📚 Concepts

📚 Trigonometry

Trigonometry is the branch of mathematics that deals with the relationship between angles and sides of right-angled triangles.

📚 Right-Angled Triangle

Trigonometric concepts apply only to right-angled triangles — triangles with one angle exactly equal to 90°.

📚 Trigonometric Ratio

A trigonometric ratio is the ratio of two sides of a right-angled triangle with respect to an acute angle (𝜃). These ratios are used to calculate unknown sides or angles.

📚 Parts of a Right-Angled Triangle

• Opposite side (Perpendicular): The side opposite the angle 𝜃

• Adjacent side (Base): The side next to angle 𝜃

• Hypotenuse: The side opposite the right angle; longest side

📚 Trigonometric Ratios Names

There are six standard trigonometric ratios:

• sin 𝜃 = Opposite ⁄ Hypotenuse

• cos 𝜃 = Adjacent ⁄ Hypotenuse

• tan 𝜃 = Opposite ⁄ Adjacent

• cosec 𝜃, sec 𝜃, and cot 𝜃 are their reciprocals

📚 Reciprocal Ratios

Each of the primary trigonometric ratios has a reciprocal:

• cosec 𝜃 = 1 ⁄ sin 𝜃

• sec 𝜃 = 1 ⁄ cos 𝜃

• cot 𝜃 = 1 ⁄ tan 𝜃

  
  

📚 Quotient Identities

• tan 𝜃 = sin 𝜃 ⁄ cos 𝜃

• cot 𝜃 = cos 𝜃 ⁄ sin 𝜃

📚 Range of Values (For 0° < 𝜃 < 90°)

• 0 < sin 𝜃 ≤ 1

• 0 < cos 𝜃 ≤ 1

• 0 < tan 𝜃 < ∞

• cot 𝜃 > 0

• sec 𝜃 ≥ 1

• cosec 𝜃 ≥ 1

📚 Trigonometric Ratios of Standard Angles

The values of trigonometric ratios for angles 0°, 30°, 45°, 60°, and 90° are commonly used and form the trigonometric table.

✍️ Formulas

✍️ Basic Trigonometric Ratios 

In ∆ABC, right-angled at B, angle 𝜃 at A

• 𝑠𝑖𝑛 𝜃 = 𝐵𝐶 ⁄ 𝐴𝐶

• 𝑐𝑜𝑠 𝜃 = 𝐴𝐵 ⁄ 𝐴𝐶

• 𝑡𝑎𝑛 𝜃 = 𝐵𝐶 ⁄ 𝐴𝐵

✍️ Reciprocal Ratios

• 𝑐𝑜𝑠𝑒𝑐 𝜃 = 𝐴𝐶 ⁄ 𝐵𝐶 = 1 ⁄ 𝑠𝑖𝑛 𝜃

• 𝑠𝑒𝑐 𝜃 = 𝐴𝐶 ⁄ 𝐴𝐵 = 1 ⁄ 𝑐𝑜𝑠 𝜃

• 𝑐𝑜𝑡 𝜃 = 𝐴𝐵 ⁄ 𝐵𝐶 = 1 ⁄ 𝑡𝑎𝑛 𝜃

✍️ Quotient Identities

• 𝑡𝑎𝑛 𝜃 = 𝑠𝑖𝑛 𝜃 ⁄ 𝑐𝑜𝑠 𝜃

• 𝑐𝑜𝑡 𝜃 = 𝑐𝑜𝑠 𝜃 ⁄ 𝑠𝑖𝑛 𝜃

✍️ Fundamental Trigonometric Identities

• 𝑠𝑖𝑛² 𝜃 + 𝑐𝑜𝑠² 𝜃 = 1

• 1 + 𝑡𝑎𝑛² 𝜃 = 𝑠𝑒𝑐² 𝜃

• 1 + 𝑐𝑜𝑡² 𝜃 = 𝑐𝑜𝑠𝑒𝑐² 𝜃

✍️ Trigonometric Table