Question

Express $\sin \dfrac{x}{2}$ in terms of $\cos x$ using the double angle identity?

Answer 1

We use the formula for the trigonometric function of multiple angles where $\cos x=1-2{{\sin }^{2}}\dfrac{x}{2}$. We simplify the equation to get the value of $\sin \dfrac{x}{2}$ in terms of $\cos x$. The final expression is the solution of the problem.

Complete step by step answer:
We have the formula for the trigonometric function of submultiple angles where $\cos x=1-2{{\sin }^{2}}\dfrac{x}{2}$. We need to find the inverse form of $\cos x=1-2{{\sin }^{2}}\dfrac{x}{2}$ to get the value of $\sin \dfrac{x}{2}$ in terms of $\cos x$.
Simplifying the equation, we get
$\cos x=1-2{{\sin }^{2}}\dfrac{x}{2} \\\ \Rightarrow 2{{\sin }^{2}}\dfrac{x}{2}=1-\cos x \\\$
We now divide both sides by 2 to get
$2{{\sin }^{2}}\dfrac{x}{2}=1-\cos x \\\ \Rightarrow {{\sin }^{2}}\dfrac{x}{2}=\dfrac{1-\cos x}{2} \\\$
Now we take the square root value of the equation and get $\sqrt{{{\sin }^{2}}\dfrac{x}{2}}=\pm \sqrt{\dfrac{1-\cos x}{2}}$.
Expressing $\sin \dfrac{x}{2}$ in terms of $\cos x$, we get $\sin \dfrac{x}{2}=\pm \sqrt{\dfrac{1-\cos x}{2}}$.

Hence, the simplified form will be $\sin \dfrac{x}{2}=\pm \sqrt{\dfrac{1-\cos x}{2}}$.

Note: The multiple angles work as $\cos 2x=1-2{{\sin }^{2}}x$. We just change the angle from $2x$ to $x$ to get $\cos x=1-2{{\sin }^{2}}\dfrac{x}{2}$. The trigonometric functions of multiple angles are the multiple angle formula. Double and triple angles formulas are there under the multiple angle formulas. Sine, tangent and cosine are the general functions for the multiple angle formula.