Trigonometric functions¶
Part 2 · Functions · Chapter 5 · lecture notes by Fabio Furini · Chapter PDF
1. Trigonometric functions¶
Definition 1: sine function and cosine function
The trigonometric functions:
\[\begin{align}
\label{seno} f: \mathbb{R} \rightarrow \mathbb{R},~~~ f: x \mapsto \sin x \\[2ex]
\label{coseno}
f: \mathbb{R} \rightarrow \mathbb{R},~~~ f: x \mapsto \cos x
\end{align}\]
are called sine and cosine.
-
The sine is an odd function, while the cosine is an even function. Hence we have:
\[ \sin(-x)=-\sin x, ~~~\forall x \in \R {\rm ~~~~and~~~~} \cos(-x)=\cos x, ~~~\forall x \in \R \] -
The sine and the cosine are periodic functions with period:
\[ T=2\: \pi {\rm ~~~~and~we~have~the~fundamental~identity~~~} \sin^2 x + \cos^2 x = 1,~~ \forall x \in \mathbb{R} \]
We have
\[ \cos\left(x -\frac{\pi}{2}\right) = \sin x,~~ \forall x \in \mathbb{R} \] -
From the sign and the monotonicity of the sine function we have:
\[ \sin x = 0 \Longleftrightarrow x= k\;\pi,~ \forall k \in \Z \qquad \begin{cases} \sin x > 0 & {\rm if~~~} x \in (2\;k\;\pi~,~ 2\;k\;\pi + \pi),~ \forall k \in \Z \\[3ex] \sin x < 0 & {\rm if~~~} x \in (2\;k\;\pi-\pi~,~ 2\;k\;\pi),~ \forall k \in \Z \end{cases} \]\[ \begin{cases} {\rm if~~~} x \in (2\;k\;\pi- \frac{\pi}{2},~ 2\;k\;\pi + \frac{\pi}{2}),~ \forall k \in \Z & f {\rm ~~is~increasing} \\[3ex] {\rm if~~~} x \in (2\;k\;\pi+ \frac{\pi}{2}~,~ 2\;k\;\pi + \frac{3\;\pi}{2}),~ \forall k \in \Z & f {\rm ~~is~decreasing} \end{cases} \] -
From the sign and the monotonicity of the cosine function we have:
\[ \cos x = 0 \Longleftrightarrow x= k\;\pi-\frac{\pi}{2},~ \forall k \in \Z \qquad \begin{cases} \cos x > 0 & {\rm if~~~} x \in (2\;k\;\pi~-\frac{\pi}{2},~ 2\;k\;\pi + \frac{\pi}{2}),~ \forall k \in \Z \\[3ex] \cos x < 0 & {\rm if~~~} x \in (2\;k\;\pi+\frac{\pi}{2}~,~ 2\;k\;\pi +\frac{3\;\pi}{2}),~ \forall k \in \Z \end{cases} \]\[ \begin{cases} {\rm if~~~} x \in (2\;k\;\pi+ \pi,~ 2\;k\;\pi + 2\; \pi),~ \forall k \in \Z & f {\rm ~~is~increasing} \\[3ex] {\rm if~~~} x \in (2\;k\;\pi ~,~ 2\;k\;\pi + \pi),~ \forall k \in \Z & f {\rm ~~is~decreasing} \end{cases} \]
The trigonometric functions:
\[\begin{align}
\label{tangente} f: \mathbb{R} \setminus \{k\;\pi-\frac{\pi}{2},~ \forall k \in \Z\} \rightarrow \mathbb{R},~~~ f: x \mapsto \tan x = \frac{\sin x}{\cos x} \\[2ex]
\label{cotangente}
f: \mathbb{R}\setminus \{k\;\pi,~ \forall k \in \Z\} \rightarrow \mathbb{R},~~~ f: x \mapsto \cot x = \frac{\cos x}{\sin x}
\end{align}\]
are called tangent and cotangent; they are both odd and periodic with period \(T=\pi\).
The trigonometric functions:
\[\begin{align}
\label{secante} f: \mathbb{R} \setminus \{k\;\pi-\frac{\pi}{2},~ \forall k \in \Z\} \rightarrow \mathbb{R},~~~ f: x \mapsto \sec x = \frac{1}{\cos x} \\[2ex]
\label{cosecante}
f: \mathbb{R}\setminus \{k\;\pi,~ \forall k \in \Z\} \rightarrow \mathbb{R},~~~ f: x \mapsto \csc x = \frac{1}{\sin x}
\end{align}\]
are called secant and cosecant and they are periodic with period \(T=2\;\pi\). The secant function is even, while the cosecant function is odd.
