Express sin(theta), cos(theta) and tan(theta) in terms of (theta/2)

Expressing \( \sin\theta \), \( \cos\theta \), and \( \tan\theta \) in Terms of \( \frac{\theta}{2} \)

In trigonometry, it's often useful to express trigonometric functions like \( \sin\theta \), \( \cos\theta \), and \( \tan\theta \) in terms of \( \frac{\theta}{2} \), especially in integration, transformation, and angle simplification problems.

1. Using Half-Angle Identities

(a) Expressing \( \sin\theta \)

Using the identity: \[ \sin\theta = 2 \sin\left(\frac{\theta}{2}\right) \cos\left(\frac{\theta}{2}\right) \]

(b) Expressing \( \cos\theta \)

Using the identity: \[ \cos\theta = \cos^2\left(\frac{\theta}{2}\right) - \sin^2\left(\frac{\theta}{2}\right) \quad\] or, \[\quad \cos\theta = 1 - 2\sin^2\left(\frac{\theta}{2}\right) \quad\] or, \[\quad \cos\theta = 2\cos^2\left(\frac{\theta}{2}\right) - 1 \] All three are equivalent and can be used depending on the known quantity.

(c) Expressing \( \tan\theta \)

Using the identity: \[ \tan\theta = \frac{2 \tan\left(\frac{\theta}{2}\right)}{1 - \tan^2\left(\frac{\theta}{2}\right)} \] This is particularly helpful when solving equations or simplifying expressions involving tangent.

2. Summary of Results

• \( \sin\theta = 2 \sin\left(\frac{\theta}{2}\right) \cos\left(\frac{\theta}{2}\right) \)
• \( \cos\theta = \cos^2\left(\frac{\theta}{2}\right) - \sin^2\left(\frac{\theta}{2}\right) \)
• \( \tan\theta = \frac{2 \tan\left(\frac{\theta}{2}\right)}{1 - \tan^2\left(\frac{\theta}{2}\right)} \)

3. When Are These Useful?

These identities are especially helpful when:

• Simplifying trigonometric integrals or derivatives
• Finding solutions to expressions with sine, cosine, or tangent
• Converting between full-angle and half-angle expressions
• Working with geometric problems involving triangles