TrigFcns - 4-masala yechimi | MathModel.uz

1.3-bo'lim : Trig funktsiyalari

4. Kalkulyatordan foydalanmasdan $\displaystyle \cos \left( { - \frac{{2\pi }}{3}} \right)$ ning aniq qiymatini aniqlang.

Yechim

Avval biz $ - \pi + \frac{\pi }{3} = - \frac{{2\pi }}{3}$ ning (manfiy burchaklar soat yoʻnalishi boʻyicha, musbat burchaklar esa soat yoʻnalishi boʻyicha teskari yoʻnalishda aylanishini eslatib oʻtamiz) $ - \frac{{2\pi }}{3}$ uchun terminal chizigʻi uchinchi kvadrantda manfiy $x$ oʻqi bilan $\frac{\pi }{3}$ burchagini hosil qilishini payqashimiz mumkin va bu muammo uchun quyidagi birlik doiraga ega boʻlamiz.

$A unit circle with the angles $\frac{\pi }{6}$, $\frac{\pi }{4}$ and $\frac{\pi }{3}$ drawn in the first quadrant and labeled with the coordinates $\left( {\frac{{\sqrt 3 }}{2},\frac{1}{2}} \right)$, $\left( {\frac{{\sqrt 2 }}{2},\frac{{\sqrt 2 }}{2}} \right)$ and $\left( {\frac{1}{2},\frac{{\sqrt 3 }}{2}} \right)$ where they meet the circle. The axes are labeled with the angles 0 and $2\pi $, $\frac{\pi }{2}$, $\pi $ and $\frac{{3\pi }}{2}$ and the points (1,0), (0,1), (-1,0) and (0,-1). The angle $ - \frac{{2\pi }}{3}$ is drawn in the third quadrant and lies on the same line through the origin as the angle $\frac{\pi }{3}$, so the coordinates of the two points on the circle differ only in sign.$

2-qadam

$ - \frac{{2\pi }}{3}$ni ifodalovchi chiziq $\frac{\pi }{3}$ ni ifodalovchi chiziqning oyna tasviridir va shuning uchun $ - \frac{{2\pi }}{3}$ koordinatalari $\frac{\pi }{3}$ koordinatalari bilan bir xil bo‘ladi, bundan tashqari ikkala koordinata ham salbiy bo‘ladi. Shunday qilib, bizning yangi koordinatalarimiz $\left( { - \frac{1}{2}, - \frac{{\sqrt 3 }}{2}} \right)$ bo'ladi va shuning uchun javob:

\[\cos \left( { - \frac{{2\pi }}{3}} \right) = - \frac{1}{2}\]