Find the exact value

Question

Cos(x)=1/4 

 

csc(x) > 0

 

find sin (x/2) 

Answer 1

Moe,

 

The key to solving this problem is to look at the trigonometric definitions and identities.

 

The sin 1/2 angle identity (http://www.wyzant.com/resources/lessons/math/trigonometry/half-angle-double-angle-formulas) is

 

sin (x/2) = ±√((1-cos(x))/2)

 

So sin (x/2) = ± √(3/8) = ±√(3/8) x √(8/8) = ±√(24)/8 = ±2√(6)/8 = ±√(6)/4

 

Now in this case the +/- is not that both answers are right, rather that we need more information to figure out whether the sin is positive or negative in this case.  Here we are told the csc(x)>0, and the
csc(x)= 1/sin(x) so if the csc(x) is positive, so must the sine of x, so the exact answer is √(6)/4

 

The article at the webpage has a good explanation if you have more questions.  I hope this helps.  John

Answer 2

sin (x/2) = ± [(1-cos x)/2]^1/2                    (half-angle identity)

 

= ± [(1-1/4)/2]^1/2                                        (cos x = 1/4 as stated in the problem)

 

= ± (3/8)^1/2 = ±(1/2)√(3/2) = ±(√6)/4       (Simplify the radical)

 

Since csc x = (1/sin x) > 0 as stated in the problem, then sin (x) > 0 which eliminates the -√6/4 term.  Hence:

 

sin(x/2) = +(√6)/4