# Trigonometric Ratios

• Trigonometry deals with the relationship between the angles and sides of a triangle.
• It’s most commonly used with right-angled triangles, where one angle is 90 degrees.

## Definitions and Ratios

• The three main trigonometric ratios are: sine (sin), cosine (cos), and tangent (tan).
• In any right-angle triangle, the hypotenuse is the longest side opposite the right angle, the opposite is the side opposite the angle under consideration, and the adjacent is the side lying next to (or adjoining) the angle under consideration.
• Sine of an angle (θ) is calculated as the ratio of the length of the side opposite the angle (O) to the length of the hypotenuse (H): sin θ = O/H.
• Cosine of an angle (θ) is calculated as the ratio of the length of the adjacent side (A) to the length of the hypotenuse (H): cos θ = A/H.
• Tangent of an angle (θ) is calculated as the ratio of the opposite side (O) to the adjacent side (A): tan θ = O/A.

## The Sine Rule and Cosine Rule

• The Sine Rule states that the ratio of a side of a triangle to the sine of its opposite angle is the same for all three sides. It helps you to find missing sides/angles in a triangle if you know one angle and two sides, or two angles and one side.
• The Cosine Rule relates the lengths of the sides of a triangle to the cosine of one of the angles. Use the Cosine Rule when you know the lengths of two sides and the size of the enclosed angle, or the lengths of three sides and you want to find one of the angles.

## Simple Trigonometric Equations

• Trigonometric equations typically involve the unknown number being inside the trigonometric function (like `sin x = 0.5`), meaning you need to manipulate the equation a bit differently.
• You can solve trigonometric equations by using graphs and exact values, using trigonometric identities, or using the ‘CAST’ diagram.