Geometry: Trigonometry

This section covers trigonometric ratios, the Pythagorean theorem, and the sine and cosine rules. These are fundamental tools for solving problems involving angles and sides of triangles.

Trigonometric Ratios

Sine, Cosine, and Tangent

For a right-angled triangle with angle $\theta$, the trigonometric ratios are defined as follows:

• Sine (sin): $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
• Cosine (cos): $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
• Tangent (tan): $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

These ratios relate the angles of a right-angled triangle to the lengths of its sides.

Trigonometric Ratios for Common Angles

Here's a table of common trigonometric ratios:

Angle (θ)	0°	30°	45°	60°	90°
sin(θ)	0	1/2	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	1
cos(θ)	1	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	1/2	0
tan(θ)	0	$\frac{1}{\sqrt{3}}$	1	$\sqrt{3}$	undefined

Pythagoras Theorem

The Pythagorean theorem applies to right-angled triangles. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Mathematically, this is expressed as: $a^2 + b^2 = c^2$, where 'c' is the hypotenuse and 'a' and 'b' are the other two sides.

Sine and Cosine Rules

The sine and cosine rules are used to solve triangles when you know the lengths of two sides and the included angle (SAS) or when you know the lengths of all three sides (SSS).

Sine Rule

The sine rule states that in any triangle:

$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

where 'a', 'b', and 'c' are the sides opposite angles 'A', 'B', and 'C' respectively.

Cosine Rule

The cosine rule relates the sides of a triangle to one of its angles.

For the case where you know all three sides (SSS):

$c^2 = a^2 + b^2 - 2ab \cos(C)$

You can rearrange this to find the cosine of any angle:

$ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} $

For the case where you know two sides and the included angle (SAS):

$c^2 = a^2 + b^2 - 2ab \cos(C)$