1.1: The Trigonometric Functions

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The Pythagorean trigonometric identity is \[\sin^2 x+\cos^2 x=1,\nonumber\] and the addition theorems are \[\begin{aligned}\sin(x+y)&=\sin(x)\cos(y)+\cos(x)\sin(y), \\ \cos(x+y)&=\cos(x)\cos(y)-\sin(x)\sin(y).\end{aligned}\]

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 \[\sin(-x)=-\sin(x),\quad\cos(-x)=\cos(x).\nonumber\]