Trigonometric identities Revision Notes for AQA GCSE Further Maths

Trigonometric identities

What are trigonometric identities?

Trigonometric identities are mathematical equations involving trigonometric functions that remain true for all possible values of the variables involved. These powerful relationships allow us to transform complex trigonometric expressions into simpler forms and solve equations that would otherwise be very difficult.

The beauty of trigonometric identities lies in their universal truth - they work for any angle, whether it's measured in degrees or radians. Understanding these identities is crucial for solving advanced trigonometric problems and forms the foundation for many areas of higher mathematics.

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Trigonometric identities are different from regular equations because they are true for ALL values of the variable, not just specific solutions. This is why we often use the equivalence symbol (≡) instead of the equals sign (=).

The fundamental identities

The Pythagorean identity

The most important trigonometric identity comes directly from Pythagoras' theorem applied to the unit circle. When we consider a point P with coordinates $(\cos \theta, \sin \theta)$ on a circle of radius 1, we can see that:

$\sin^2\theta + \cos^2\theta = 1$

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This identity is incredibly versatile because it can be rearranged into different forms depending on what we need:

$\sin^2\theta = 1 - \cos^2\theta$
$\cos^2\theta = 1 - \sin^2\theta$

These rearranged forms are particularly useful when we need to eliminate one trigonometric function in favour of another.

The tangent identity

The second fundamental identity defines tangent in terms of sine and cosine:

$\tan \theta = \frac{\sin \theta}{\cos \theta}$

This relationship is essential for converting between different trigonometric functions and simplifying expressions. From this identity, we can also derive:

$\sin \theta = \cos \theta \tan \theta$
$\cos \theta = \frac{\sin \theta}{\tan \theta}$

Using identities to solve equations

Trigonometric identities become powerful tools when solving equations. The key strategy is to use identities to rewrite equations so they involve only one trigonometric function, making them much easier to solve.

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Worked Example: Mixed Function Equation

Let's solve the equation: $2\cos^2\theta - \sin \theta - 1 = 0$ for values of $\theta$ between 0° and 360°.

Step 1: Use the Pythagorean identity to eliminate $\cos^2\theta$ Since $\cos^2\theta = 1 - \sin^2\theta$ , we can substitute: $2(1 - \sin^2\theta) - \sin \theta - 1 = 0$

Step 2: Expand and rearrange $2 - 2\sin^2\theta - \sin \theta - 1 = 0$ $-2\sin^2\theta - \sin \theta + 1 = 0$ $2\sin^2\theta + \sin \theta - 1 = 0$

Step 3: Factor the quadratic expression This factors as: $(2\sin \theta - 1)(\sin \theta + 1) = 0$

Step 4: Solve each factor Either $\sin \theta = \frac{1}{2}$ or $\sin \theta = -1$ This gives us $\theta = 30°, 150°$ , or $270°$

This example shows how the Pythagorean identity transforms a complex mixed equation into a simple quadratic that we can solve using factorisation.

Proving trigonometric identities

When proving identities, we need to show that one side of an equation can be transformed into the other through valid algebraic steps. The key is to work with one side at a time and use known identities to simplify.

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Worked Example: Double Angle Proof

Let's prove that $\cos^2x - \sin^2x \equiv 2\cos^2x - 1$

Starting with the left side: $\cos^2x - \sin^2x$

Using the Pythagorean identity ( $\sin^2x = 1 - \cos^2x$ ): $\cos^2x - (1 - \cos^2x)$ $= \cos^2x - 1 + \cos^2x$
$= 2\cos^2x - 1$

This matches the right side, proving the identity. The equivalence symbol (≡) indicates this relationship is true for all values of $x$ .

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Worked Example: Complex Fraction Simplification

Let's show that $\frac{\tan x \cos x}{\sqrt{1 - \cos^2x}}$ simplifies to 1.

Step 1: Use the tangent identity $\tan x \cos x = \frac{\sin x}{\cos x} \times \cos x = \sin x$

Step 2: Simplify the denominator using the Pythagorean identity $\sqrt{1 - \cos^2x} = \sqrt{\sin^2x} = \sin x$

Step 3: Complete the simplification $\frac{\tan x \cos x}{\sqrt{1 - \cos^2x}} = \frac{\sin x}{\sin x} = 1$

Special angle values

Certain angles have exact trigonometric values that you should memorise, as they appear frequently in problems:

For 30° ( $\frac{\pi}{6}$ radians):

$\sin 30° = \frac{1}{2}$
$\cos 30° = \frac{\sqrt{3}}{2}$
$\tan 30° = \frac{1}{\sqrt{3}}$

For 45° ( $\frac{\pi}{4}$ radians):

$\sin 45° = \frac{1}{\sqrt{2}}$
$\cos 45° = \frac{1}{\sqrt{2}}$
$\tan 45° = 1$

For 60° ( $\frac{\pi}{3}$ radians):

$\sin 60° = \frac{\sqrt{3}}{2}$
$\cos 60° = \frac{1}{2}$
$\tan 60° = \sqrt{3}$

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These values come from the ratios in 30-60-90 and 45-45-90 right triangles, and knowing them will save you significant time in examinations.

Key exam techniques

Strategy for solving trigonometric equations

The following systematic approach will help you tackle any trigonometric equation:

Identify the functions present - Look for sine, cosine, and tangent terms
Choose your target function - Decide which single function to express everything in terms of
Apply appropriate identities - Use the Pythagorean or tangent identities to eliminate unwanted functions
Solve the resulting equation - This often becomes a quadratic equation in your chosen function
Find all solutions in the given range - Remember that trigonometric functions are periodic

Strategy for proving identities

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Key Steps for Proving Identities:

Work with one side only - Never manipulate both sides simultaneously
Start with the more complex side - This usually gives you more options for simplification
Look for opportunities to use fundamental identities - The Pythagorean identity is often key
Factor when possible - Factorisation often reveals the path to the solution
Check your final result - Ensure both sides are identical

Common exam traps to avoid

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Warning: Common Mistakes to Avoid

Don't divide by trigonometric functions without checking they're not zero
Remember the range restrictions - Solutions must fall within the specified interval
Watch out for extraneous solutions - Always verify your answers in the original equation
Be careful with square roots - Remember that $\sqrt{\sin^2x} = |\sin x|$ , not necessarily $\sin x$

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Key Points to Remember:

The Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ is your most powerful tool for eliminating one trigonometric function in favour of another
The tangent identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$ allows you to convert between tangent and the other functions
When solving equations, aim to express everything in terms of a single trigonometric function
For proving identities, work with one side at a time and use algebraic manipulation to transform it into the other side
Memorise the exact values for 30°, 45°, and 60° - they appear constantly in examinations

Trigonometric identities (AQA GCSE Further Maths): Revision Notes