Question

What is $\cos \left( \dfrac{\theta }{2} \right)$ in terms of trigonometric functions of a unit $\theta$ ?

Answer 1

We must know the trigonometric identity for the calculation of cosine of double angle, $\cos 2\theta ={{\cos }^{2}}\theta -{{\sin }^{2}}\theta$ , and then express the RHS part in terms of $\cos \theta$ only. Then, we must use the RHS part to find out the value of $\cos \left( \dfrac{\theta }{2} \right)$ by replacing $\theta$ with $\dfrac{\theta }{2}$ .

Complete step-by-step solution:
We know that the identity for calculation the value of cosine of double angle is
$\cos 2\theta ={{\cos }^{2}}\theta -{{\sin }^{2}}\theta$ .
We all are very well aware of the trigonometric identity that the sum of the squares of sine and cosine of the same angle is 1, i.e.,
${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$
We can also write the above equation as
${{\sin }^{2}}\theta =1-{{\cos }^{2}}\theta$
Now, we can replace the value of sine squared, in the identity for calculating the cosine of double angle. Hence, we get
$\cos 2\theta ={{\cos }^{2}}\theta -\left( 1-{{\cos }^{2}}\theta \right)$
We can also express this equation as
$\cos 2\theta =2{{\cos }^{2}}\theta -1...\left( i \right)$
Now, we know that we have to find the value of $\cos \left( \dfrac{\theta }{2} \right)$ . So, let us use the above equation to calculate $\cos \theta$ in terms of $\cos \left( \dfrac{\theta }{2} \right)$ .
Using equation (i), we can easily write,
$\cos \theta =2{{\cos }^{2}}\dfrac{\theta }{2}-1$
Let us now add 1 on both sides of the equation. Hence, we get
$\cos \theta +1=2{{\cos }^{2}}\dfrac{\theta }{2}-1+1$
Or, we can write this down as
$\cos \theta +1=2{{\cos }^{2}}\dfrac{\theta }{2}$
We need to find the value of $\cos \left( \dfrac{\theta }{2} \right)$ , so let us bring this term on the left hand side (LHS).
$2{{\cos }^{2}}\dfrac{\theta }{2}=\cos \theta +1$
Let us now multiply both sides of the equation by $\dfrac{1}{2}$ ,
$\dfrac{1}{2}\times 2{{\cos }^{2}}\dfrac{\theta }{2}=\dfrac{1}{2}\times \left( \cos \theta +1 \right)$
We can rewrite this as,
${{\cos }^{2}}\dfrac{\theta }{2}=\dfrac{\cos \theta +1}{2}$
Let us now take square roots on both sides of this equation,
$\sqrt{{{\cos }^{2}}\dfrac{\theta }{2}}=\sqrt{\dfrac{\cos \theta +1}{2}}$
Hence, we get
$\cos \dfrac{\theta }{2}=\sqrt{\dfrac{\cos \theta +1}{2}}$.
Thus, we can express $\cos \left( \dfrac{\theta }{2} \right)$ as $\sqrt{\dfrac{\cos \theta +1}{2}}$.

Note: We must remember the trigonometric identities correctly to be able to solve this problem. We can also use other methods, but we will eventually get the same result. It is important to note that we don’t have any direct formula to calculate the value of $\cos \left( \dfrac{\theta }{2} \right)$.