Question

Qual deve ser o valor do seno de um ângulo sabendo que Ele se encontra no primeiro quadrante e que o cosseno desse mesmo ângulo é 3/5

Answer 1

Resposta:

Explicação passo-a-passo:

A relação fundamental da trigonometria diz que

sen²x + cos²x = 1

sen² + (3/5)² = 1

sen²x + 9/25 = 1

sen² = 1 - 9/25

sen²x = (25 - 9)/25

sen²x = 16/25

sen x = ± √16/25

sen x = ± 4/5

Como o ângulo é do primeiro quadrante, logo sen x = 4/5

Answer 2

O seno do ângulo α é 4/5.

• Considere o ângulo α do primeiro quadrante na figura a seguir (ou anexa) e as razões trigonométricas do triângulo retângulo, seno e cosseno.

$\setlength{\unitlength}{1cm}\begin{picture}(6,5)(0,0) \thicklines % \thicklines ou \thinlines \put(0,0){\line(1,0){3}} \put(0,0){\line(3,4){3}} \put(3,0){\line(0,1){4}} \qbezier [30](0.6,0.8)(1,0.5)(1,0) \put(2.6,0.4){\line(1,0){0.4}} \put(2.6,0){\line(0,1){0.4}} \put(2.8,0.2){\circle*{0.07}} \put(1.1,0.5){ \alpha } \put(1.4,-0.5){\sf CA (3)} \put(3.3,1.5){\sf CO} \put(.4,2.3){\large \sf {hip (5)}}\end{picture}$$\setlength{\unitlength}{1cm} \begin{picture}(6,5)(0,0) \thicklines % \thicklines ou \thinlines\put(0,0){\line(1,0){3}}\put(0,0){\line(3,4){3}}\put(3,0){\line(0,1){4}}\qbezier [30](0.6,0.8)(1,0.5)(1,0)\put(2.6,0.4){\line(1,0){0.4}}\put(2.6,0){\line(0,1){0.4}} \put(2.8,0.2){\circle*{0.07}}\put(1.1,0.5){ \alpha }\put(1.4,-0.5){\sf CA (3)}\put(3.3,1.5){\sf CO}\put(.4,2.3){\large \sf {hip (5)}}\end{picture}$$\setlength{\unitlength}{.95cm}\begin{picture}(6,5)(0,0) \thicklines % \thicklines ou \thinlines \put(0,0){\line(1,0){3}} \put(0,0){\line(3,4){3}} \put(3,0){\line(0,1){4}} \qbezier [30](0.6,0.8)(1,0.5)(1,0) \put(2.6,0.4){\line(1,0){0.4}} \put(2.6,0){\line(0,1){0.4}} \put(2.8,0.2){\circle*{0.07}} \put(1.1,0.5){ \alpha } \put(1.4,-0.5){\sf CA (3)} \put(3.3,1.5){\sf CO} \put(.4,2.3){\large \sf {hip (5)}}\end{picture}$$\large \sf sen\ \alpha = \dfrac{cateto \ oposto}{hipotenusa}$

$\large \sf cos\ \alpha = \dfrac{cateto \ adjacente }{hipotenusa}$

$\setlength{\unitlength}{1cm}\begin{picture}(6,5)(0,0)\thicklines\put(0,0){\line(1,0){3}}\put(0,0){\line(3,4){3}}\put(3,0){\line(0,1){4}}\qbezier [30](0.6,0.8)(1,0.5)(1,0)\put(2.6,0.4){\line(1,0){0.4}}\put(2.6,0){\line(0,1){0.4}} \put(2.8,0.2){\circle*{0.07}}\put(1.1,0.5){ \alpha }\put(1.4,-0.5){\sf CA (3)}\put(3.3,1.5){\sf CO}\put(.4,2.3){\large \sf {hip (5)}}\end{picture}$$\setlength{\unitlength}{1cm}\begin{picture}(6,5)(0,0) \thicklines % \thicklines ou \thinlines \put(0,0){\line(1,0){3}} \put(0,0){\line(3,4){3}} \put(3,0){\line(0,1){4}} \qbezier [30](0.6,0.8)(1,0.4)(1,0) \put(2.6,0.4){\line(1,0){0.4}} \put(2.6,0){\line(0,1){0.4}} \put(2.8,0.2){\circle*{0.07}} \put(1.1,0.5){ \alpha } \put(1.4,-0.5){\sf CA (3)} \put(3.3,1.5){\sf CO} \put(.4,2.3){\large \sf {hip (5)}}\end{picture}$$\setlength{\unitlength}{1cm}\begin{picture}(6,5)(0,0) \thicklines % \thicklines ou \thinlines \put(0,0){\line(1,0){3}} \put(0,0){\line(3,4){3}} \put(3,0){\line(0,1){4}} \qbezier [30](0.6,0.8)(1,0.5)(1,0) \put(2.6,0.4){\line(1,0){0.4}} \put(2.6,0){\line(0,1){0.4}} \put(2.8,0.2){\circle*{0.07}} \put(1.1,0.5){ \alpha } \put(1.4,-0.5){\sf CA (3)} \put(3.3,1.5){\sf CO} \put(.4,2.3){\large \sf {hip (5)}}\end{picture}$

• Se o cosseno do ângulo α é 3/5, então o cateto adjacente a α mede 3 unidades e a hipotenusa mede 5 unidades.
• Determine a medida do cateto oposto a α usando o teorema de Pitágoras.

hip² = CO² + CA²

5² = CO² + 3²

25 = CO² + 9

CO² = 25 − 9

CO² = 16

CO = 4

• Determine o valor do seno do ângulo α.

$\large \sf sen\ \alpha = \dfrac{CO}{hip}$

$\boxed {\large \sf sen\ \alpha =\dfrac {4}{5}}$

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