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»TRIGONOMETRIE

Trigonometria, ca ramură a matematicii, se ocupă cu măsurarea unghiurilor şi

a lungimilor de segmente, cu ajutorul funcţiilor sin, cos, tg şi ctg.

Instrumentarul necesar pentru indeplinirea acestor activităţi conţine

numeroase teoreme şi identităţi trigonometrice fundamentale.

Acestea sunt:

2) APLICATIA-1

Data publicării : 01.09.2010

Suport teoretic:

Ecuatii trigonometrice, identitati trigonometrice remarcabile.

Enunt:

Sa se afle suma S a solutiilor ecuatiei trigonometrice

${4{sin}^3}{x}+3{cos2x}-6{sinx}+1=0,\;{x}\in{[0,2\pi]}.$ {4{sin}^3}{x}+3{cos2x}-6{sinx}+1=0,\;{x}\in{[0,2\pi]}.

Raspuns:

$S=\frac{5\pi}{2}.$ S=\frac{5\pi}{2}.

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1) TEORIE

Data publicării : 16.10.2008

Identitati remarcabile:

${\sin ^2}{a}+{\cos^2}{a}=1,\forall{a}\in{\mathbb{R}}.$ {\sin ^2}{a}+{\cos^2}{a}=1,\forall{a}\in{\mathbb{R}}.
$\sin(-x)=-\sin{x} ,\forall{x}\in{\mathbb{R}}.$ \sin(-x)=-\sin{x} ,\forall{x}\in{\mathbb{R}}.
$\cos{(-x)}=\cos{x} ,\forall{x}\in{\mathbb{R}}.$ \cos{(-x)}=\cos{x} ,\forall{x}\in{\mathbb{R}}.
${tgx} = \frac{\sin{x}}{\cos{x}},\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix} (2k+1)\frac{\pi}{2}|k\in{\mathbb{Z}}\end{Bmatrix}}.$ {tgx} = \frac{\sin{x}}{\cos{x}},\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix} (2k+1)\frac{\pi}{2}|k\in{\mathbb{Z}}\end{Bmatrix}}.
${tg(-x)} = {- tgx}, \forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1)\frac{\pi}{2}|k\in{\mathbb{Z}}\end{Bmatrix}}.$ {tg(-x)} = {- tgx}, \forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1)\frac{\pi}{2}|k\in{\mathbb{Z}}\end{Bmatrix}}.
${ctg(-x)}={- ctgx},\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}k\pi|k\in{\mathbb{Z}}\end{Bmatrix}}.$ {ctg(-x)}={- ctgx},\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}k\pi|k\in{\mathbb{Z}}\end{Bmatrix}}.
$\cos{(a+b)}=\cos{a}\cos{b}-\sin{a}\sin{b}, \forall{a,b}\in{\mathbb{R}}.$ \cos{(a+b)}=\cos{a}\cos{b}-\sin{a}\sin{b}, \forall{a,b}\in{\mathbb{R}}.
$\cos{2a}={{\cos}^2}{a}-{{\sin}^2}{a} = 2{{\cos}^2}{a} - 1 = 1 - 2{{\sin}^2}{a},\forall{a}\in{\mathbb{R}}.$ \cos{2a}={{\cos}^2}{a}-{{\sin}^2}{a} = 2{{\cos}^2}{a} - 1 = 1 - 2{{\sin}^2}{a},\forall{a}\in{\mathbb{R}}.
$\cos{3a} =\ cos{a}(4{\cos^2}{a}-3),\forall{a}\in{\mathbb{R}}.$ \cos{3a} =\ cos{a}(4{\cos^2}{a}-3),\forall{a}\in{\mathbb{R}}.
$\cos{(a-b)}=\cos{a}\cos{b}+\sin{a}\sin{b}, \forall{a,b}\in{\mathbb{R}}.$ \cos{(a-b)}=\cos{a}\cos{b}+\sin{a}\sin{b}, \forall{a,b}\in{\mathbb{R}}.
$\cos{(\frac{\pi}{2}-x)}=\sin{x},\forall{x}\in{\mathbb{R}}.$ \cos{(\frac{\pi}{2}-x)}=\sin{x},\forall{x}\in{\mathbb{R}}.
$\cos{(\pi-x)}= -\ cos{x}, \forall{x}\in{\mathbb{R}}.$ \cos{(\pi-x)}= -\ cos{x}, \forall{x}\in{\mathbb{R}}.
$\cos{(x+2k\pi)} =\ cos{x},\forall{x}\in{\mathbb{R}},\forall{k}\in{\mathbb{Z}}.$ \cos{(x+2k\pi)} =\ cos{x},\forall{x}\in{\mathbb{R}},\forall{k}\in{\mathbb{Z}}.
$\sin{(a+b)} =\ sin{a}\cos{b} +\ cos{a}\sin{b},\forall{a,b}\in{\mathbb{R}}.$ \sin{(a+b)} =\ sin{a}\cos{b} +\ cos{a}\sin{b},\forall{a,b}\in{\mathbb{R}}.
$\sin{2a} = 2\sin{a}\cos{a},\forall{a}\in{\mathbb{R}}.$ \sin{2a} = 2\sin{a}\cos{a},\forall{a}\in{\mathbb{R}}.