Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
1  
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
1  
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

2 Answers 2

up vote 3 down vote accepted

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 $\dfrac{1}{8}.$

share|improve this answer
    
sorry, I just edited question! –  Sb Sangpi Apr 17 '12 at 10:22

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 $2\pi$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.