Question

How do you express $\sin \dfrac{x}{2}$ in terms of $\cos x$ using the double angle identity?

Answer 1

Sine and cosine are two most basic trigonometric functions. Sine is the ratio of height and hypotenuse and cosine is the ratio of base and the hypotenuse. These two can be interconverted very easily and this property will be useful for us in this question. . Using half-angle identity we can express it in form of the both sine function and cosine function at once.

Complete step by step solution:
According to the question we have to express $\sin \dfrac{x}{2}$ in terms of $\cos x$ using the double angle identity
So we know different trigonometric formulas, so we will use this formula in this question to get our answer
$\Rightarrow \cos 2x = 1 - 2{\sin ^2}x$
But in the question it is $\cos x$ instead of $\cos 2x$ so we have to convert $\cos x$ in this formula
$\Rightarrow \cos x = 1 - 2{\sin ^2}\dfrac{x}{2}$ (As we already know this formula)
$\Rightarrow 2{\sin ^2}\dfrac{x}{2} = 1 - \cos x$ (Taking sine function to left hand side and taking cosine function to the right hand side)
$\Rightarrow {\sin ^2}\dfrac{x}{2} = \dfrac{{1 - \cos x}}{2}$ (Dividing both sides with 2)

$\Rightarrow \sin \dfrac{x}{2} = \pm \sqrt {\dfrac{{1 - \cos x}}{2}}$ (Square rooting both sides)

Note:
We can observe that both positive and negative signs are given when it is square rooted because if we didn’t it will lead to an error. We will miss one solution in this case. Using the formula of converting the sine function into a cosine function is very important. We can convert the cosine function to another cosine function of half the angle by using the half-angle identity as we converted the sine function from the cosine function. Using half-angle identity we can express it in form of the both sine function and cosine function at once.