Question

Use the sum or different to find he exact value of each trigonometric function
a) tan(π)/(12)
b) cos 150
c) cos(7π)/(12)

Answer 1

c. cos (pi/4 + pi/3) = cos pi/4 cos pi/3 - sin pi/4 sin pi/3)
= 1/sqrt2 * 1/2 - 1/sqrt2 * sqrt3 / 2

b. cos ( 90 + 60) = cos 90 cos 60 - sin 90 sin 60
= 0 * 1/2 - 1 * sqrt3 / 2

Answer 2

a. use the identity \[\tan(x-y)=\frac{\tan(x)-\tan(y)}{1+\tan(x) \tan(y)}\]with \[x=\frac{4\pi}{12}; \ \ y=\frac{3\pi}{12}\]\[\tan \left( \frac{\pi}{12} \right)=\tan \left( \frac{4\pi}{12}-\frac{3\pi}{12} \right)\]\[=\frac{\tan(4\pi/12)-\tan{3\pi/12)}}{1+\tan(4\pi/12)\tan(3\pi/12)}\]\[=\frac{\sqrt{3}-1}{1+\sqrt{3}}\]if you want rationalize the denominator\[=\frac{3-2\sqrt{3}+1}{3-1}\]\[=\frac{4-2\sqrt{3}}{2} \]\[=2-\sqrt{3}\]