Question

Quais são os quadrantes do círculo trigonomêtrico no qual o cosseno é positivo?​

Answer 1

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⠀⠀☞ Dos quatro quadrantes do círculo trigonométrico somente no primeiro e no quarto o cosseno é positivo.  ✅

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⠀⠀ O círculo trigonométrico possui raio unitário (ou seja, igual à 1) e centro na origem (ou seja, em (0, 0)). Com sua origem angular na parte positiva do eixo x e sua rotação angular no sentido anti-horário temos quatro quadrantes:

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$\setlength{\unitlength}{0.95cm}\begin{picture}(6,5)\thicklines\put(0,0){\vector(1,0){5}}\put(0,0){\vector(0,1){5}}\put(0,0){\vector(-1,0){5}}\put(0,0){\vector(0,-1){5}}\put(4.8,0.2){x}\put(0.2,4.8){y}\bezier(-3,0)(-2.77,2.77)(0,3)\bezier(3,0)(2.77,2.77)(0,3)\bezier(-3,0)(-2.77,-2.77)(0,-3)\bezier(3,0)(2.77,-2.77)(0,-3)\put(-1.8,1.5){\Huge\bf{Q_2}}\put(0.9,1.5){\Huge\bf{Q_1}}\put(-1.7,-1.8){\Huge\bf{Q_3}}\put(0.9,-1.8){\Huge\bf{Q_4}}\put(-6,3.5){\dashbox{0.1}(3.4,1){ \sf 90^{\circ} < Q_2 < 180^{\circ} }}\put(3,3.5){\dashbox{0.1}(3.4,1){ \sf 0^{\circ} < Q_1 < 90^{\circ} }}\put(3,-4.2){\dashbox{0.1}(3.4,1){ \sf 270^{\circ} < Q_4 < 360^{\circ} }}\put(-6,-4.2){\dashbox{0.1}(3.4,1){ \sf 180^{\circ} < Q_3 < 270^{\circ} }}\bezier(0.7,0)(0.65,0.65)(0,0.7)\put(0.0,0.67){\line(7,28){0.38}}\bezier(-0.7,0)(-0.65,0.65)(0,0.7)\put(-0.29,0.4){\line(-4,-40){0.38}}\bezier(-0.7,0)(-0.65,-0.65)(0,-0.7)\put(0,-0.3){\line(-4,-22){0.38}}\bezier(0.7,0)(0.65,-0.65)(0,-0.7)\put(1.07,0){\line(-4,-31){0.38}}\put(5.4,-0.2){\LARGE\text{ \mathbf {0^{\circ}} }}\put(-0.4,5.5){\LARGE\text{ \mathbf {90^{\circ}} }}\put(-6.3,-0.1){\LARGE\text{ \mathbf {180^{\circ}} }}\put(-0.5,-5.8){\LARGE\text{ \mathbf {270^{\circ}} }}\put(-3.6,-0.45){\bf{-1}}\put(3.2,-0.45){\bf{1}}\put(0.2,-3.5){\bf{-1}}\put(0.2,3.4){\bf{1}}\end{picture}$

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$\sf (Esta~imagem~n\tilde{a}o~\acute{e}~visualiz\acute{a}vel~pelo~App~Brainly$ ☹ )

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⠀⠀Com um ângulo qualquer nós podemos traçar um segmento de reta que parte da origem até um ponto específico P da circunferência de forma que, neste ponto, outro segmento de reta seja traçado até o eixo x, formando assim um triângulo retângulo de hipotenusa unitária:

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$\setlength{\unitlength}{0.95cm}\begin{picture}(6,5)\thicklines\put(0,0){\vector(1,0){5}}\put(0,0){\vector(0,1){5}}\put(0,0){\vector(-1,0){5}}\put(0,0){\vector(0,-1){5}}\put(4.8,0.2){x}\put(0.2,4.8){y}\bezier(-3,0)(-2.77,2.77)(0,3)\bezier(3,0)(2.77,2.77)(0,3)\bezier(-3,0)(-2.77,-2.77)(0,-3)\bezier(3,0)(2.77,-2.77)(0,-3)\put(0,0){\circle*{0.13}}\put(1.33,2.68){\circle*{0.13}}\put(0,0){\line(1,2){1.3}}\put(1.35,0){\line(0,1){2.6}}\put(1.05,0){\line(0,1){0.3}}\put(1.05,0.3){\line(1,0){0.3}}\put(1.2,0.15){\circle*{0.13}}\bezier(0.6,0)(0.55,0.4)(0.29,0.5)\put(0.18,0.1){ \sf \Theta }\put(0.4,-0.4){ \sf \Delta x }\put(1.5,1.3){ \sf \Delta y }\put(0.4,1.4){1}\put(-3.6,-0.45){\bf{-1}}\put(3.2,-0.45){\bf{1}}\put(0.2,-3.5){\bf{-1}}\put(0.2,3.4){\bf{1}}\end{picture}$

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$\sf (Esta~imagem~n\tilde{a}o~\acute{e}~visualiz\acute{a}vel~pelo~App~Brainly$ ☹ )

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⠀⠀Relembrando que o cosseno é a divisão do cateto adjacente (Δx) pela hipotenusa (1) então temos que o cosseno só é positivo quando Δx > 0, o que só acontece nos quadrantes 1 e 4. ✅

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⠀⠀☀️ + exercícios sobre círculo trigonométrico:

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⠀⠀✈https://brainly.com.br/tarefa/37962049

⠀⠀✈https://brainly.com.br/tarefa/37007230

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