2. Values of the trigonometric functions¶
| \(\cos\) | \(\sin\) | \(\tan\) | \(\cot\) | \(\sec\) | \(\csc\) | |
| 0 | 1 | 0 | 0 | \(\pm \infty\) | 1 | \(\pm \infty\) |
| \(x=\frac{\pi}{6}\) (\(30^{\circ}\)) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{2}\) | \(\frac{\sqrt{3}}{3}\) | \(\sqrt{3}\) | \(\frac{2}{3}\:\sqrt{3}\) | 2 |
| \(x=\frac{\pi}{4}\) (\(45^{\circ}\)) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{2}}{2}\) | 1 | 1 | \(\sqrt{2}\) | \(\sqrt{2}\) |
| \(x=\frac{\pi}{3}\) (\(60^{\circ}\)) | \(\frac{1}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(\sqrt{3}\) | \(\frac{\sqrt{3}}{3}\) | 2 | \(\frac{2}{3}\: \sqrt{3}\) |
| \(x=\frac{\pi}{2}\) (\(90^{\circ}\)) | 0 | 1 | \(\pm \infty\) | 0 | \(\pm \infty\) | 1 |
| \(x=\pi\) (\(180^{\circ}\)) | \-1 | 0 | 0 | \(\pm \infty\) | \-1 | \(\pm \infty\) |
| \(\phantom{-} \cos\) | \(\phantom{-} \sin\) | \(\phantom{-} \tan\) | \(\phantom{-} \cot\) | |
| \(\phantom{-}x\) | \(\phantom{-} \cos x\) | \(\phantom{-} \sin x\) | \(\phantom{-} \tan x\) | \(\phantom{-} \cot x\) |
| \(-x\) | \(\phantom{-} \cos x\) | \(- \sin x\) | \(-\tan x\) | \(-\cot x\) |
| \(\frac{\pi}{2} +x\) | \(-\sin x\) | \(\phantom{-} \cos x\) | \(-\cot x\) | \(-\tan x\) |
| \(\frac{\pi}{2} - x\) | \(\phantom{-} \sin x\) | \(\phantom{-} \cos x\) | \(\phantom{-} \cot x\) | \(\phantom{-} \tan x\) |
| \(\pi+x\) | \(-\cos x\) | \(-\sin x\) | \(\phantom{-} \tan x\) | \(\phantom{-} \cot x\) |
| \(\pi-x\) | \(-\cos x\) | \(\phantom{-} \sin x\) | \(-\tan x\) | \(-\cot x\) |
3. Main trigonometric formulas¶
Relations among the trigonometric functions
| \(\sin x\) | \(\cos x\) | \(\tan x\) | \(\cot x\) | |
| \(\sin x\) | - | \(\pm \sqrt{1 - \cos^2 x}\) | \(\pm \sqrt{\frac{\tan^2 x}{1 + \tan^2 x}}\) | \(\pm \sqrt{\frac{1}{1 + \cot^2 x}}\) |
| \(\cos x\) | \(\pm \sqrt{1 - \sin^2 x}\) | - | \(\pm \sqrt{\frac{1}{1 + \tan^2 x}}\) | \(\pm \sqrt{\frac{\cot^2 x}{1 + \cot^2 x}}\) |
| \(\tan x\) | \(\pm\sqrt{\frac{\sin^2 x}{1 - \sin^2 x}}\) | \(\pm\sqrt{\frac{1- \cos^2 x}{\cos^2 x}}\) | - | \(\frac{1}{\cot x}\) |
| \(\cot x\) | \(\pm\sqrt{\frac{1- \sin^2 x}{\sin^2 x}}\) | \(\pm\sqrt{\frac{\cos^2 x}{1 - \cos^2 x}}\) | \(\frac{1}{\tan x}\) | - |
where the symbol \(\pm\) means that the sign depends on the quadrant in which \(x\) lies
Addition and subtraction
\[\begin{align}
\label{trig:add}
\sin ( x_1 \pm x_2) &= \sin x_1 \: \cos x_2 \pm \sin x_2 \: \cos x_1\\[2ex]
\cos ( x_1 \pm x_2) &= \cos x_1 \: \cos x_2 \mp \sin x_1 \: \sin x_2
