Calcule o valor de:
a) sen 2pi/3
b) cos 2pi/3
c) sen 7pi/6
d) cos 7pi/6
e) sen 5pi/3
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a) ![\mathrm{sen\,}\dfrac{2\pi}{3} \mathrm{sen\,}\dfrac{2\pi}{3}](https://tex.z-dn.net/?f=%5Cmathrm%7Bsen%5C%2C%7D%5Cdfrac%7B2%5Cpi%7D%7B3%7D)
![=\mathrm{sen}\left(\pi-\dfrac{2\pi}{3}\right )\\ \\ =\mathrm{sen}\left(\dfrac{3\pi-2\pi}{3}\right )\\ \\ =\mathrm{sen\,}\dfrac{\pi}{3}\\ \\ =\dfrac{\sqrt{3}}{2} =\mathrm{sen}\left(\pi-\dfrac{2\pi}{3}\right )\\ \\ =\mathrm{sen}\left(\dfrac{3\pi-2\pi}{3}\right )\\ \\ =\mathrm{sen\,}\dfrac{\pi}{3}\\ \\ =\dfrac{\sqrt{3}}{2}](https://tex.z-dn.net/?f=%3D%5Cmathrm%7Bsen%7D%5Cleft%28%5Cpi-%5Cdfrac%7B2%5Cpi%7D%7B3%7D%5Cright+%29%5C%5C+%5C%5C+%3D%5Cmathrm%7Bsen%7D%5Cleft%28%5Cdfrac%7B3%5Cpi-2%5Cpi%7D%7B3%7D%5Cright+%29%5C%5C+%5C%5C+%3D%5Cmathrm%7Bsen%5C%2C%7D%5Cdfrac%7B%5Cpi%7D%7B3%7D%5C%5C+%5C%5C+%3D%5Cdfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D)
b)![\cos \dfrac{2\pi}{3} \cos \dfrac{2\pi}{3}](https://tex.z-dn.net/?f=%5Ccos+%5Cdfrac%7B2%5Cpi%7D%7B3%7D)
![=-\cos\left(\pi-\dfrac{2\pi}{3} \right )\\ \\ =-\cos\left(\dfrac{3\pi-2\pi}{3} \right )\\ \\ =-\cos \dfrac{\pi}{3}\\ \\ =-\dfrac{1}{2} =-\cos\left(\pi-\dfrac{2\pi}{3} \right )\\ \\ =-\cos\left(\dfrac{3\pi-2\pi}{3} \right )\\ \\ =-\cos \dfrac{\pi}{3}\\ \\ =-\dfrac{1}{2}](https://tex.z-dn.net/?f=%3D-%5Ccos%5Cleft%28%5Cpi-%5Cdfrac%7B2%5Cpi%7D%7B3%7D+%5Cright+%29%5C%5C+%5C%5C+%3D-%5Ccos%5Cleft%28%5Cdfrac%7B3%5Cpi-2%5Cpi%7D%7B3%7D+%5Cright+%29%5C%5C+%5C%5C+%3D-%5Ccos+%5Cdfrac%7B%5Cpi%7D%7B3%7D%5C%5C+%5C%5C+%3D-%5Cdfrac%7B1%7D%7B2%7D)
c)![\mathrm{sen\,}\dfrac{7\pi}{6} \mathrm{sen\,}\dfrac{7\pi}{6}](https://tex.z-dn.net/?f=%5Cmathrm%7Bsen%5C%2C%7D%5Cdfrac%7B7%5Cpi%7D%7B6%7D)
![=-\mathrm{sen}\left(\dfrac{7\pi}{6}-\pi \right )\\ \\ =-\mathrm{sen}\left(\dfrac{7\pi-6\pi}{6} \right )\\ \\ =-\mathrm{sen\,}\dfrac{\pi}{6}\\ \\ =-\dfrac{1}{2} =-\mathrm{sen}\left(\dfrac{7\pi}{6}-\pi \right )\\ \\ =-\mathrm{sen}\left(\dfrac{7\pi-6\pi}{6} \right )\\ \\ =-\mathrm{sen\,}\dfrac{\pi}{6}\\ \\ =-\dfrac{1}{2}](https://tex.z-dn.net/?f=%3D-%5Cmathrm%7Bsen%7D%5Cleft%28%5Cdfrac%7B7%5Cpi%7D%7B6%7D-%5Cpi+%5Cright+%29%5C%5C+%5C%5C+%3D-%5Cmathrm%7Bsen%7D%5Cleft%28%5Cdfrac%7B7%5Cpi-6%5Cpi%7D%7B6%7D+%5Cright+%29%5C%5C+%5C%5C+%3D-%5Cmathrm%7Bsen%5C%2C%7D%5Cdfrac%7B%5Cpi%7D%7B6%7D%5C%5C+%5C%5C+%3D-%5Cdfrac%7B1%7D%7B2%7D)
d)![\cos \dfrac{7\pi}{6} \cos \dfrac{7\pi}{6}](https://tex.z-dn.net/?f=%5Ccos+%5Cdfrac%7B7%5Cpi%7D%7B6%7D)
