Half Angle Trigonometric Identities

What are Half Angle Identities?

Half angle identities are trigonometric formulas that express the sine, cosine, and tangent of half an angle (θ/2) in terms of trigonometric functions of the original angle (θ). These identities are essential in various mathematical and physical applications, simplifying complex trigonometric expressions and solving advanced problems.

The Half Angle Formulas

The primary half angle identities are:

\[ \begin{align*} \sin \frac{\theta}{2} &= \pm \sqrt{\frac{1 - \cos \theta}{2}} \\ \cos \frac{\theta}{2} &= \pm \sqrt{\frac{1 + \cos \theta}{2}} \\ \tan \frac{\theta}{2} &= \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} = \frac{\sin \theta}{1 + \cos \theta} \end{align*} \]

Where:

• \(\theta\) is the original angle
• \(\frac{\theta}{2}\) is half the original angle
• \(\sin\), \(\cos\), and \(\tan\) are the sine, cosine, and tangent functions, respectively

Derivation and Explanation

These identities can be derived using the double angle formulas and solving for the half angle. The choice of sign (+ or -) depends on the quadrant in which the angle θ/2 lies.

Example Calculation

Let's calculate the half angle identities for θ = 60°:

\[ \begin{align*} \cos 60° &= \frac{1}{2} \\[10pt] \sin \frac{60°}{2} &= \sqrt{\frac{1 - \cos 60°}{2}} = \sqrt{\frac{1 - \frac{1}{2}}{2}} = \frac{1}{2} \\[10pt] \cos \frac{60°}{2} &= \sqrt{\frac{1 + \cos 60°}{2}} = \sqrt{\frac{1 + \frac{1}{2}}{2}} = \frac{\sqrt{3}}{2} \\[10pt] \tan \frac{60°}{2} &= \frac{\sin 60°}{1 + \cos 60°} = \frac{\frac{\sqrt{3}}{2}}{1 + \frac{1}{2}} = \frac{1}{\sqrt{3}} \end{align*} \]

Visual Representation

This diagram illustrates the relationship between θ (60°) and θ/2 (30°) on the unit circle, visually representing the half angle concept.