Trigonometria Folha de Apoio
Identidades básicas
\tan(x) = \frac{\sin(x)}{\cos(x)}
\tan(x) = \frac{1}{\cot(x)}
\cot(x) = \frac{1}{\tan(x)}
\cot(x) = \frac{\cos(x)}{\sin(x)}
\sec(x) = \frac{1}{\cos(x)}
\csc(x) = \frac{1}{\sin(x)}
Identidades pitagoreanas
\cos^2(x)+\sin^2(x) = 1
\sec^2(x)-\tan^2(x) = 1
\csc^2(x)-\cot^2(x) = 1
Identidades de ângulo duplo
\sin(2x)=2\sin(x)\cos(x)
\cos(2x)=1-2\sin^2(x)
\cos(2x) = 2\cos^2(x)-1
\cos(2x) = \cos^2(x)-\sin^2(x)
\tan(2x) = \frac{2\tan(x)}{1-\tan^2(x)}
Identidades de soma/diferença
\sin(s+t) = \sin(s)\cos(t)+\cos(s)\sin(t)
\sin(s-t) = \sin(s)\cos(t)-\cos(s)\sin(t)
\cos(s+t) = \cos(s)\cos(t)-\sin(s)\sin(t)
\cos(s-t) = \cos(s)\cos(t)+\sin(s)\sin(t)
\tan(s+t) = \frac{\tan(s)+\tan(t)}{1-\tan(s)\tan(t)}
\tan(s-t) = \frac{\tan(s)-\tan(t)}{1+\tan(s)\tan(t)}
Identidades de soma para produto
\cos(s)\cos(t)=\frac{\cos(s-t)+\cos(s+t)}{2}
\sin(s)\sin(t)=\frac{\cos(s-t)-\cos(s+t)}{2}
\sin(s)\cos(t)=\frac{\sin(s+t)+\sin(s-t)}{2}
\cos(s)\sin(t)=\frac{\sin(s+t)-\sin(s-t)}{2}
Identidades de ângulo triplo
\sin(3x)=-\sin^3(x)+3\cos^2(x)\sin(x)
\sin(3x)=-4\sin^3(x)+3\sin(x)
\cos(3x)=\cos^3(x)-3\sin^2(x)\cos(x)
\cos(3x)=4\cos^3(x)-3\cos(x)
\tan(3x)=\frac{3\tan(x)-\tan^3(x)}{1-3\tan^2(x)}
\cot(3x)=\frac{3\cot(x)-\cot^3(x)}{1-3\cot^2(x)}
Imagens de função
y = \sin(x)
-1\le y\le 1
y = \cos(x)
-1\le y\le 1
y = \tan(x)
-\infty < y <\infty
y = \cot(x)
-\infty < y <\infty
y = \csc(x)
-\infty < y\le -1\:\bigcup \:1\le y < \infty
y = \sec(y)
-\infty < y\le -1\:\bigcup \:1\le y < \infty
y = \arcsin(x)
-\frac{\pi \:}{2}\:\le y\le \:\:\frac{\pi \:}{2}\:
y = \arccos(x)
0\:\le \:y\:\le \:\pi
y = \arctan(x)
-\frac{\pi \:}{2} < \:y < \frac{\pi \:}{2}:
y = \arccot(x)
0 < x < \pi
y = \arccsc(x)
0\le y <\frac{\pi }{2}\:\bigcup \:\pi\le y <\frac{3\pi }{2}
y = \arcsec(x)
-\pi < y\le -\frac{\pi }{2}\:\bigcup \:0 < y < \frac{\pi }{2}<\infty
Valores de função
sin(x)
cos(x)
tan(x)
cot(x)
0
0
1
0
\mathrm{Indefinido}
\frac{π}{6}
\frac{1}{2}
\frac{\sqrt{3}}{2}
\frac{\sqrt{3}}{3}
\sqrt{3}
\frac{π}{4}
\frac{\sqrt{2}}{2}
\frac{\sqrt{2}}{2}
1
1
\frac{π}{3}
\frac{\sqrt{3}}{2}
\frac{1}{2}
\sqrt{3}
\frac{\sqrt{3}}{3}
\frac{π}{2}
1
0
\mathrm{Indefinido}
0
\frac{2π}{3}
\frac{\sqrt{3}}{2}
-\frac{1}{2}
-\sqrt{3}
-\frac{\sqrt{3}}{3}
\frac{3π}{4}
\frac{\sqrt{2}}{2}
-\frac{\sqrt{2}}{2}
-1
-1
\frac{5π}{6}
\frac{1}{2}
-\frac{\sqrt{3}}{2}
-\frac{\sqrt{3}}{3}
-\sqrt{3}
π
0
-1
0
\mathrm{Indefinido}
\frac{7π}{6}
-\frac{1}{2}
-\frac{\sqrt{3}}{2}
\frac{\sqrt{3}}{3}
\sqrt{3}
\frac{5π}{4}
-\frac{\sqrt{2}}{2}
-\frac{\sqrt{2}}{2}
1
1
\frac{4π}{3}
-\frac{\sqrt{3}}{2}
-\frac{1}{2}
\sqrt{3}
\frac{\sqrt{3}}{3}
\frac{3π}{2}
-1
0
\mathrm{Indefinido}
0
\frac{5π}{3}
-\frac{\sqrt{3}}{2}
\frac{1}{2}
-\sqrt{3}
-\frac{\sqrt{3}}{3}
\frac{7π}{4}
-\frac{\sqrt{2}}{2}
\frac{\sqrt{2}}{2}
-1
-1
\frac{11π}{6}
-\frac{1}{2}
\frac{\sqrt{3}}{2}
-\frac{\sqrt{3}}{3}
-\sqrt{3}
2π
0
1
0
\mathrm{Indefinido}
Álgebra
