Find the value for sin(x/2) if the following conditions hold: tan(x) = -4/3 and x is in Quadrant IV.

Question

Find the value for sin(x/2) if the following conditions hold: tan(x) = -4/3 and x is in Quadrant IV.

 

 

Answer 1

Since x is in Q IV, you know that cos x is positive. BUT, you do not know what the quadrant of x/2 is unless you have some more information about the measure of x.

 

You will need the identity

 

sin (x/2) = ±√[(1 - cos x)/2],

 

where the sign is chosen based upon the quadrant of x/2. You cannot know this unless you know two quadrantal angles between which x lies.

 

Unless you have another identity for sin (x/2) which involves the angle x only and no square roots or sign choices, you will not be able to solve this and get a single correct answer.

 

The previous responder used a calculator, but problems like this usually must be done without one.

Answer 2

Tan (x) = - 4/3    Tan (x) - Sin (x) / Cos (x)  in Quad IV

 

Tan^-1 (- 1.333) = -53.12 degrees

 

x = -53.12 degrees so x/2 = -26.56 degrees

 

Sin (- 26.56) = - 0.447