Half Angle Calculator

Half-angle identities derive directly from double-angle formulas in trigonometry. The core relationships express the trigonometric functions of an angle x/2 in terms of the functions of the full angle x. For sine, the formula is \(\sin(x/2) = \pm\sqrt{\frac{1 - \cos(x)}{2}}\). For cosine, it is \(\cos(x/2) = \pm\sqrt{\frac{1 + \cos(x)}{2}}\). For tangent, it can be expressed as \(\tan(x/2) = \frac{1 - \cos(x)}{\sin(x)} = \frac{\sin(x)}{1 + \cos(x)}\). The choice of the positive or negative sign depends entirely on the quadrant in which the terminal side of the half angle \(x/2\) lies.

Let us calculate the exact value of \(\cos(15^\circ)\) using the half-angle formula. We know that \(15^\circ\) is half of \(30^\circ\), so we set \(x = 30^\circ\). First, recall the cosine of the base angle: \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\). Next, substitute this into the half-angle identity for cosine: \(\cos(15^\circ) = \sqrt{\frac{1 + \cos(30^\circ)}{2}}\). This simplifies to \(\sqrt{\frac{1 + \sqrt{3}/2}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}}\). Simplifying the denominator yields \(\frac{\sqrt{2 + \sqrt{3}}}{2}\), which gives the precise numerical value approximately equal to 0.9659.

Always verify the quadrant of your resulting half angle before finalizing your calculations, as the quadrant dictates whether your radical expression takes a positive or negative sign. When dealing with radians, remember that dividing an angle by two scales its position on the unit circle proportionally. Double-check your input values to ensure your base angle matches the domain requirements of standard inverse trigonometric functions.