Question

Solve the equation cos(x/2 + pi/6) = 1.

Answer 1

The range is \[-\frac{ 1 }{ 2 } \pi \le x \le \frac{ 1 }{ 2 } \pi\]

Answer 2

When I use the identity cos(A+B) = cosAcosB - sinAsinB, I get \[\cos \frac{ x }{ 2 }\cos \frac{ \pi }{ 6 } - \sin \frac{ x }{ 2 }\sin \frac{ \pi }{ 6 } = 1\]This then simplifies to \[\frac{ \sqrt{3} }{ 2 }\cos \frac{ x }{ 2 } - \frac{ 1 }{ 2 }\sin \frac{ x }{ 2 } = 1\] Am I on the right track here? If so, now where do I go?

Answer 3

It's easier than that. You have:

\(\Large \cos(x/2 + pi/6) = 1\)

What if, instead, you had: \(\Large \cos(z) = 1\)

What would z be?

Answer 4

z would be equal to 0, then.

Answer 5

so x/2 + pi/6 = 0 then

Answer 6

Very good.

Now, instead of z, you have:

\(\Large x/2 + pi/6 = 0\)

Answer 7

exactly. :) Now find x! :)

Answer 8

Therefore x = - pi/3. Of course yes. :D I got it, thanks!

Answer 9

you're welcome. :)