TrigFcns - 11-masala yechimi | MathModel.uz

1.3-bo'lim : Trig funktsiyalari

11. Kalkulyatordan foydalanmasdan $\displaystyle \sec \left( {\frac{{29\pi }}{4}} \right)$ ning aniq qiymatini aniqlang.

Yechim

Avval $\frac{{5\pi }}{4} + 6\pi = \frac{{29\pi }}{4}$ ekanligini va $6\pi $ uchta to'liq aylanish ekanligini eslab, $\frac{{29\pi }}{4}$ va $\frac{{5\pi }}{4}$ bir xil burchakni ifodalashini ko'rishimiz mumkin. Keyin, $\pi + \frac{\pi }{4} = \frac{{5\pi }}{4}$ va shunga o'xshash $\frac{{5\pi }}{4}$ni ifodalovchi chiziq uchinchi kvadrantda manfiy $x$ o'qi bilan $\frac{\pi }{4}$ burchagini hosil qilishini va biz ushbu muammo uchun quyidagi birlik doiraga ega bo'lishini unutmang.

$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 third quadrant is labeled with both $\frac{{29\pi }}{4}$ and $\frac{{5\pi }}{4}$ and lies on the same line through the origin as the angle $\frac{\pi }{4}$, so the coordinates of the two points on the circle differ only in sign.$

2-qadam

$\frac{{95\pi }}{4}$ ni ifodalovchi chiziq $\frac{\pi }{4}$ ni ifodalovchi chiziqning oyna tasviridir va shuning uchun $\frac{{29\pi }}{4}$ koordinatalari $\frac{\pi }{4}$ koordinatalari bilan bir xil bo‘ladi, bundan tashqari ikkala koordinata endi manfiy bo‘ladi. Shunday qilib, bizning yangi koordinatalarimiz $\left( { - \frac{{\sqrt 2 }}{2}, - \frac{{\sqrt 2 }}{2}} \right)$ bo'ladi va shuning uchun javob:

\[\sec \left( \frac{29\pi }{4} \right)=\frac{1}{\cos \left( \frac{29\pi }{4} \right)}=\frac{1}{-{}^{\sqrt{2}}/{}_{2}}=-\frac{2}{\sqrt{2}}=-\sqrt{2}\]