0
$\begingroup$

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

  • 0
    sorrry I just edited question!2012-04-17

2 Answers 2

3

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 $\dfrac{1}{8}.$

  • 0
    sorry, I just edited question!2012-04-17
2

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 $2\pi$.