Question

What is the exact value of cos 112.5°?

Answer 1

We have 112.5 = 225/2; cos112.5 < 0 because 90 < 112.5 < 180;
cos 112.5 = cos(225/2);
We use the trigonometric formula: cos(x/2) = - $\sqrt{ \frac{1+cosx}{2} }$;
Then, cos(225/2) = - $\sqrt{ \frac{1+cos225}{2} }$;
We have the formula: cos(270 - x) = -sinx (because cos(a-b) = cosacosb + sinasinb);
Then, cos225 = cos(270-45) = -sin45 = -$\frac{ \sqrt{2} }{2} ;$
Finally, cos112.5 = -$\frac{ \sqrt{2- \sqrt{2} } }{2} ;$