# Find all values of $x$ between 0 and $2\pi$ for which $\sin x=0.6$

Find all values of $x$ between 0 and $2\pi$ for which $\sin x=0.6$.

this is what I found out from yahoo answer site
$\sin^{-1} (0.6) = 0.64$
$\pi-0.64= 2.5$
the answer is right because I used the calculator but I would like to know how to manually solve this question without calculator.

from the comment I find out the using calculator is the best but second question

$\sin x=-0.45$
$\sin^-1 (-0.45) = -0.4667$ by using calculator
but the right answer is $3.61,5.816$

can you show me where did I make mistake?

thx

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What do you mean by "manually solve this question"? Manually calculate the value of $\arcsin(0.6)$? It is transcendental as far as i can say, so there is no way to write it other than $\arcsin(0.6)$ (or, equivalently, $\arccos(0.8)$, $\arctan(3/4)$ etc). – penartur Apr 17 '12 at 9:51
so, I must use calculator to solve this question? thx – Sb Sangpi Apr 17 '12 at 9:59
There are ways of solving this without a calculator, but using one is the most efficient way. Of course, you could, e.g., solve it using the geometric definition of the sine (draw a unit circle, draw a vertical line $A$ 0.6 to the right of the center, draw a line $B$ perendicular to $A$ through the center and a line $C$ from the center to one of the intersections of $A$ with the circle, measure the angle between $B$ and $C$), look it up in a table, using some numeric approximation method or whatever, but using a calculator is actually your best bet here. – Johannes Kloos Apr 17 '12 at 10:05
You could also expand the function $sin^{-1}(x)$ in taylor series and find an approximation by polynomial combinations. – chemeng Apr 17 '12 at 10:17
sorrry I just edited question! – Sb Sangpi Apr 17 '12 at 10:21

In a 3,4,5 triangle, the angle values are roughly 37,53, and 90 degrees. So since $\sin(t)= 0.6,$ the angle is about $37^\circ$. The other solution would be $(180-37)^\circ = 143^\circ$. You can convert the acute angle to radians as $37 \pi/180$. So to rough this out a little more, $37 \times 22 /(7 \times 180) \simeq 5.28 \times 11/90 \simeq 5.28/8 = .66$. I did not use a calculator in any of this calculation. My value is larger than yours since I ran $\dfrac{11}{90}$ as roughly $\dfrac{1}{8}.$
As for your edited question, note that $5.816+0.4667 = 2\pi$ (adjusted for a computational error).
Remember that $\sin(x) = \sin(x+2\pi) = sin(\pi-x)$. The calculator just gave you one of the infinite number of answers, and using these formulas you can deduce all the answers between $0$ and $2\pi$.