0.1: The Trigonometric Functions
The Pythagorean trigonometric identity is
\[\sin^2x+\cos^2x=1,\nonumber \]
and the addition theorems are
\[\begin{aligned}\sin(x+y)&=\sin(x)\cos(y)+\cos(x)\sin(y) \\ \cos(x+y)&=\cos(x)\cos(y)-\sin(x)\sin(y).\end{aligned} \nonumber \]
Also, the values of \(\sin x\) in the first quadrant can be remembered by the rule of quarters, with \(0^◦ = 0,\: 30^◦ = π/6,\: 45^◦ = π/4,\: 60^◦ = π/3,\: 90^◦ = π/2\):
\[\begin{array}{ccccc}\sin 0^{\circ}=\sqrt{\frac{0}{4}}&&\sin 30^{\circ}=\sqrt{\frac{1}{4}},&&\sin 45^{\circ}=\sqrt{\frac{2}{4}}, \\ &\sin 60^{\circ}=\sqrt{\frac{3}{4}}&&\sin 90^{\circ}=\sqrt{\frac{4}{4}}.\end{array}\nonumber \]
The following symmetry properties are also useful:
\[\sin (\pi/2-x)=\cos x,\quad\cos(\pi /2-x)=\sin x;\nonumber \]
and
\[\sin (-x)=-\sin (x),\quad\cos (-x)=\cos(x).\nonumber \]