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

Find \(\sin30^\circ\) and \(\cos60^\circ\).

Answer: Both are \(\frac12\).

In a right triangle, opposite to \(\theta\) is 9 and hypotenuse is 15. Find \(\sin\theta\).

Answer: \(\sin\theta=\frac{9}{15}=\frac35\).

Adjacent is 7, hypotenuse is 25. Find \(\cos\theta\).

Answer: \(\cos\theta=\frac{7}{25}\).

If \(\tan\theta=0.75\), find \(\theta\) (to 1 d.p.).

Answer: \(\theta=\tan^{-1}(0.75)\approx 36.9^\circ\).

A ladder leans against a wall. Foot is 4 m from wall and ladder length is 6 m. Find angle with ground.

Answer: \(\cos\theta=\frac{4}{6}=\frac23\Rightarrow \theta\approx 48.2^\circ\).

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.