SOLUTION: If csc (theta) <0 and cos(theta)=5/13, find sin (theta/2) and cos (theta/2).

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Question 1076776: If csc (theta) <0 and cos(theta)=5/13, find sin (theta/2) and cos (theta/2).
Answer by ikleyn(52786) About Me  (Show Source):
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If csc (theta) <0 and cos(theta)=5/13, find sin (theta/2) and cos (theta/2).
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csc%28theta%29 < 0 means sin%28theta%29 < 0.

Together with cos%28theta%29 = 5%2F13, it means that theta is the angle in QIV.

Hence,  theta%2F2 is the angle in QII.


Then  sin%28theta%2F2%29 = sqrt%28%281-cos%28theta%29%29%2F2%29 = sqrt%28%281-%285%2F13%29%29%2F2%29 = sqrt%28%2813-5%29%2F%2813%2A2%29%29 = sqrt%288%2F26%29 = 2%2Fsqrt%2813%29 = %282%2Asqrt%2813%29%29%2F13.

The sign is "+" since sine is positive in QII.


cos%28theta%2F2%29 = -sqrt%28%281%2Bcos%28theta%29%29%2F2%29 = -sqrt%28%281%2B%285%2F13%29%29%2F2%29 = -sqrt%28%2813%2B5%29%2F%2813%2A2%29%29 = -sqrt%2818%2F26%29 = -sqrt%289%2F13%29 = %28-3%29%2Fsqrt%2813%29.

The sign is "-" since cosine is negative in QII.

To see more solved similar problems on calculating trig functions look into the lessons
    - Calculating trigonometric functions of angles
    - Advanced problems on calculating trigonometric functions of angles
in this site.

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Trigonometry: Solved problems".