Trigonometry Study Guide — AQA A-Level

Overview

Trigonometry at A-Level is a significant step up from GCSE. It forms the mathematical backbone for describing any periodic, wave-like phenomena, from sound and light to alternating current and harmonic motion. For the AQA specification, your fluency in this topic is critical, as it integrates heavily with algebra, calculus, and geometric reasoning. Examiners will test your ability not just to solve equations, but to construct rigorous proofs and apply trigonometric models to abstract and real-world problems. A typical exam question might ask you to prove a complex identity, solve an equation involving reciprocal functions for a specific range, or find the maximum displacement of a modelled tide. Mastering the content in this guide will equip you to handle these challenges with confidence.

Key Concepts

Concept 1: The Unit Circle and Radian Measure

The unit circle is your new best friend. It's a circle with a radius of 1 centered at the origin, and it provides a visual foundation for all trigonometric functions. For any point (x, y) on the circle, the angle ( heta ) (measured in radians counter-clockwise from the positive x-axis) is linked to the coordinates by ( x = \cos( heta) ) and ( y = \sin( heta) ). This is why ( \cos^2 heta + \sin^2 heta = 1 ) – it's just Pythagoras's theorem on the circle! Radians are the natural unit for measuring angles at this level, especially in calculus. Remember that ( 2\pi ) radians is a full circle (360°).

Example: To find the exact value of (\cos(5\pi/4) ), you would locate the angle on the unit circle. It lies in the third quadrant, where both x and y are negative. The reference angle is (\pi/4 ), and (\cos(\pi/4) =\sqrt{2}/2 ). Therefore, (\cos(5\pi/4) = -\sqrt{2}/2 ).

Concept 2: Reciprocal and Pythagorean Identities

Beyond sine, cosine, and tangent, you must master the reciprocal functions: secant (sec), cosecant (cosec), and cotangent (cot).

( \sec(x) = 1 /\cos(x) )
( \csc(x) = 1 /\sin(x) ) (often written as cosec)
( \cot(x) = 1 / an(x) =\cos(x) /\sin(x) )

These lead to two more essential Pythagorean identities, derived by dividing ( \sin^2 heta + \cos^2 heta = 1 ) by ( \cos^2 heta ) and ( \sin^2 heta ) respectively:

( 1 + an^2 heta = \sec^2 heta )
( 1 + \cot^2 heta = \csc^2 heta )

Examiners award marks for using these identities to simplify equations into a single trigonometric function.

Concept 3: Compound and Double Angle Formulas

These formulas (which you must memorise) are used to expand expressions like ( \sin(A+B) ) or ( \cos(2A) ). They are fundamental for solving more complex equations and for calculus involving trigonometric functions.

Compound Angle Formulas:

(\sin(A \pm B) =\sin A \cos B \pm \cos A \sin B)
(\cos(A \pm B) =\cos A \cos B \mp \sin A \sin B) (Note the sign flip!)
( an(A \pm B) = (an A \pm an B) / (1 \mp an A an B) )

Double Angle Formulas (derived by setting A=B):

(\sin(2A) =2\sin A \cos A)
(\cos(2A) =\cos^2 A -\sin^2 A =2\cos^2 A - 1 = 1 -2\sin^2 A )
( an(2A) = 2 an A / (1 - an^2 A) )

Concept 4: Harmonic Form (R-alpha method)

Expressions of the form ( a\cos x + b\sin x ) can be written as a single sinusoidal function, ( R\cos(x - \alpha) ) or ( R\sin(x + \alpha) ). This is crucial for finding maximum/minimum values and solving equations.

To express ( a\cos x + b\sin x ) in the form ( R\cos(x - \alpha) ):

( R =\sqrt{a^2 + b^2} )
Expand (R\cos(x -\alpha) = R(\cos x \cos \alpha+\sin x \sin \alpha) )
Compare coefficients: ( a =R\cos \alpha) and ( b =R\sin \alpha). This leads to (an \alpha= b/a ).

