Sin Theta Formula

Sin Theta Formula (sin θ)

The Sin Theta formula is the foundational ratio in trigonometry. It relates the angle of a right-angled triangle to the ratio of its Perpendicular (opposite side) to its Hypotenuse. Mastering the sine formulas, values, and identities is essential for CBSE Class 10 & 11 board exams, as well as JEE and NEET physics.

Class: 10, 11 & 12
Topic: Trigonometry
Exams: CBSE · JEE · NEET

1. Basic Sin Theta Formula (Right-Angled Triangle)

In a right-angled triangle, the sine of an angle θ is defined as the ratio of the length of the Perpendicular (the side opposite to angle θ) to the length of the Hypotenuse (the longest side of the triangle).

Right-Angled Triangle Formula
sin θ = PerpendicularHypotenuse  =  PH
✓ Trick to Remember (NCERT Style)
Indian students often use the mnemonic "Pandit Badri Prasad, Hari Hari Bol" (PBP / HHB). The first fraction P/H corresponds to Sin θ, B/H to Cos θ, and P/B to Tan θ.

2. Sin Theta in terms of Other Trigonometric Ratios

Sin θ can be expressed using other trigonometric identities. These are crucial for solving Class 11 and 12 integration and trigonometry proving questions.

Pythagorean Identity
sin θ = ± √(1 − cos2θ)
Reciprocal Identity: sin θ = 1 / cosec θSine and Cosecant are reciprocals of each other.
Quotient Identity: sin θ = tan θ × cos θDerived from tan θ = sin θ / cos θ.
Complementary Angle: sin θ = cos(90° − θ)Applicable for acute angles in the first quadrant.

3. Sin Theta Value Table (0° to 360°)

Memorizing the standard values of sin θ is mandatory for fast calculation in competitive exams. Notice how the values range purely between -1 and 1.

Angle (θ) in Degrees Angle (θ) in Radians sin θ Value
0° 0 0
30° π / 6 1 / 2
45° π / 4 1 / √2
60° π / 3 √3 / 2
90° π / 2 1
180° π 0
270° 3π / 2 −1
360° 2π 0

4. Advanced Formulas: Sin 2θ, Sin 3θ & Addition

For Class 11 Trigonometry and JEE Mains, the multiple angle and sum/difference formulas are critical:

Double Angle (Sin 2θ)
sin 2θ = 2 sin θ cos θ
Triple Angle (Sin 3θ)
sin 3θ = 3 sin θ − 4 sin3θ
  • Sum Formula: sin(A + B) = sin A cos B + cos A sin B
  • Difference Formula: sin(A − B) = sin A cos B − cos A sin B

Common Mistakes to Avoid

× Mixing up Perpendicular and Base
The "Perpendicular" is always the side opposite to the angle θ you are calculating. If you change the angle to the other acute angle in the triangle, the Perpendicular and Base swap!
× Radians vs. Degrees in Calculators
When calculating sin θ values in physics problems, ensure you know whether θ is given in degrees or radians. Calculating sin(30 radians) instead of sin(30°) is a very common exam error.

Solved Examples on Sin Theta Formula

Example 1: Class 10 Level

In a right-angled triangle, the perpendicular is 3 cm and the base is 4 cm. Find the value of sin θ.

First, find the Hypotenuse (H) using Pythagoras Theorem:
H2 = P2 + B2 = 32 + 42 = 9 + 16 = 25
H = √25 = 5 cm
Now, apply the formula: sin θ = P / H
sin θ = 3 / 5
Answer: sin θ = 0.6

Example 2: Identity Application

If cos θ = 12/13 and θ is an acute angle, find the value of sin θ.

Using the Pythagorean identity: sin2θ + cos2θ = 1
sin2θ = 1 − (12/13)2
sin2θ = 1 − 144/169 = (169 − 144) / 169 = 25/169
Since θ is acute, sin θ is positive.
sin θ = √(25/169)
Answer: sin θ = 5/13

Practice Questions

Test your grasp of trigonometry with these practice problems:

  1. If sin θ = 8/17, find the value of cos θ. (Ans: 15/17)
  2. Evaluate: 2 sin 30° cos 30°. (Ans: √3 / 2)
  3. In triangle ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine sin A. (Ans: 7/25)

Frequently Asked Questions (FAQs)

What is the sin theta formula?
The primary sin theta formula in a right-angled triangle is sin θ = Perpendicular / Hypotenuse (P/H). It relates the opposite side of the angle to the longest side of the triangle.
What is the relationship between sin and cosec?
Sine and cosecant are reciprocals of each other. The formula is sin θ = 1 / cosec θ, meaning if you know one value, you can flip the fraction to find the other.
What are the maximum and minimum values of sin theta?
The value of sin θ always oscillates between a maximum of 1 (at 90°) and a minimum of -1 (at 270°). It can never be greater than 1 or less than -1.
Is sin(-θ) equal to -sin(θ)?
Yes, the sine function is an odd function. This means that sin(-θ) = -sin(θ). For example, sin(-30°) = -sin(30°) = -0.5.