Mini-Lesson

Geometry & Trigonometry

This HL mini-lesson extends trigonometry: the reciprocal ratios (sec, csc, cot), the three Pythagorean identities, the compound-angle and double-angle formulae, harder equations, and inverse trig functions.

Answer as you go and collect ⭐ stars. Press Start when ready.

Reciprocal ratios

Reciprocal trigonometric ratios

Three further ratios are the reciprocals of the familiar ones:

sec θ = 1/cos θ · cosec θ = 1/sin θ · cot θ = 1/tan θ

Quick check

Reciprocal ratio

?cosec θ is defined as:

Pythagorean identities

The Pythagorean identities

Dividing sin²θ + cos²θ = 1 by cos²θ or sin²θ gives two more identities:

sin²θ + cos²θ = 11 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ

Calculate

Compound angle

✎Using sin75° = sin(45° + 30°), find sin75° to 4 decimal places.

Hint: sin45°cos30° + cos45°sin30° = (√6 + √2)/4.

Compound angles

Compound-angle formulae

These let you split or combine angles:

sin(A ± B) = sinA cosB ± cosA sinBcos(A ± B) = cosA cosB ∓ sinA sinB

Worked example

sin75° = sin(45°+30°) = sin45°cos30° + cos45°sin30° = (√6 + √2)/4 ≈ 0.966.

Quick check

Pythagorean identity

?Which identity is correct?

Calculate

Double angle (sin)

✎θ is acute with sinθ = 3/5 and cosθ = 4/5. Find sin 2θ.

Hint: sin 2θ = 2 sinθ cosθ = 2(3/5)(4/5).

Double angles

Double-angle formulae

Setting B = A gives the double-angle formulae:

sin 2θ = 2 sinθ cosθcos 2θ = 2cos²θ − 1 = 1 − 2sin²θ = cos²θ − sin²θ

Example: if sinθ = 3/5, cosθ = 4/5 then sin2θ = 2(3/5)(4/5) = 24/25 = 0.96.

Sort it

Reciprocal, Pythagorean or double-angle?

Tap an identity, then its family.

🔁 Reciprocal ratio

△ Pythagorean identity

✌️ Double-angle

Equations

Solving trigonometric equations

Use identities to reduce an equation to a single ratio, then find all solutions in the given interval.

Watch: a quadratic in sin θ (say) may have two values of sin θ, each giving two angles per revolution.

Calculate

Double angle (cos)

✎With sinθ = 3/5, find cos 2θ using cos 2θ = 1 − 2sin²θ.

Hint: 1 − 2(3/5)² = 1 − 18/25.

Quick check

Double angle

?Which is a correct form of cos 2θ?

Inverse trig

Inverse trigonometric functions

arcsin, arccos and arctan invert the ratios on restricted domains so they are one-to-one.

arcsin: range [−π/2, π/2] · arccos: [0, π] · arctan: (−π/2, π/2)

Calculate

Reciprocal ratio

✎If cos θ = 0.5, find sec θ.

Hint: sec θ = 1/cos θ = 1/0.5.

Match it

Match each expression to its identity

Tap a statement on the left, then its matching partner on the right.

Expression

Equals

Modelling

Modelling with trigonometry

Periodic phenomena use y = a sin(b(x − c)) + d, with amplitude |a|, period 2π/b, horizontal shift c and vertical shift d.

Calculate

Pythagorean identity

✎Given tan θ = 2, find sec²θ using 1 + tan²θ = sec²θ.

Hint: 1 + 2².

Quick check

Compound angle

?sin(A + B) expands to:

Strategy

Choosing an identity

To simplify, aim for a single ratio: swap sec/cosec/cot for 1/cos, 1/sin, 1/tan, and use a Pythagorean identity to remove squares. For 2θ terms, expand with the double-angle formulae.

Exam habit: always list every solution in the stated interval, not just the principal value.

Recap

The big ideas to know

Reciprocals: sec=1/cos, cosec=1/sin, cot=1/tan

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

Compound: sin(A±B)=sinAcosB±cosAsinB

Double angle: sin2θ=2sinθcosθ; cos2θ=2cos²θ−1

Inverse trig: arcsin/arccos/arctan on restricted ranges

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