HALF ANGLE FORMULA

Half Angle Formula

Definition Of Half Angle Formula

In Half Angle Formula, trigonometric function of ?/2 is expressed in term of trigonometric functions of ?.

More About Half Angle Formula

Video Examples: Acute and Obtuse Angles in Geometry

Example of Half Angle Formula

Derivations of Half Angle Formula:

Finding Sin θ/2
1 - 2 Sin²θ = cos ²θ [Double-angle formula.]
1 - 2 Sin² θ/2= cos θ [Replace θ by θ/2 and 2θ by θ.]
Sin² θ/2 = 1 - Cosθ /2 [Solve for Sin² θ/2 ]
Sin θ/2 = [Take square root on both the sides.]

Finding Cos θ/2
2 Cos²θ -1 = Cos ²θ [Double-angle formula.]
2 Cos² θ/2 - 1 = cos θ [Replace θ by θ/2 and 2θ by θ.]
Cos² θ/2 = 1 - Cosθ /2 [Solve for Cos² θ/2 ]
Cos θ/2 = [Take square root on both the sides.]

Solved Example on Half Angle Formula

Ques: Find the exact value of the expression sin 120° by using the half angle formula.

Choices:

Correct Answer: A

Solution:

Quick Summary

Half angle formulas allow calculation of trigonometric functions of angles that are half of known angles.
The sign (+ or -) depends on the quadrant in which θ/2 lies.

\[ \sin{\frac{\theta}{2}} = \pm \sqrt{\frac{1 - \cos{\theta}}{2}}, \cos{\frac{\theta}{2}} = \pm \sqrt{\frac{1 + \cos{\theta}}{2}} \]

🍎 Teacher Insights

Emphasize the importance of determining the correct quadrant for θ/2. Provide ample practice problems involving various angles and quadrants. Connect the derivations back to the double-angle formulas for better understanding.

🎓 Prerequisites

Trigonometric functions (sine, cosine, tangent)
Double-angle formulas
Unit circle
Algebraic manipulation

Check Your Knowledge

Q1: What is the value of sin(15°) using the half-angle formula?

A) (√6 + √2)/4 B) (√6 - √2)/4 C) (√2 - √6)/4 D) (-√6 - √2)/4

Frequently Asked Questions

Q: How do I choose the correct sign when using the half angle formulas?
A: Determine the quadrant in which θ/2 lies. This will determine whether sin(θ/2) and cos(θ/2) are positive or negative.

Q: What are the half angle formulas used for?
A: They are used to find exact values of trigonometric functions for angles that are half of common angles (e.g., 15°, 22.5°, 67.5°), and in calculus for integration and differentiation.