Example: The maximum value of (3\cos x+4\sin x) is ( R =\sqrt{3^2 + 4^2} = 5 ), because the maximum value of the cosine function is 1.

Mathematical Relationships

Formula	Status	Use Case
( \sin^2 heta + \cos^2 heta = 1 )	Must memorise	The fundamental identity. Used in almost all proofs and equation solving.
( 1 + an^2 heta = \sec^2 heta )	Must memorise	Solving equations involving tan and sec.
( 1 + \cot^2 heta = \csc^2 heta )	Must memorise	Solving equations involving cot and cosec.
Compound Angle Formulas	Must memorise	Expanding expressions, proofs, calculus.
Double Angle Formulas	Must memorise	Simplifying expressions, solving equations, integration.
Small Angle Approximations: ( \sin heta \approx heta ), ( \cos heta \approx 1 - heta^2/2 ), ( an heta \approx heta )	Given on formula sheet	Used for finding limits and approximating values for small angles (in RADIANS).
Harmonic Form: ( a\cos x + b\sin x = R\cos(x-\alpha) )	Must memorise method	Finding max/min values, solving equations, modelling waves.

Practical Applications

Trigonometry is essential in many fields:

Physics: Modelling simple harmonic motion (e.g., a pendulum), analysing AC circuits, and describing wave phenomena like light and sound.
Engineering: Designing structures, calculating forces in mechanics, and in signal processing.
Computer Graphics: Rotating and transforming objects in 2D and 3D space.
Geography & Navigation: Calculating distances and positions using triangulation.
"

Solve the equation ( 2\cos^2 x + \sin x = 1 ) for ( 0 \le x < 2\pi ). Give your answers in terms of ( \pi ).",
"marks": 5,
"solution": "Step 1: The equation involves both cos and sin. We must use an identity to express it in terms of a single function. Using the identity ( \cos^2 x = 1 - \sin^2 x ):
( 2(1 - \sin^2 x) + \sin x = 1 )

Step 2: Expand and rearrange into a quadratic equation in ( \sin x ):
( 2 - 2\sin^2 x + \sin x = 1 )
( 0 = 2\sin^2 x - \sin x - 1 )

Step 3: Solve the quadratic. Let ( y = \sin x ). ( 2y^2 - y - 1 = 0 ). This factorises to ( (2y+1)(y-1) = 0 ).
So, ( y = -1/2 ) or ( y = 1 ).
This means ( \sin x = -1/2 ) or ( \sin x = 1 ).

Step 4: Find the principal values and all solutions in the range ( 0 \le x < 2\pi ).
For ( \sin x = 1 ), the principal value is ( x = \arcsin(1) = \pi/2 ). This is the only solution in the range.
For ( \sin x = -1/2 ), the principal value is ( x = \arcsin(-1/2) = -\pi/6 ). This is outside the range. We use the symmetry of the sine graph (or the unit circle) to find solutions in the range. The solutions are in the 3rd and 4th quadrants. ( x = \pi + \pi/6 = 7\pi/6 ) and ( x = 2\pi - \pi/6 = 11\pi/6 ).

Final answer: ( x = \pi/2, 7\pi/6, 11\pi/6 )",
"examiner_commentary": "This is a standard question style that earns full marks for a systematic approach. M1 is awarded for correctly substituting the Pythagorean identity. A1 for forming a correct quadratic equation. M1 for a valid attempt to solve the quadratic. A1 for finding at least two correct solutions. A1 for all three correct solutions and no extras. A common error is failing to find all secondary solutions for ( \sin x = -1/2 ). Sketching the sine wave is a good way to ensure all solutions are found."

Trigonometry

Study Notes

Overview

Key Concepts

Concept 1: The Unit Circle and Radian Measure

Concept 2: Reciprocal and Pythagorean Identities

Concept 3: Compound and Double Angle Formulas

Concept 4: Harmonic Form (R-alpha method)

Mathematical Relationships

Practical Applications

Worked Examples

Practice Questions

In this Guide

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