![=-\cos \left(\dfrac{7\pi}{6}-\pi \right )\\ \\ =-\cos \left(\dfrac{7\pi-6\pi}{6} \right )\\ \\ =-\cos \dfrac{\pi}{6}\\ \\ =-\dfrac{\sqrt{3}}{2} =-\cos \left(\dfrac{7\pi}{6}-\pi \right )\\ \\ =-\cos \left(\dfrac{7\pi-6\pi}{6} \right )\\ \\ =-\cos \dfrac{\pi}{6}\\ \\ =-\dfrac{\sqrt{3}}{2}](https://tex.z-dn.net/?f=%3D-%5Ccos+%5Cleft%28%5Cdfrac%7B7%5Cpi%7D%7B6%7D-%5Cpi+%5Cright+%29%5C%5C+%5C%5C+%3D-%5Ccos+%5Cleft%28%5Cdfrac%7B7%5Cpi-6%5Cpi%7D%7B6%7D+%5Cright+%29%5C%5C+%5C%5C+%3D-%5Ccos+%5Cdfrac%7B%5Cpi%7D%7B6%7D%5C%5C+%5C%5C+%3D-%5Cdfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D)
e)![\mathrm{sen\,}\dfrac{5\pi}{3} \mathrm{sen\,}\dfrac{5\pi}{3}](https://tex.z-dn.net/?f=%5Cmathrm%7Bsen%5C%2C%7D%5Cdfrac%7B5%5Cpi%7D%7B3%7D)
![=-\mathrm{sen}\left(2\pi-\dfrac{5\pi}{3}\right)\\ \\ =-\mathrm{sen}\left(\dfrac{6\pi-5\pi}{3}\right)\\ \\ =-\mathrm{sen\,}\dfrac{\pi}{3}\\ \\ =-\dfrac{\sqrt{3}}{2} =-\mathrm{sen}\left(2\pi-\dfrac{5\pi}{3}\right)\\ \\ =-\mathrm{sen}\left(\dfrac{6\pi-5\pi}{3}\right)\\ \\ =-\mathrm{sen\,}\dfrac{\pi}{3}\\ \\ =-\dfrac{\sqrt{3}}{2}](https://tex.z-dn.net/?f=%3D-%5Cmathrm%7Bsen%7D%5Cleft%282%5Cpi-%5Cdfrac%7B5%5Cpi%7D%7B3%7D%5Cright%29%5C%5C+%5C%5C+%3D-%5Cmathrm%7Bsen%7D%5Cleft%28%5Cdfrac%7B6%5Cpi-5%5Cpi%7D%7B3%7D%5Cright%29%5C%5C+%5C%5C+%3D-%5Cmathrm%7Bsen%5C%2C%7D%5Cdfrac%7B%5Cpi%7D%7B3%7D%5C%5C+%5C%5C+%3D-%5Cdfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D)
b)
c)
d)
e)
Respondido por
0
O valor dos ângulos, são:
- a) √3/2
- b) - √2/2
- c) - 1/2
- d) - √3/2
- e) - √3/2
Ângulos notáveis
Os ângulos notáveis são ângulos que são bastante conhecidos os seus valores de seno, cosseno e tangente, onde eles são os ângulos de 30º, 45º e 60º, são, respectivamente:
- 1/2, √2/2, √3/2
- √3/2, √2/2, 1/2
- √3/3, 1, √3
Para calcularmos o valor de cada ângulo, devemos converter para graus. Temos:
a) sen 2pi/3
π está para 180º
2π/3 está para x
x*π = 180º*2π/3
x = 120º
sen 2π/3 = √3/2
b) cos 2pi/3
cos 120º = - √2/2
c) sen 7pi/6
π está para 180º
7π/6 está para x
x*π = 180º*7π/6
x = 210º
sen 210º = - sen 30º = - 1/2
d) cos 7pi/6
cos 210º = - cos 30º = - √3/2
e) sen 5pi/3
π está para 180º
5π/3 está para x
x*π = 180º*5π/3
x = 300º
sen 300º = - sen 60º = - √3/2
Aprenda mais sobre ângulos notáveis aqui:
https://brainly.com.br/tarefa/51366377
#SPJ3
Anexos:
![](https://pt-static.z-dn.net/files/d8f/2b387e746e297f49dfeaee1f26947bd1.jpg)
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