SOLUTION: Find sin(x/2), cos(x/2), and tan(x/2) from the given information tan(x)=1; 0 degrees< x <90 degrees

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Question 1056788: Find sin(x/2), cos(x/2), and tan(x/2) from the given information
tan(x)=1; 0 degrees< x <90 degrees
Answer by ikleyn(52782) About Me  (Show Source):
You can put this solution on YOUR website!
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Find sin(x/2), cos(x/2), and tan(x/2) from the given information
tan(x)=1; 0 degrees< x <90 degrees
~~~~~~~~~~~~~~~~~~~~~~~~~ 

The condition "tan(x)=1; 0° < x < 90° " implies

x = 45°.


Then x/2 = 22°30'.


To calculate  sin(22°30'), apply the formula of half argument for sines  (see the lesson  Trigonometric functions of half argument in this site):

sin(22°30') = sqrt%28%281-cos%28pi%2F4%29%29%2F2%29 = sqrt%28%281-sqrt%282%29%2F2%29%2F2%29 = sqrt%28%282-sqrt%282%29%29%2F4%29 = sqrt%282-sqrt%282%29%29%2F2.



Similarly, 

cos(22°30') = sqrt%28%281%2Bcos%28pi%2F4%29%29%2F2%29 = sqrt%28%281%2Bsqrt%282%29%2F2%29%2F2%29 = sqrt%28%282%2Bsqrt%282%29%29%2F4%29 = sqrt%282%2Bsqrt%282%29%29%2F2.


Hence,

tan(22°30') = sin(22°30')/cos(22°30') = sqrt%28%282-sqrt%282%29%29%2F%282%2Bsqrt%282%29%29%29 =  = sqrt%28+%28%282-sqrt%282%29%29%5E2%29%2F%282%5E2-%28sqrt%282%29%29%5E2%29+%29 = %282-sqrt%282%29%29%2Fsqrt%282%29 = sqrt%282%29-1.

See the lesson  Trigonometric functions of half argument - Examples in this site.