How would you find sinx=.3

Question

I dont get how to do this, the problme is 

sinx=.3

you have to take the inverse of n.3 and you get .305 (round) but the i dont know where to go from there, you need 2 solutions and .305 is the first solution for x then i dont know why you add pi. I dont get when to add pi and when to subtract pi 

Answer 1

Find x if sin(x) = 0.3.

Let’s change the variable to T, for theta, because we want to use a graph of the unit circle on the x-y plane.

sin(T) = 0.3

Every point on the unit circle, (x,y), is used to DEFINE the trig function cos(T) and sin(T):

(x,y) := ( cos(T), sin(T) )

So y = sin(T).

We want to know where on the unit circle sin(T) = 0.3

Let’s draw the horizontal line y = 0.3 and observe that it intersects the unit circle in two points; one in Quadrant I and the other in Quadrant II.

The inverse sine function, arcsine, will only return angle values between -pi/2 and +pi/2; i.e., points on the right half of the unit circle.

Arcsin(0.3) ≈ 0.30469 radians ≈ T_1, our Quadrant I angle.

The angle in Quadrant II, T_2 = pi - T_1 ≈ 2.83690 radians.

But since no domain was specified for sin(T) = 0.3, it is All Real Numbers. Since sine is periodic with period 2 pi, we can add any number of 2 pi’s to our two answers.

So all the angles that satisfy sin(x) = 0.3 are:

x ≈ 0.30469 + n(2 pi) and x ≈ 2.83690 + n(2 pi); where n is any integer.

Answer 2

 Set your calculator at Radian

 

   Look for Sin^-1( 0.3) = 0.30469 / read in calculator.

 

       This is principle angle. You know that Supplement angle has the same sine value as principle

 

          So the other angle is ( π - sin^-1 0.3 )

 

           If you read in calculator:

                                                Sin ( π - sin ^-10.3 ) =0.3  / what calculator reads

 

        However:

                      If read  from the calculator:

                           Sin ( π - 0.30469) = 0. 299997468     /can be rounded off to 0.3 

 

 

Answer 3

Hi Jennifer,

    The inverse sin function, x=sin^-1(θ) is defined (single valued) for -π/2 <θ<π/2. 

When θ=.3 x=.30469 rad

 

To find other angles whose sine is .3 you should use the relation for the sum of two angles i.e.

 

sin(x+y) = sin(x)cos(y) + cos(x)sin(y) notice that if y=2π then sin(x+2π) = sin(x) so another angle that has the same sin is .30468 + 2π = 6.5878 rad Note that there a many angles the have the same value and they are all 2π apart.

 

Hope this helps

Jim