Chapter 1.8, Problem 21E

a.

To determine

To check the identity

  $\cos \left(\frac{\theta }{2}\right)=\sqrt{\frac{1+\cos \theta }{2}}$

At $\theta =0$

Answer to Problem 21E

The given identity is true at $\theta =0$

Explanation of Solution

Given:

  $\theta =0$

Concept Used:

  $\cos \left(0\right)=1$

Calculation:

To check the identity

  $\cos \left(\frac{\theta }{2}\right)=\sqrt{\frac{1+\cos \theta }{2}}$

At $\theta =0$

Put $\theta =0$ in left hand side of given identity respectively and prove that they are equal, as

  $\begin{array}{c}\cos \left(\frac{0}{2}\right)=\cos \left(0\right)\\ =1\end{array}$

Similarly;

  $\begin{array}{c}\sqrt{\frac{1+\cos \theta }{2}}=\sqrt{\frac{1+\cos 0}{2}}\\ =\sqrt{\frac{1+1}{2}}\\ =\sqrt{\frac{2}{2}}\\ =\sqrt{1}\\ =1\end{array}$

This, implies that left hand side and right hand side of given identity are true for $\theta =0$ .

Hence, the given identity is true at $\theta =0$

b.

To determine

To check the identity

  $\cos \left(\frac{\theta }{2}\right)=\sqrt{\frac{1+\cos \theta }{2}}$

At $\theta =\pi /4$

Answer to Problem 21E

The given identity is true at $\theta =\pi /4$

Explanation of Solution

Given:

  $\theta =\pi /4$

Concept Used:

  $\cos \left(\pi /4\right)=1/\sqrt{2}$

Calculation:

To check the identity

  $\cos \left(\frac{\theta }{2}\right)=\sqrt{\frac{1+\cos \theta }{2}}$

At $\theta =\pi /4$

Put $\theta =\pi /4$ in left hand side of given identity respectively and check whether they are equal, as

  $\begin{array}{c}\cos \left(\frac{\pi /4}{2}\right)=\cos \left(\frac{\pi }{8}\right)\\ =0.923879533,\text{ using calculator}\end{array}$

Similarly;

  $\begin{array}{c}\sqrt{\frac{1+\cos \theta }{2}}=\sqrt{\frac{1+\cos \left(\pi /4\right)}{2}}\\ =\sqrt{\frac{1+\left(1/\sqrt{2}\right)}{2}}\\ =\sqrt{\frac{\frac{\sqrt{2}+1}{\sqrt{2}}}{2}}\\ =\sqrt{\frac{\sqrt{2}+1}{2\sqrt{2}}}\\ =0.923879533\end{array}$

This, implies that left hand side and right hand side of given identity are true for $\theta =\pi /4$ .

Hence, the given identity is true at $\theta =\pi /4$

c.

To determine

To check the identity

  $\cos \left(\frac{\theta }{2}\right)=\sqrt{\frac{1+\cos \theta }{2}}$

At $\theta =\pi /2$

Answer to Problem 21E

The given identity is true at $\theta =\pi /2$

Explanation of Solution

Given:

  $\theta =\pi /2$

Concept Used:

  $\cos \left(\pi /4\right)=1/\sqrt{2},\text{ and }\cos \left(\pi /2\right)=0$

Calculation:

To check the identity

  $\cos \left(\frac{\theta }{2}\right)=\sqrt{\frac{1+\cos \theta }{2}}$

At $\theta =\pi /2$

Put $\theta =\pi /2$ in left hand side of given identity respectively and check whether they are equal, as

  $\begin{array}{c}\cos \left(\frac{\pi /2}{2}\right)=\cos \left(\frac{\pi }{4}\right)\\ =\frac{1}{\sqrt{2}}\end{array}$

And;

  $\begin{array}{c}\sqrt{\frac{1+\cos \theta }{2}}=\sqrt{\frac{1+\cos \left(\pi /2\right)}{2}}\\ =\sqrt{\frac{1+0}{2}}\\ =\sqrt{\frac{1}{2}}\\ =\frac{1}{\sqrt{2}}\end{array}$

This, implies that left hand side and right hand side of given identity are true for $\theta =\pi /2$ .

Hence, the given identity is true at $\theta =\pi /2$

d.

To determine

To check the identity

  $\cos \left(\frac{\theta }{2}\right)=\sqrt{\frac{1+\cos \theta }{2}}$

At $\theta =\pi$

Answer to Problem 21E

The given identity is true at $\theta =\pi$

Explanation of Solution

Given:

  $\theta =\pi$

Concept Used:

  $\cos \left(\pi /2\right)=0,\text{ and }\cos \left(\pi \right)=-1$

Calculation:

To check the identity

  $\cos \left(\frac{\theta }{2}\right)=\sqrt{\frac{1+\cos \theta }{2}}$

At $\theta =\pi$

Put $\theta =\pi /2$ in left hand side of given identity respectively and check whether they are equal, as

  $\cos \left(\frac{\pi }{2}\right)=0$

And;

  $\begin{array}{c}\sqrt{\frac{1+\cos \theta }{2}}=\sqrt{\frac{1+\cos \left(\pi \right)}{2}}\\ =\sqrt{\frac{1-1}{2}}\\ =\sqrt{\frac{0}{2}}\\ =0\end{array}$

This, implies that left hand side and right hand side of given identity are true for $\theta =\pi$ .

Hence, the given identity is true at $\theta =\pi$ .