Precalculus question need help solving?

Question

If csc(X)= -13/5
&
3pi/2 < X < 2pi
find sin x/2

Answer 1

csc(x) is by definition 1/sin(x), so we know that sin(x) = -5/13.

To find sin(x/2), we need to use the half-angle formula

sin(x/2) = +- sqrt[(1 - cos(x))/2].

In this case, cos(x) = +- sqrt[1 - sin^2(x)] = +-12/13. Using the given fact that x is between 3pi/2 and 2pi, we know that cos(x) is positive, so it is 12/13.

Putting this into the above half-angle formula, we get

sin(x/2) = +- sqrt(1/26).

I'll leave it to you to figure out whether the answer should be positive or negative.

Answer 2

If csc(X)= -13/5 & 3pi/2 < X < 2pi, find sin x/2
sin x = 1/csc(X) = -5/13
sin(X/2) =+/- sqrt{[1-cos(x)]/2)}
Since cos (X) = 12/13, we have :
sin(X/2) = +/-sqrt{[1-(12/13)]/2)}
sin (X/2) = -.1961161351
The sine is minus since X/2 is in the 3rd quadrant.

Answer 3

sin(x) = 1/csc(x) = -5/13
cos(x) = sqrt( 1 - 25/169) = sqrt(144/169) = 12/13
( x in 3pi/2 .. 2pi tells you to take +ve sqrt)
cos(x) = 1 - 2sin^2(x/2)
so sin^2(x/2) = (1-cos(x))/2 = 1/26
sin(x/2) = -1 / sqrt(26)
[again sign comes from the 3pi/2..2pi range]

Answer 4

Assuming the area is 0<=x<=2pi Multiply by way of with the help of (2/3) , so it suits the arguments of the trig function. 0<=x<=4pi/3 Now, from the graph of the cosine function. cos(x) =-a million whilst x= pi , 3pi , 5pi and so forth and so forth , strange multiples of pi. In out area, there is basically one strange multiply of pi , that's basically pi as a result (2x/3) = pi ---> x= 3pi / 2 basically SOLN interior the era ------------------------- with connection with " .there happens to no longer be from now on angles for this, are you able to nevertheless clarify the thank you to define the distinctive angles" evaluate a similiar occasion, sparkling up cos(2x) = -a million interior the area 0<=x<=2pi Now, once you regulate the area you will would desire to multiply with the help of two 0<=2x<=4pi Now, there are 2 values of x for which the cosine function equals -a million , in this area. x= pi and x=3pi as a result , 2x = pi , 3pi x=pi/2 and x= 3pi/2