Calculate x? Sin(x+60)=2Sinx

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Calculate x? Sin(x+60)=2Sinx

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Answer 1 · 2018-05-27T11:22:46

$x = \frac{π}{3} + 2 k π$ $x=pi/3+2kpi$

Explanation:

We have
$sin (x + \frac{π}{3}) = sin (x) cos (\frac{π}{3}) + cos (x) sin (\frac{π}{3}) = 2 sin (x)$ $sin(x+pi/3)=sin(x)cos(pi/3)+cos(x)sin(pi/3)=2sin(x)$
Dividing by $sin (x)$ $sin(x)$
$cos (\frac{π}{3}) + cot (x) sin (\frac{π}{3}) = 2$ $cos(pi/3)+cot(x)sin(pi/3)=2$
$cot (x) = \frac{2 - cos (\frac{π}{3})}{sin (\frac{π}{3})}$ $cot(x)=(2-cos(pi/3))/sin(pi/3)$
so

$tan (x) = \frac{sin (\frac{π}{3})}{2 - cos (\frac{π}{3})} = \frac{1}{\sqrt{3}}$ $tan(x)=sin(pi/3)/(2-cos(pi/3))=1/sqrt(3)$

Answer 2 · 2018-05-27T11:28:19

$x = 30 + 360 n$ $x = 30 + 360n$

Explanation:

First, we apply compound angle formula on $sin (x + 60)$ $sin(x+60)$ .

$sin (x + 60) = sin (x) cos (60) + sin (60) cos (x) = \frac{1}{2} sin (x) + \frac{\sqrt{3}}{2} cos (x)$ $sin(x+60) = sin(x)cos(60) + sin(60)cos(x) = 1/2sin(x) + sqrt(3)/2cos(x)$

We now have:
$2 sin (x) = \frac{1}{2} sin (x) + \frac{\sqrt{3}}{2} cos (x)$ $2sin(x) = 1/2sin(x) + sqrt(3)/2cos(x)$

Since $sin (x)$ $sin(x)$ is not equal to 0 (if $sin (x)$ $sin(x)$ is equal to 0, it is not possible for $sin (x + 60)$ $sin(x+60)$ to be equal to 0 as well), we can divide both sides of the equation by $sin (x)$ $sin(x)$ .

$2 = \frac{1}{2} + \frac{\sqrt{3}}{2 tan (x)}$ $2 = 1/2 + sqrt(3)/(2tan(x))$

Making $tan (x)$ $tan(x)$ the subject,

$\frac{3}{2} = \frac{\sqrt{3}}{2 tan (x)}$ $3/2 = sqrt(3)/(2tan(x))$
$tan (x) = \frac{1}{\sqrt{3}}$ $tan(x) = 1/sqrt(3)$ .

Therefore,

$x = 30 + 360 n$ $x = 30 + 360n$

The $360 n$ $360n$ is because trigonometric functions are periodic about 360 degrees, or 2 $π$ $pi$ radians, which means the equation will still hold no matter how much you add or subtract 360 degrees from x.

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Calculate x? Sin(x+60)=2Sinx

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