SOLUTION: if csc x=3, 90 degrees< x < 180 degrees then, sin (x/2) cos (x/2) tan (x/2)

Algebra ->  Trigonometry-basics -> SOLUTION: if csc x=3, 90 degrees< x < 180 degrees then, sin (x/2) cos (x/2) tan (x/2)      Log On


   

Question 1204691: if csc x=3, 90 degrees< x < 180 degrees
then,
sin (x/2)
cos (x/2)
tan (x/2)
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Hint #1
When 90 < x < 180, csc%28x%29+=+3 leads to cos%28x%29+=+-2%2Asqrt%282%29%2F3


Hint #2
90 < x < 180 has all sides cut in half to 45 < x/2 < 90
Angle x/2 is in quadrant 1 where all 6 trig functions are positive


Hint #3
The half angle identities are
sin%28x%2F2%29+=+%22%22%2B-sqrt%28+%281-cos%28x%29%29%2F2+%29

cos%28x%2F2%29+=+%22%22%2B-sqrt%28+%281%2Bcos%28x%29%29%2F2+%29

tan%28x%2F2%29+=+%22%22%2B-sqrt%28+%281-cos%28x%29%29%2F%281%2Bcos%28x%29%29+%29

However, keep the 2nd hint in mind, so we can drop the plus minus to write






Or once you know what sin(x/2) and cos(x/2) are, you can compute tangent like so
tan%28x%2F2%29+=+%28sin%28x%2F2%29%29%2F%28cos%28x%2F2%29%29

More trig identities can be found here
https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf