Trigonometric Identities and Equations

Trigonometric Identities and Equations

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable. At A-Level you must be able to prove, manipulate and apply these identities to solve equations.

Fundamental Identities

The two Pythagorean identities you must know are:

• sin squared plus cos squared equals one: sin²θ + cos²θ = 1
• tan theta equals sin theta over cos theta: tanθ = sinθ / cosθ

From the first identity, dividing through by cos²θ gives 1 + tan²θ = sec²θ, and dividing through by sin²θ gives 1 + cot²θ = cosec²θ.

These derived identities are essential for simplifying expressions involving reciprocal trigonometric functions.

Double Angle Formulae

The double angle formulae express trig functions of 2A in terms of A:

• sin2A = 2sinAcosA
• cos2A = cos²A - sin²A (this has three equivalent forms)
• cos2A = 2cos²A - 1
• cos2A = 1 - 2sin²A
• tan2A = 2tanA / (1 - tan²A)

Worked Example: Solve sin2θ = sinθ for 0 ≤ θ < 360 degrees.

Step 1: Replace sin2θ using the double angle formula: 2sinθcosθ = sinθ

Step 2: Rearrange: 2sinθcosθ - sinθ = 0

Step 3: Factorise: sinθ(2cosθ - 1) = 0

Step 4: Either sinθ = 0, giving θ = 0, 180; or cosθ = 1/2, giving θ = 60, 300.

Solutions: θ = 0, 60, 180, 300 degrees.

Addition Formulae

The addition (compound angle) formulae are:

• sin(A ± B) = sinAcosB ± cosAsinB
• cos(A ± B) = cosAcosB ∓ sinAsinB
• tan(A ± B) = (tanA ± tanB) / (1 ∓ tanAtanB)

Note the sign pattern: in the cosine formula, the sign in the result is opposite to the sign in the argument.

Worked Example: Find the exact value of sin75°.

sin75° = sin(45° + 30°) = sin45°cos30° + cos45°sin30°

= (√2/2)(√3/2) + (√2/2)(1/2)

= (√6 + √2) / 4

The R-formula (Harmonic Form)

Expressions of the form asinθ + bcosθ can be written as Rsin(θ + α) where:

• R = √(a² + b²)
• tanα = b/a (for Rsin(θ + α)) or tanα = a/b (for Rcos(θ - α))

This is used to find the maximum and minimum values of combined trig expressions and to solve equations.

Worked Example: Write 3sinθ + 4cosθ in the form Rsin(θ + α).

Step 1: R = √(3² + 4²) = √25 = 5

Step 2: tanα = 4/3, so α = 53.13° (to 2 d.p.)

Result: 3sinθ + 4cosθ = 5sin(θ + 53.13°)

The maximum value is 5 (when θ + 53.13° = 90°) and the minimum value is -5.

Solving Trigonometric Equations

When solving trig equations over a given interval:

1. Reduce the equation to a single trig function using identities

2. Find the principal value using inverse trig functions

3. Use the CAST diagram or symmetry to find all solutions in range

4. If the argument is modified (e.g. 2θ or θ + 30°), adjust the interval accordingly

Key technique: If the equation involves sin²θ and cosθ (or similar mixes), use sin²θ = 1 - cos²θ to convert everything to one function, then solve the resulting quadratic.

Small Angle Approximations

For small angles measured in radians:

• sinθ ≈ θ
• cosθ ≈ 1 - θ²/2
• tanθ ≈ θ

These are useful for simplifying expressions when θ is close to zero.

Exam Tips

• Always check your interval and whether the question asks for degrees or radians.
• When using double angle formulae, choose the version of cos2A that simplifies the equation best.
• In R-formula questions, state R and α clearly. Marks are awarded for the method.
• Never divide both sides of an equation by sinθ or cosθ — you lose solutions where that function equals zero. Always factorise instead.
• Show every step when finding all solutions in a given range; examiners check that you have not missed any.