Question

sin (0.77x) = -1/2
How did they get 0.77x = 3.665 and 5.760?
Someone please help show me the steps of their work so I can see it.

Answer 1

I think there are two ways.

The first is memorizing what you take the sine of to get -1/2.

The other ways is using the inverse sine function.
Because\[sin(77x)=-\frac{1}{2}\], you can say
\[77x = sin^{-1}(-\frac{1}{2})\]

Answer 2

This only helps if you have a calculator..

Answer 3

Hello, thank you for answering. This is what I initially thought as well. however, the inverse sine of negative one half gave me: -0.5235987756 radians. Above it shows different answers for the solution.

Answer 4

So I am baffled how my text came up with such a number.

Answer 5

Huh! I'm foggy on trigonometry, sorry!

Answer 6

Thanks anyways.

Answer 7

I used wolfram alpha to solve for \[a\]in\[sin(a)=-\frac{1}{2}\]
and it produced
\[x=\frac{7\pi}{6}+2\pi n_1\]and\[x=\frac{11\pi}{6}+2\pi n_2\]

Answer 8

The first is 3.6651914291880921115397506138260866982300309659376234...

Answer 9

when \[n_1=0}\]

Answer 10

Oh I see.. The 3.665 and 5.760 are just decimal equivalents of the fraction. Thanks for shedding some light on it.

Answer 11

when \[n_1=0\] I mean

Answer 12

If you got it, then congratulations! I'm still a bit confused! :)

Answer 13

The second resolves to 5.7595865315812876038481795360124219543614772321876940... when \[n_2=0\]

Answer 14

Haha. I got it. It is saying. The sine function is equal to -1/2 in two different quadrants. In a circle (degrees) it is located at 7pi/6 and 11pi/6 These are equivalent to a 30 degree angle (pi/6) in the first quadrant.

7pi/6 = 3.665

Answer 15

Excuse me, those are radians!

Answer 16

Well thanks for guiding me. I got it. Medaled.

Answer 17

Cool! Would you just have to memorize \[\frac{7\pi}{6}\] and \[\frac{11\pi}{6}\]? I think you would have to...

Answer 18

Ah well, I'll recap another day! I'm glad I could be of some service! Take care!

Answer 19

I wouldn't really memorize, but it's not a bad idea if you have that capacity! :)

If anything it would be convenient. However, there is a method to finding it with logic. If the Y-coordinate is -1/2 then the x must be \[\sqrt{3}\div2\] this is assuming we understand the basic triangles.

This information tells us with an x-coord of 1/2 and a y-coord of \[\sqrt{3}\div2\] it is a 30 degree angle (in quadrant I). However the problem needs a solution for where Y is negative.

It is true in two quadrants. The formula for DEGREES to RADIANS is:( DEGREE*PI )/ 180

since quadrant III begins from 180 and ends at 270 AND we knew it is essentially a 30 degree angle then we can add 30 DEGREES to 180. This becomes a 210 angle. Divide 270 by 180 and you get 7/6. Don't forget to multiply it by Pi.

Hopefully that shows how to get an angle in another quadrant. There are of course other tricks to get there.


You could see this visually. You could just directly reflect the angle into an opposite quadrant by making a straight line out of it. Here it has 180 degrees added to it. So you don't need to just simply memorize radians.