TrigFcns - 8-masala yechimi | MathModel.uz

1.3-bo'lim : Trig funktsiyalari

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

Yechim

Ushbu muammoni hal qilish uchun biz shuni ta'kidlashimiz kerakki, $ - \frac{\pi }{3}$ to'rtinchi kvadrantda musbat $x$ o'qi bilan $\frac{\pi }{3}$ burchagini hosil qiladi va biz ushbu 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). The angle $ - \frac{\pi }{3}$ is drawn in the fourth quadrant and a dashed vertical line connects the point where it meets the circle to the point for $\frac{\pi }{3}$, showing the two points have the same $x$-coordinate and $y$-coordinates that differ only in sign.$

2-qadam

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

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