Introduction to trigonometry

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.

History

Trigonometry grew from astronomy, navigation, and surveying. Mathematicians in ancient Greece, India, and the Islamic Golden Age developed angle and chord/sine tables for practical measurement.

Derivation and reasoning

Why trig ratios depend only on angle

Any two right triangles with the same acute angle are similar, so corresponding side ratios are equal. That is why \(\sin\theta,\cos\theta,\tan\theta\) are functions of \(\theta\) only.

Checkpoints

Applications

• Surveying: height and distance estimation without direct measurement.
• Navigation: direction and angle-based positioning.
• Engineering: force components and incline analysis.

References

• Standard high-school trigonometry curricula (right-triangle ratios).
• Geometry and trigonometry texts on SOH-CAH-TOA applications.