TR-41: Half Angle Identities (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial provides a comprehensive guide to understanding the Half Angle Identities in trigonometry. The Half Angle Identities are essential in simplifying trigonometric expressions and solving equations. This guide will break down the concepts into manageable steps, supported by algebraic and geometric proofs, making it suitable for students preparing for exams in International A-level or Cambridge curricula.

Step 1: Understand the Half Angle Identities

The Half Angle Identities express trigonometric functions of half angles in terms of full angles. The identities are as follows:

• Sine Half Angle Identity:
  [ \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}} ]

• Cosine Half Angle Identity:
  [ \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}} ]

• Tangent Half Angle Identity:
  [ \tan\left(\frac{\theta}{2}\right) = \frac{\sin(\theta)}{1 + \cos(\theta)} = \frac{1 - \cos(\theta)}{\sin(\theta)} ]

Practical Advice

• Remember that the sign (positive or negative) depends on the quadrant in which the angle (\frac{\theta}{2}) lies.

Step 2: Derive the Half Angle Identities Algebraically

To derive these identities, we can use the double angle formulas and some algebraic manipulation.

Sine Half Angle Derivation

1. Start with the double angle identity for cosine: [ \cos(2x) = 1 - 2\sin^2(x) ] Set (x = \frac{\theta}{2}): [ \cos(\theta) = 1 - 2\sin^2\left(\frac{\theta}{2}\right) ]
2. Rearranging gives: [ 2\sin^2\left(\frac{\theta}{2}\right) = 1 - \cos(\theta) ]
3. Dividing by 2: [ \sin^2\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{2} ]
4. Taking the square root leads to the sine half angle identity.

Cosine Half Angle Derivation

1. Use the double angle identity for sine: [ \sin(2x) = 2\sin(x)\cos(x) ] Set (x = \frac{\theta}{2}): [ \sin(\theta) = 2\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right) ]
2. Solve for (\cos\left(\frac{\theta}{2}\right)) using the identity derived from sine.

Tangent Half Angle Derivation

1. Use the sine and cosine half-angle identities: [ \tan\left(\frac{\theta}{2}\right) = \frac{\sin\left(\frac{\theta}{2}\right)}{\cos\left(\frac{\theta}{2}\right)} ]
2. Substitute the identities to express tangent in terms of sine and cosine.

Step 3: Geometric Interpretation

Understanding the Half Angle Identities can also be visualized geometrically.

• Consider a unit circle where an angle (\theta) creates a point on the circumference.
• The coordinates of this point give the sine and cosine values.
• By halving the angle (\theta), you can visualize how the corresponding coordinates change, thus reinforcing the algebraic identities derived.

Conclusion

The Half Angle Identities are powerful tools in trigonometry, useful for simplifying expressions and solving equations. By understanding their derivations and geometric interpretations, you can confidently apply these identities in various mathematical contexts. For further exploration, consider practicing problems involving these identities and their applications in calculus and geometry.