If csc(x)=9, for 90*<x<180*, then:

Question

Sin (x/2)=

Cos (x/2)=

Tan (x/2)=

Can someone explain me the process of this please? I have been doing the procedure explained in this video yet I continue to get it wrong.

https://www.youtube.com/watch?v=wfXsREOO_pE&feature=emb_title&ab_channel=Mathispower4u

Answer 1

Your 1/2 angle formulas require cosine values so you have to figure out the cosine of the angle which has a cosecant of 9 which is in quadrant 2. Draw a right triangle. since csc is hypotenuse/opposite can solve for adjacent using pythagorean theorem which is( sqroot80)

So cosine of this angle is -(sqroot80)/9 Cosine is negative in quadrant 2

Sin x/2 = sqroot[(1-cos x)/2]= sq root [(1- (-sqrot80/9)/2] = sqroot (9+sqroot80)/18}=.998

For the other questions just insert the correct value of cosine x into your equation and solve

Answer 2

There is more than one way to do this; here is one of them. Csc(x) = 1/sin(x) = 9 then sin(x) = 1/9 or 0.111. Using my TI30X calculator to find the angle gives me 6.4°, but this is in the first quadrant. We are looking for an angle in the second quadrant. The angle in the second quadrant giving this sine value is 180°- 6.4° = 173.6°. x/2 is then 86.8° which is in the first quadrant. Now you should be able to use a calculator to determine the results you need.

Answer 3

This can be answered with certainty. Since sine and cosecant are reciprocals, the fact that csc(x) = 9 implies that sin(x) = 1/9. That result is unambiguous; you don't even need to know which quadrant you're in to get that far.

But because we are interested in finding the trigonometric functions of x/2, we really want to know cos(x). Now for any value of x, sine squared plus cosine squared equals one. So with a little algebra, we can determine that cos(x) must either equal SQRT(80)/9 or -SQRT(80)/9.

We are also told that x lies between 90 degrees and180 degrees, so our angle lies in the 2nd Quadrant. There cosine values are negative (because they are the x-coordinate of a point on the Unit Circle). So it must be that cos(x) = -SQRT(80)/9.

After that, you can use the Half Angle formulas to determine sin(x/2) and cos(x/2), noting that x/2 lies in the First Quadrant (because 90 < x < 180 ---> 45 < x/2 < 90); and therefore sin(x/2) and cox(x/2) will be positive values.

Lastly, tan(x/2) is their ratio.

Hoping this helps,

Michael H