TrigFcns - 7-masala yechimi | MathModel.uz

1.3-bo'lim : Trig funktsiyalari

7. Kalkulyatordan foydalanmasdan $\displaystyle \cos \left( {\frac{{8\pi }}{3}} \right)$ ning aniq qiymatini aniqlang.

Yechim

Birinchidan, biz $2\pi + \frac{{2\pi }}{3} = \frac{{8\pi }}{3}$ ekanligini va $2\pi $ bitta to'liq inqilob bo'lgani uchun $\frac{{8\pi }}{3}$ va $\frac{{2\pi }}{3}$ burchaklari bir xil burchak ekanligini payqashimiz mumkin. Shuni ham yodda tutingki, $\pi - \frac{\pi }{3} = \frac{{2\pi }}{3}$ va shunga o'xshash $\frac{{8\pi }}{3}$ uchun terminal liniyasi ikkinchi kvadrantda manfiy $x$ o'qi bilan $\frac{\pi }{3}$ burchagini hosil qiladi va biz bu muammo uchun quyidagi birlik doirasiga ega bo'lamiz.

$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). A line in the second quadrant is labeled with both $\frac{{8\pi }}{3}$ and $\frac{{2\pi }}{3}$ and a dashed horizontal line connects the point where it meets the circle to the point for $\frac{\pi }{3}$, showing the two points have the same $y$-coordinate and $x$-coordinates that differ only in sign.$

2-qadam

$\frac{{8\pi }}{3}$ ni ifodalovchi chiziq koordinatalari $\frac{\pi }{3}$ ni ifodalovchi chiziq koordinatalari bilan bir xil bo‘ladi, bundan tashqari $x$ koordinatasi endi manfiy bo‘ladi. Shunday qilib, bizning yangi koordinatalarimiz $\left( { - \frac{1}{2},\frac{{\sqrt 3 }}{2}} \right)$ bo'ladi va shuning uchun javob:

\[\cos \left( {\frac{{8\pi }}{3}} \right) = - \frac{1}{2}\]