Question

if cosxcos(pi/7)+sinxsin(pi/7)=root2/2

Answer 1

Answer:

$\displaystyle x=\frac{11\pi}{28}$

$\displaystyle x=-\frac{3\pi}{28}$

Step-by-step explanation:

Cosine Of A Sum Of Angles

The cosine of the sum of angles can be expressed in terms of the individual angles as follows

$cos(a+b)=cos\ a\ cos\ b-sin\ a\ sin\ b$

The cosine of the subtraction of angles is

$cos(a-b)=cos\ a\ cos\ b+sin\ a\ sin\ b$

Since we have

$\displaystyle cosx\ cos(\frac{\pi}{7})+sinx\ sin(\frac{\pi}{7})=\frac{\sqrt{2}}{2}$

We can see it's equivalent to the cosine of the subtraction of angles, thus

$\displaystyle cosx\ cos(\frac{\pi}{7})+sinx\ sin(\frac{\pi}{7})=cos(x-\frac{\pi}{7})$

Completing the equation we have

$\displaystyle cos(x-\frac{\pi}{7})=\frac{\sqrt{2}}{2}$

We know

$\displaystyle cos\ \frac{\pi}{4}=\frac{\sqrt{2}}{2}$

And also

$\displaystyle cos\ (-\frac{\pi}{4})=\frac{\sqrt{2}}{2}$

So we have two possible solutions

$\displaystyle x-\frac{\pi}{7}=\frac{\pi}{4}$

$\displaystyle x-\frac{\pi}{7}=-\frac{\pi}{4}$

Thus, the first solution is

$\displaystyle x=\frac{\pi}{7}+\frac{\pi}{4}$

$\displaystyle x=\frac{11\pi}{28}$

And the second solution is

$\displaystyle x=\frac{\pi}{7}-\frac{\pi}{4}$

$\displaystyle x=-\frac{3\pi}{28}$