Exercise 5.3.1 solution
- \(\sin75^\circ= \sin ( 45^\circ + 30^\circ ) = \sin45^\circ\cos30^\circ + \cos45^\circ\sin30^\circ =\dfrac{\sqrt{6}+\sqrt{2}}{4}\)
- \(\cos75^\circ = \cos ( 45^\circ + 30^\circ ) =\cos45^\circ\cos30^\circ - \sin45^\circ\sin30^\circ =\dfrac{\sqrt{6}-\sqrt{2}}{4}\)
- \(\tan75^\circ = \sin75^\circ / \cos75^\circ =\dfrac{\sqrt{6}+\sqrt{2}}{\sqrt{6}-\sqrt{2}}=2+\sqrt{3}\)
- \(\csc75^\circ = 1 / \sin75^\circ =\sqrt{6}-\sqrt{2}\)
- \(\sec75^\circ = 1 / \cos75^\circ =\sqrt{6}+\sqrt{2}\)
- \(\cot75^\circ = 1 / \tan75^\circ =2-\sqrt{3}\)
Since \(15^\circ\) is the complementary angle to \(75^\circ\), we know that
- \(\sin15^\circ = \cos75^\circ =\dfrac{\sqrt{6}-\sqrt{2}}{4}\)
- \(\cos15^\circ = \sin75^\circ =\dfrac{\sqrt{6}+\sqrt{2}}{4}\)
- \(\tan15^\circ = \cot75^\circ =2-\sqrt{3}\)
- \(\csc15^\circ = \sec75^\circ =\sqrt{6}+\sqrt{2}\)
- \(\sec15^\circ = \csc75^\circ =\sqrt{6}-\sqrt{2}\)
- \(\cot15^\circ = \tan75^\circ =2+\sqrt{3}\)