Summary (Grade 12 NSC Matric Mathematics): Revision Notes

Summary

Fundamental identities

Trigonometry is built on several core relationships that connect the basic trigonometric functions. Understanding these fundamental identities is essential for solving complex trigonometric problems.

The Pythagorean identities form the foundation of trigonometric relationships. The most important identity states that the sum of the squares of sine and cosine for any angle equals one: $\cos^2\theta + \sin^2\theta = 1$. This relationship comes directly from the Pythagorean theorem applied to the unit circle.

From this fundamental relationship, we can derive two additional forms:

• $\cos^2\theta = 1 - \sin^2\theta$ (useful when you know sine and need cosine)
• $\sin^2\theta = 1 - \cos^2\theta$ (useful when you know cosine and need sine)

The ratio identities show how tangent relates to sine and cosine. These relationships are:

• $\tan\theta = \frac{\sin\theta}{\cos\theta}$ (tangent equals sine divided by cosine)
• $\frac{\cos\theta}{\sin\theta} = \frac{1}{\tan\theta}$ (the reciprocal relationship)

These identities allow you to convert between different trigonometric functions and are particularly useful in simplifying complex expressions.

Special angle values

Certain angles appear frequently in trigonometry problems, and knowing their exact values is crucial for efficient problem-solving.

Two special right triangles provide the foundation for these values:

The 30-60-90 triangle has sides in the ratio $1 : \sqrt{3} : 2$. In this triangle:

• The side opposite to 30° has length 1
• The side opposite to 60° has length $\sqrt{3}$
• The hypotenuse has length 2

The 45-45-90 triangle has sides in the ratio $1 : 1 : \sqrt{2}$. In this triangle:

• Both legs have length 1
• The hypotenuse has length $\sqrt{2}$

These triangles give us the exact trigonometric values for the standard angles:

For 0°: $\cos 0° = 1$, $\sin 0° = 0$, $\tan 0° = 0$

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

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

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

For 90°: $\cos 90° = 0$, $\sin 90° = 1$, $\tan 90°$ = undefined

These values are essential for solving trigonometric equations without a calculator.

Quadrant analysis and reduction formulae

Understanding how trigonometric functions behave in different quadrants is crucial for solving problems involving angles greater than 90°.

Each quadrant has specific reduction formulae that help you find the trigonometric values of any angle:

First Quadrant (0° to 90°):

• $\sin(360° + \theta) = \sin\theta$
• $\cos(360° + \theta) = \cos\theta$
• $\tan(360° + \theta) = \tan\theta$

Second Quadrant (90° to 180°):

• $\sin(180° - \theta) = \sin\theta$
• $\cos(180° - \theta) = -\cos\theta$
• $\tan(180° - \theta) = -\tan\theta$

Third Quadrant (180° to 270°):

• $\sin(180° + \theta) = -\sin\theta$
• $\cos(180° + \theta) = -\cos\theta$
• $\tan(180° + \theta) = \tan\theta$

Fourth Quadrant (270° to 360°):

• $\sin(360° - \theta) = -\sin\theta$
• $\cos(360° - \theta) = \cos\theta$
• $\tan(360° - \theta) = -\tan\theta$

Additionally, the co-function relationships are:

• $\sin(90° - \theta) = \cos\theta$
• $\cos(90° - \theta) = \sin\theta$
• $\sin(90° + \theta) = \cos\theta$
• $\cos(90° + \theta) = -\sin\theta$

Function properties

Trigonometric functions have specific behaviours that make calculations more predictable and help in solving equations.

Negative angle identities show how functions respond to negative inputs:

• $\sin(-\theta) = -\sin\theta$ (sine is an odd function)
• $\cos(-\theta) = \cos\theta$ (cosine is an even function)
• $\tan(-\theta) = -\tan\theta$ (tangent is an odd function)

Periodicity identities describe how trigonometric functions repeat their values:

• $\sin(\theta ± 360°) = \sin\theta$ (sine repeats every 360°)
• $\cos(\theta ± 360°) = \cos\theta$ (cosine repeats every 360°)
• $\tan(\theta ± 180°) = \tan\theta$ (tangent repeats every 180°)

Cofunction identities relate trigonometric functions at complementary angles:

• $\sin(90° - \theta) = \cos\theta$
• $\cos(90° - \theta) = \sin\theta$
• $\sin(90° + \theta) = \cos\theta$
• $\cos(90° + \theta) = -\sin\theta$

These properties are essential for simplifying expressions and solving trigonometric equations efficiently.

Triangle trigonometry

When working with triangles that are not right-angled, specific rules help us find unknown sides and angles.

Consider a triangle with vertices A, B, and C, where:

• Side $a$ is opposite angle $A$
• Side $b$ is opposite angle $B$
• Side $c$ is opposite angle $C$

Area rule gives us three ways to calculate a triangle's area using trigonometry:

• $\text{Area} = \frac{1}{2}bc \sin A$
• $\text{Area} = \frac{1}{2}ab \sin C$
• $\text{Area} = \frac{1}{2}ac \sin B$

These formulae are particularly useful when you know two sides and the included angle.

Sine rule establishes proportional relationships in any triangle: $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

This can also be written as:

• $a \sin B = b \sin A$
• $b \sin C = c \sin B$
• $a \sin C = c \sin A$

The sine rule is most useful when you know two angles and one side, or two sides and an angle opposite one of them.

Cosine rule extends the Pythagorean theorem to non-right triangles:

• $a^2 = b^2 + c^2 - 2bc \cos A$
• $b^2 = a^2 + c^2 - 2ac \cos B$
• $c^2 = a^2 + b^2 - 2ab \cos C$

The cosine rule is essential when you know three sides and need to find an angle, or when you know two sides and the included angle.

Advanced identities

Complex trigonometric problems often require compound angle and double angle identities.

Compound angle identities help us find trigonometric values for sums and differences of angles:

For sine:

• $\sin(\theta + \beta) = \sin\theta \cos\beta + \cos\theta \sin\beta$
• $\sin(\theta - \beta) = \sin\theta \cos\beta - \cos\theta \sin\beta$

For cosine:

• $\cos(\theta + \beta) = \cos\theta \cos\beta - \sin\theta \sin\beta$
• $\cos(\theta - \beta) = \cos\theta \cos\beta + \sin\theta \sin\beta$

Double angle identities are special cases where $\beta = \theta$:

For sine: $\sin(2\theta) = 2 \sin\theta \cos\theta$

For cosine (multiple forms):

• $\cos(2\theta) = \cos^2\theta - \sin^2\theta$
• $\cos(2\theta) = 1 - 2\sin^2\theta$
• $\cos(2\theta) = 2\cos^2\theta - 1$

For tangent: $\tan(2\theta) = \frac{2 \tan\theta}{1 - \tan^2\theta}$

These identities are crucial for solving complex trigonometric equations and simplifying expressions involving multiple angles.