TrigFcns - 5-masala yechimi | MathModel.uz

1.3-bo'lim : Trig funktsiyalari

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

Yechim

Birinchidan, biz $\pi - \frac{\pi }{4} = \frac{{3\pi }}{4}$ va shunga o'xshash (manfiy burchaklar soat yo'nalishi bo'yicha aylantirilishini eslab qolishimiz mumkin) biz $\frac{{3\pi }}{4}$ uchun terminal chizig'i ikkinchi kvadrantda manfiy $x$ o'qi bilan $\frac{\pi }{4}$ burchagini hosil qilishini ko'rishimiz mumkin 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). The angle $\frac{{3\pi }}{4}$ is drawn in the second quadrant and a dashed horizontal line connects the point where it meets the circle to the point for $\frac{\pi }{4}$, showing the two points have the same $y$-coordinate and $x$-coordinates that differ only in sign.$

2-qadam

$\frac{{3\pi }}{4}$ ni ifodalovchi chiziq koordinatalari $\frac{\pi }{4}$ ni ifodalovchi chiziq koordinatalari bilan bir xil bo‘ladi, bundan tashqari $x$ 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:

\[\tan \left( \frac{3\pi }{4} \right)=\frac{\sin \left( \frac{3\pi }{4} \right)}{\cos \left( \frac{3\pi }{4} \right)}=\frac{{}^{\sqrt{2}}/{}_{2}}{-{}^{\sqrt{2}}/{}_{2}}=-1\]