Note:

• \(\sin (\theta )= \) sine of theta.
• This definition is very similar to the definition of the tangent ratio, \( \tan(\theta) = \dfrac{\mathrm{opp}}{\mathrm{adj}}\).

Note:

• \(\cos (\theta) = \) cosine of theta.
• Each ratio uses a different pair of sides.

• A sine ratio of \(\dfrac{3}{5}\) can be achieved in multiple ways. A given sine (or cosine ratio) only determines the shape of the triangle. It is not affected by the scale. This is the same result we found for the tangent ratio.
• If \(\sin( \theta )= \dfrac{3}{5}\), then \(\cos (\theta) = \dfrac{4}{5}\) and vice versa. Both ratios specify the same shape.
• Corresponding angles in similar right-angled triangles have the same sine and cosine ratios.

Trigonometry is the study of relationships involving the side lengths and angles of triangles.

In a right-angled triangle, if one angle other than the right angle is known, then the shape of the triangle is fixed. An additional knowledge of one side length fixes the scale and determines the other two side lengths.

The trigonometric ratios sine, cosine, and tangent relate side lengths and angles of triangles.

Each trigonometric ratio is defined as a quotient of two side lengths in a right-angled triangle.

\[\sin( \theta) = \frac{\mathrm{opp}}{\mathrm{hyp}} \qquad \cos (\theta) = \frac{\mathrm{adj}}{\mathrm{hyp}} \qquad \tan( \theta )= \frac{\mathrm{opp}}{\mathrm{adj}} \qquad \]

To help remember these definitions, it is common to form the mnemonic SOH-CAH-TOA.

Consider the following example: