Trigonometry — Trigonometric Ratios and Identities

Overview

Trigonometry is a fundamental branch of mathematics that deals with the relationships between angles and sides of triangles, particularly right-angled triangles. For the JKTET Paper II, this topic forms a crucial component of the mathematics section and tests your understanding of basic trigonometric ratios, their relationships, and standard identities.

Mastery of trigonometry is essential because it connects geometry with algebra and has practical applications in measurement, surveying, and physics problems. Questions typically test your ability to calculate ratio values for standard angles, apply identities to simplify expressions, and prove trigonometric relationships. Students who thoroughly understand the six ratios and the three fundamental identities can handle most exam questions with confidence.

The scope for JKTET is limited to right-triangle trigonometry and standard identities — you are not expected to deal with trigonometric equations, graphs, or inverse functions at this level.

Key Concepts

• **Right-angled triangle reference**: All six trigonometric ratios are defined with respect to an acute angle (θ) in a right-angled triangle, using three sides — opposite, adjacent, and hypotenuse.

• **Six trigonometric ratios**: sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot). The last three are reciprocals of the first three.

• **Complementary angle relationship**: The trigonometric ratio of an angle equals the co-ratio of its complement. For example, sin(90° − θ) = cos θ.

• **Standard angles**: You must memorise exact values for 0°, 30°, 45°, 60°, and 90°. These appear repeatedly in calculations.

• **Pythagorean identities**: Three fundamental identities derived from the Pythagorean theorem connect the squares of ratios.

• **Ratio relationships**: tan θ = sin θ / cos θ and cot θ = cos θ / sin θ. These help convert between ratios.

• **Range restrictions**: sin θ and cos θ always lie between −1 and 1. For acute angles in a right triangle, all six ratios are positive.

Formulas / Key Facts

### Definitions of Six Ratios (for acute angle θ in right triangle)

| Ratio | Formula | |-------|---------| | sin θ | Opposite / Hypotenuse | | cos θ | Adjacent / Hypotenuse | | tan θ | Opposite / Adjacent | | cosec θ | Hypotenuse / Opposite = 1 / sin θ | | sec θ | Hypotenuse / Adjacent = 1 / cos θ | | cot θ | Adjacent / Opposite = 1 / tan θ |

### Standard Angle Values

| Angle | sin | cos | tan | |-------|-----|-----|-----| | 0° | 0 | 1 | 0 | | 30° | 1/2 | √3/2 | 1/√3 | | 45° | 1/√2 | 1/√2 | 1 | | 60° | √3/2 | 1/2 | √3 | | 90° | 1 | 0 | undefined |