SOLUTION: If sinx = 15/17 an X is in quadrant 1, find the exact values of sin X/2 and cos x/2

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Question 1081528: If sinx = 15/17 an X is in quadrant 1, find the exact values of sin X/2 and cos x/2
Found 2 solutions by math_helper, ikleyn:
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!

Note: In the figure b = |BC|, h=|BD|
+sin%28x%29+=+15%2F17+=+b%2Fc+ —> set b=15 and c=17
Solve for a: +a=sqrt%28c%5E2+-+b%5E2%29+=+sqrt%2817%5E2-15%5E2%29+=+sqrt%28289-225%29+=+sqrt%2864%29+=+8+
—
Because x is bisected: +%28b-h%29%2Fh+=+c%2Fa+ <<—<<< angle bisector theorem
++%2815-h%29%2Fh+=+17%2F8+ —> +25h+=+120+ —> +h=24%2F5+
—
We need 'd':

Finally, ready to solve:

+highlight%28sin%28x%2F2%29+=+%283%2F34%29%2Asqrt%2834%29%29+
—
I will leave it to you to find cos(x/2). You can use +cos%28x%2F2%29+=+a%2Fd+ and plug in a & d (easiest way).

Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
.
If sinx = 15/17 an X is in quadrant 1, find the exact values of sin X/2 and cos x/2
~~~~~~~~~~~~~~~

The standard technique for solving such problems is THIS: 

1.  cos(x) = sqrt%281-sin%5E2%28x%29%29 = sqrt%281-%2815%2F17%29%5E2%29 = sqrt%281+-+225%2F289%29 = sqrt%28%28289-225%29%2F289%29 = sqrt%2864%2F289%29 = 8%2F17


    notice sqrt is taken with the sign "+" since the angle "x" is in Q1 by the condition.



2.  Now  sin%28x%2F2%29 =    (apply the Trigonometry formula for half-argument)

         = sqrt%28%281-cos%28x%29%29%2F2%29 = sqrt%28%281-8%2F17%29%2F2%29 = sqrt%28%2817-8%29%2F%2817%2A2%29%29 = sqrt%289%2F34%29 = 3%2Fsqrt%2834%29 = %283%2Asqrt%2834%29%29%2F34.

    Again, the sign is "+" at the sqrt since x/2 is in Q1 together with x.



3.  Next  cos%28x%2F2%29 =    (apply the Trigonometry formula for half-argument)

         = sqrt%28%281%2Bcos%28x%29%29%2F2%29 = sqrt%28%281%2B8%2F17%29%2F2%29 = sqrt%28%2817%2B8%29%2F%2817%2A2%29%29 = sqrt%2825%2F34%29 = 5%2Fsqrt%2834%29 = %285%2Asqrt%2834%29%29%2F34.

    Again, the sign is "+" at the sqrt since x/2 is in Q1 together with x.

Solved. 

For half-argument formulas of Trigonometry see the lessons
    - FORMULAS FOR TRIGONOMETRIC FUNCTIONS
    - Trigonometric functions of half argument
in this site.