How to do this one?

Question

How to do this one?

1 Answer

Impact of this question

1037 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-12-08T01:32:17

pi/3, pi, (5pi)/3

Explanation:

$3 cos (\frac{x}{2}) - 3 sin x = 0$ $3cos (x/2) - 3sin x = 0$ .
Replace sin x by $2 sin (\frac{x}{2}) . cos (\frac{x}{2})$ $2sin (x/2).cos (x/2)$
$3 cos (\frac{x}{2}) - 6 sin (\frac{x}{2}) . cos (\frac{x}{2}) = 0$ $3cos (x/2) - 6sin (x/2).cos (x/2) = 0$
$3 cos (\frac{x}{2}) (1 - 2 sin (\frac{x}{2})) = 0$ $3cos (x/2)(1 - 2sin (x/2)) = 0$
Either one of the factors should be zero.

A. $cos (\frac{x}{2}) = 0$ $cos (x/2) = 0$ . Unit circle gives 2 solutions:
$\frac{x}{2} = \frac{π}{2}$ $x/2 = pi/2$ and $\frac{x}{2} = \frac{3 π}{2}$ $x/2 = (3pi)/2$ -->
a. $\frac{x}{2} = \frac{π}{2}$ $x/2 = pi/2$ --> $x = π$ $x = pi$
b. x/2 = (3pi)/2 --> $x = 3 π$ $x = 3pi$ (rejected because outside of range)
B. $1 - 2 sin (\frac{x}{2}) = 0$ $1 - 2sin (x/2) = 0$
$sin (\frac{x}{2}) = \frac{1}{2}$ $sin (x/2) = 1/2$
Trig table and unit circle give 2 solutions:
$\frac{x}{2} = \frac{π}{6}$ $x/2 = pi/6$ and $\frac{x}{2} = \frac{5 π}{6}$ $x/2 = (5pi)/6$ -->
a. $\frac{x}{2} = \frac{π}{6}$ $x/2 = pi/6$ --> $x = \frac{π}{3}$ $x = pi/3$
b. $\frac{x}{2} = \frac{5 π}{6}$ $x/2 = (5pi)/6$ --> $x = \frac{5 π}{3}$ $x = (5pi)/3$
Answers for $[0, 2 π)$ $[0, 2pi)$ :
$\frac{π}{3}, π, \frac{5 π}{3}$ $pi/3, pi, (5pi)/3$

Science

Math

Humanities

... and beyond

How to do this one?

1 Answer

Explanation:

Related questions

Impact of this question