TrigFcns - 9-masala yechimi | MathModel.uz

1.3-bo'lim : Trig funktsiyalari

9. Kalkulyatordan foydalanmasdan $\displaystyle \tan \left( {\frac{{15\pi }}{4}} \right)$ ning aniq qiymatini aniqlang.

Yechim

Avval biz $4\pi - \frac{\pi }{4} = \frac{{15\pi }}{4}$ ekanligini va $4\pi $ ikkita to'liq aylanish ekanligini ta'kidlashimiz mumkin, shuning uchun $\frac{{15\pi }}{4}$ va $ - \frac{\pi }{4}$ uchun terminal liniyasi bir xil burchakni ifodalaydi. Shuni ham yodda tutingki, $ - \frac{\pi }{4}$ to'rtinchi kvadrantda musbat $x$ o'qi bilan $\frac{\pi }{4}$ burchagini hosil qiladi va biz ushbu muammo uchun quyidagi birlik doiraga 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 fourth quadrant is labeled with both $\frac{{15\pi }}{4}$ and $ - \frac{\pi }{4}$ and a dashed vertical line connects the point where it meets the circle to the point for $\frac{\pi }{4}$, showing the two points have the same $x$-coordinate and $y$-coordinates that differ only in sign.$

2-qadam

$\frac{{15\pi }}{4}$ ni ifodalovchi chiziq koordinatalari $\frac{\pi }{4}$ ni ifodalovchi chiziq koordinatalari bilan bir xil bo‘ladi, bundan tashqari $y$ koordinatasi 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:

\[\tan \left( \frac{15\pi }{4} \right)=\frac{\sin \left( \frac{15\pi }{4} \right)}{\cos \left( \frac{15\pi }{4} \right)}=\frac{-{}^{\sqrt{2}}/{}_{2}}{{}^{\sqrt{2}}/{}_{2}}=-1\]