Trigonometry relates angles and side lengths in right triangles. The three core ratios are sine, cosine, and tangent.

Right-triangle ratios

For angle \(\theta\) in a right triangle:

\[ \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\quad \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\quad \tan\theta=\frac{\text{opposite}}{\text{adjacent}}. \]

Remember: opposite/adjacent depend on the chosen angle.

Common exact values

For \(30^\circ,45^\circ,60^\circ\):

• \(\sin30^\circ=\frac12,\ \cos30^\circ=\frac{\sqrt3}{2},\ \tan30^\circ=\frac1{\sqrt3}\)
• \(\sin45^\circ=\cos45^\circ=\frac{\sqrt2}{2},\ \tan45^\circ=1\)
• \(\sin60^\circ=\frac{\sqrt3}{2},\ \cos60^\circ=\frac12,\ \tan60^\circ=\sqrt3\)

Finding missing sides

Example. Hypotenuse \(=10\), angle \(=37^\circ\). Find opposite side \(x\):

\[ \sin37^\circ=\frac{x}{10}\Rightarrow x=10\sin37^\circ\approx 6.02. \]

Finding missing angles

Example. Opposite \(=5\), adjacent \(=12\). Find \(\theta\):

\[ \tan\theta=\frac{5}{12}\Rightarrow \theta=\tan^{-1}\!\left(\frac{5}{12}\right)\approx 22.6^\circ. \]

Angles of elevation and depression

Angle of elevation looks upward from horizontal; angle of depression looks downward. Draw a clear diagram and label horizontal lines before forming trig equations.

Link with Pythagoras

In right triangles, trigonometry and Pythagoras work together: first find one side/angle, then use the other method to complete the triangle.

Exam strategy

• Choose ratio from known and required sides (SOH-CAH-TOA).
• Set calculator mode to degrees unless stated otherwise.
• Round only at final step unless instructed.
• Include units and context in word-problem answers.