Half angle formula

[bethoumyvision]

The problem reads;
Find the exact values of sin a/2, cos a/2, and tan a/2 using the half angle formula.
I am given: sin a=5/13 pi2/< a < pi

I've tried several approaches and I do not end up with the answer the text book gives;
sin a/2= 5 sqrt26/26
cos a/2= sqrt26/26
tan a/2= 5

Any help would be appreciated. Thanks

 

[Loren]

Find the exact values of sin a/2, cos a/2, and tan a/2 using the half angle formula.
I am given: sin a=5/13 pi2/< a < pi

I assume you mean pi/2 which has us deal with the angle in quadrant II.
Draw a right triangle in quadrant II with the ordinate = 5 and the radius (hypotenuse) = 13. Use the Pythagorean Theorem to determine the abscisa.
Now you can determine the sine, cosine and tangent of the angle. Then use the half angle formulas.

For instance, cos a = -12/13.

\(\displaystyle \sin \frac{a}{2}= \pm \sqrt{\frac{1-\cos a}{2}}=\pm \sqrt{\frac{1-(-12/13)}{2}}=\) etc.