Trigonometric Functions

What are Trigonometric Functions?

Trigonometric functions (or trig ratios) relate the angles of a right triangle to the ratios of its sides. In Class 10, we study six trig functions — sine, cosine, tangent, cosecant, secant, and cotangent — for acute angles ($0° < \theta < 90°$). These functions are the foundation for heights and distances (Class 10), and advanced trigonometry (Class 11-12).

The Six Trigonometric Ratios

For a right triangle with angle $\theta$, hypotenuse $h$, opposite side $p$ (perpendicular), and adjacent side $b$ (base):

Primary ratios:
- $\sin\theta = \frac{p}{h}$ (opposite / hypotenuse)
- $\cos\theta = \frac{b}{h}$ (adjacent / hypotenuse)
- $\tan\theta = \frac{p}{b}$ (opposite / adjacent)

Reciprocal ratios:
- $\csc\theta = \frac{1}{\sin\theta} = \frac{h}{p}$
- $\sec\theta = \frac{1}{\cos\theta} = \frac{h}{b}$
- $\cot\theta = \frac{1}{\tan\theta} = \frac{b}{p}$

Key relationship: $\tan\theta = \frac{\sin\theta}{\cos\theta}$

Standard Angle Values — How to Remember

The quickest way to generate the entire table:

For sin: Write $\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}$ for $0°, 30°, 45°, 60°, 90°$.

Simplifying: $0, \frac{1}{2}, \frac{1}{\sqrt{2}}, \frac{\sqrt{3}}{2}, 1$.

For cos: Reverse the sin row: $1, \frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}, \frac{1}{2}, 0$.

For tan: Divide sin by cos: $0, \frac{1}{\sqrt{3}}, 1, \sqrt{3}, \text{undefined}$.

Complementary angle relationships: $\sin(90° - \theta) = \cos\theta$ and $\cos(90° - \theta) = \sin\theta$. This explains why the cos row is the reverse of the sin row!