In some special cases where we add or subtract formulas for sine and cos, we get what is called as double-angle identities and half- angle identities. These half angle formulas let the trigonometric functions expressions of angles equate to x/2 in terms of x which can be later to functions and it would be easier to perform the complex calculations.
| Sin x/2 = ±√(1-cosx)/2 |
Practice questions on Sin x/2 Formula
Question: Find the value of sin 105 degrees using Sin half-angle formulas
Solution:
\(\begin{array}{l}\begin{array}{l}\text { Explanation: } \\ \text { We need to use the half angle formula: } \\ \sin \left(\frac{\theta}{2}\right)=\pm \sqrt{\frac{1-\cos \theta}{2}} \\ \text { In this case, we want to find } \sin \left(105^{\circ}\right), \text { so that’s what we want } \sin (\frac{\theta}{2}) \text { to equal. } \\ \text { To find out what our } \theta \text { is, set these to equal to each other: } \\ \sin \left(105^{\circ}\right)=\sin \left(\frac{\theta}{2}\right)\end{array}\end{array} \)
\(\begin{array}{l}\begin{array}{l}105^{\circ}=\frac{\theta}{2} \\ 210^{\circ}=\theta \\ \text { This is our } \theta \text { . Now, we can use the half angle formula: } \\ \qquad \sin \left(105^{\circ}\right) =\sin \left(\frac{210^{\circ}}{2}\right) \\ =\pm \sqrt{\frac{1-\cos \left(210^{\circ}\right)}{2}}\end{array}\end{array} \)
= √[(1 – (-√3/2)) / 2]
= √(2 + √3) /√ 4
= √ (2 + √3) / 2
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