.99995, so 1 − cos θ = 0.00005, and 2(0.0001)/0.00005 = 4.00. ✓

The trap: these fail in degrees. sin(1°) = 0.0175, not 1. Set your calculator to radians.

Calculate

Your turn — small angles

3Use the small angle approximations to find the value that (2θ²) / (1 − cos θ) approaches for small θ.

Hint: cos θ ≈ 1 − θ²/2, so 1 − cos θ ≈ θ²/2. Then 2θ² ÷ (θ²/2).

Trigonometry · identities

The identities

sin²θ + cos²θ ≡ 1 · 1 + tan²θ ≡ sec²θ · 1 + cot²θ ≡ cosec²θsec θ = 1/cos θ · cosec θ = 1/sin θ · cot θ = 1/tan θ = cos θ / sin θ

The two reciprocal identities are not separate facts — divide sin² + cos² = 1 by cos²θ and you get 1 + tan²θ = sec²θ. Divide it by sin²θ instead and you get 1 + cot²θ = cosec²θ.

Naming trap: sec goes with cos (not with sin, despite the s). Remember: the third letter of each pair matches — sec ↔ cos, cosec ↔ sin.

Quick check

Pythagorean identity

?Which identity is correct?

Trigonometry · compound angles

Compound and double angle formulae

sin(A ± B) = sin A cos B ± cos A sin Bcos(A ± B) = cos A cos B ∓ sin A sin B · note the sign FLIPS for cosine

Set B = A and the double angle formulae fall out:

sin 2A = 2 sin A cos A
cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A (three forms — choose whichever kills the term you do not want)
tan 2A = 2 tan A / (1 − tan²A)

Worked example — sin θ = 3/5, θ acute. Find sin 2θ.

3-4-5 triangle ⇒ cos θ = 4/5 = 0.8 (positive, since θ is acute).

sin 2θ = 2 sin θ cos θ = 2 × 0.6 × 0.8 = 0.96.

Calculate

Your turn — double angle

4Given sin θ = 3/5 and θ is acute, find sin 2θ.

Hint: cos θ = 4/5 (3-4-5 triangle, positive because θ is acute). Then sin 2θ = 2 × 0.6 × 0.8.

Quick check

Which form of cos 2A?

?You are solving cos 2θ + 3 sin θ = 2. Which form of cos 2θ should you use?

Trigonometry · R form

The R-harmonic form

Any expression a sin θ + b cos θ can be squeezed into a single trig function — which instantly gives you its maximum, minimum, and how to solve it.

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

Worked example — 3 sin θ + 4 cos θ

R = √(3² + 4²) = √25 = 5.

tan α = 4/3 ⇒ α = 53.1° (1 d.p.), or 0.927 rad.

∴ 3 sin θ + 4 cos θ = 5 sin(θ + 53.1°).

Maximum = 5 (when the sine equals 1); minimum = −5. Check at θ = 0: LHS = 4; RHS = 5 sin(53.1°) = 5 × 0.7997 = 4.00. ✓

Why bother? ‘Solve 3 sin θ + 4 cos θ = 2’ looks impossible — but 5 sin(θ + 53.1°) = 2 is a one-line solve. The R form is the whole point.

Calculate

Your turn — the R form

5Write 3 sin θ + 4 cos θ in the form R sin(θ + α). Find R.

R =

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

Quick check

Degrees to radians

?What is 135° in radians?

Quick check

How many solutions?

?How many solutions does sin θ = 0.5 have in the interval 0 ≤ θ < 2π?

Sort it

Exact values

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

½

√3 / 2

√3

Match it

Name that identity

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

Expression

Equals

Recap

The big ideas to know

Radians: π = 180°; s = rθ and A = ½r²θ require radians

Small angles: sin θ ≈ θ, tan θ ≈ θ, cos θ ≈ 1 − θ²/2 — radians only

Identities: sin² + cos² = 1; 1 + tan² = sec²; 1 + cot² = cosec²

Compound: sin(A ± B) and cos(A ± B) — the cosine sign flips

Double angle: sin 2A = 2 sin A cos A; cos 2A 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

That is OCR 1.05 — and every one of these identities reappears inside integration. Press Finish to see your score.

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