\end{align}\]
\[\begin{align}
\tan ( x_1 \pm x_2) &= \frac{\tan x_1 \pm \tan x_2}{1 \mp \tan x_1 \: \tan x_2}\\[2ex]
\cot ( x_1 \pm x_2) &= \frac{\cot x_1 \: \cot x_2 \mp 1}{\cot x_2 \pm \cot x_1}
\end{align}\]
Double-angle formulas
\[\begin{align}
\sin ( 2\: x) &= 2\: \sin x \: \cos x \\[2ex]
\cos ( 2\: x) &= \cos^2 x - \sin^2 x \nonumber\\[2ex]
&= 2\: \cos^2 x - 1 \nonumber\\[2ex]
&= 1 - 2\: \sin^2 x
\end{align}\]
\[\begin{align}
\tan ( 2\: x) &= \frac{2 \:\tan x}{1 - \tan^2 x}\\[2ex]
\cot ( 2\: x) &= \frac{\cot^2 x -1}{2\: \cot x}
\end{align}\]
Triple-angle formulas
\[\begin{align}
\sin ( 3\: x) &= 3\: \sin x - 4\: \sin^3 x \\[2ex]
\cos ( 3\: x) &= 4\: \cos^3 x - 3\: \cos x
\end{align}\]
Half-angle formulas
\[\begin{align}
\sin \frac{x}{2} &= \pm \sqrt{\frac{1 - \cos x}{2}}\\[2ex]
\cos \frac{x}{2} &= \pm \sqrt{\frac{1 + \cos x}{2}}
\end{align}\]
\[\begin{align}
\tan \frac{x}{2} &= \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}} \nonumber\\[2ex]
&= \frac{\sin x}{1 + \cos x} \nonumber\\[2ex]
&= \frac{1 - \cos x}{\sin x}
\end{align}\]
\[\begin{align}
\cot \frac{x}{2} &= \pm \sqrt{\frac{1 + \cos x}{1 - \cos x}} \nonumber\\[2ex]
&= \frac{\sin x}{1 - \cos x} \nonumber\\[2ex]
&= \frac{1 + \cos x}{\sin x}
\end{align}\]
Parametric formulas (tangent half-angle substitution)
\[\begin{align}
\sin {x} &= \frac{2\: \tan \frac{{x}}{2}}{1+ \tan^2 \frac{{x}}{2}}\\[2ex]
\cos {x} &= \frac{1- \tan^2 \frac{{x}}{2}}{1+ \tan^2 \frac{{x}}{2}}
\end{align}\]
Sum-to-product formulas (prosthaphaeresis)
\[\begin{align}
\sin {x_1} + \sin {x_2} &= \phantom{-} 2\: \sin \frac{x_1 + x_2}{2} \: \cos \frac{x_1 - x_2}{2}\\[2ex]
\sin {x_1} - \sin {x_2} &= \phantom{-}2\: \sin \frac{x_1 - x_2}{2} \: \cos \frac{x_1 + x_2}{2}\\[2ex]
\cos {x_1} + \cos {x_2} &= \phantom{-}2\: \cos \frac{x_1 + x_2}{2} \: \cos \frac{x_1 - x_2}{2}\\[2ex]
\cos {x_1} - \cos {x_2} &= - 2\: \sin \frac{x_1 + x_2}{2} \: \sin \frac{x_1 - x_2}{2}
\end{align}\]
Product-to-sum formulas (Werner's formulas)
\[\begin{align}
\sin {x_1} \: \sin {x_2} &= \frac{1}{2} \bigg(\cos(x_1-x_2) - \cos(x_1 +x_2) \bigg)\\[2ex]
\sin {x_1} \: \cos {x_2} &= \frac{1}{2} \bigg(\sin(x_1-x_2) + \sin(x_1 +x_2) \bigg)\\[2ex]
\cos {x_1} \: \cos {x_2} &= \frac{1}{2} \bigg(\cos(x_1-x_2) + \cos(x_1 +x_2) \bigg)
\end{align}\]
Additional formulas
\[\begin{align}
\sin^2 x &= \frac{1 - \cos (2\:x)}{2}\\[2ex]
\cos^2 x &= \frac{1 + \cos (2\:x)}{2}\\[2ex]
\sin x \:\cos x &= \frac{\sin (2\:x)}{2}
\end{align}\]