Bài 3 trang 15

Đề bài

Tính các giá trị lượng giác (nếu có) có mỗi góc sau:

a)     $\frac{\pi }{3}+k2\pi \left(k\in Z\right)$

b)     $\frac{\pi }{3}+\left(2k+1\right)\pi \left(k\in Z\right)$

c)     $k\pi \left(k\in Z\right)$

d)     $\frac{\pi }{2}+k\pi (k\in Z)$

Lời giải chi tiết

a)

$\begin{array}{c}\cos \left(\frac{\pi }{3}+k2\pi \right)=\cos \left(\frac{\pi }{3}\right)=\frac{1}{2}\\ \sin \left(\frac{\pi }{3}+k2\pi \right)=\sin \left(\frac{\pi }{3}\right)=\frac{\sqrt{3}}{2}\\ \tan \left(\frac{\pi }{3}+k2\pi \right)=\frac{\sin \left(\frac{\pi }{3}+k2\pi \right)}{\cos \left(\frac{\pi }{3}+k2\pi \right)}=\sqrt{3}\\ \cot \left(\frac{\pi }{3}+k2\pi \right)=\frac{1}{\tan \left(\frac{\pi }{3}+k2\pi \right)}=\frac{\sqrt{3}}{3}\end{array}$

b) Các giá trị lượng giác của góc lượng giác $\frac{\pi }{3}+\left(2k+1\right)\pi \left(k\in Z\right)$

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

$\sin \left[\frac{\pi }{3}+(2k+1)\pi \right]=\sin \left(\frac{\pi }{3}+\pi +2k\pi \right)=\sin \left(\frac{\pi }{3}+\pi \right)=-\sin \frac{\pi }{3}=-\frac{\sqrt{3}}{2}$

$\tan \left[\frac{\pi }{3}+(2k+1)\pi \right]=\tan \frac{\pi }{3}=\sqrt{3}$;

$\tan \left[\frac{\pi }{3}+(2k+1)\pi \right]=\cot \frac{\pi }{3}=\frac{\sqrt{3}}{3}$

c)

$\begin{array}{c}\cos \left(k\pi \right)=[\begin{array}{c}-1;k=2n+1\\ 1;k=2n\end{array}\\ \sin \left(k\pi \right)=0\\ \tan \left(k\pi \right)=\frac{\sin \left(k\pi \right)}{\cos \left(k\pi \right)}=0\\ \cot \left(k\pi \right)\end{array}$

d)

$\begin{array}{c}\cos \left(\frac{\pi }{2}+k\pi \right)=0\\ \sin \left(\frac{\pi }{2}+k\pi \right)=[\begin{array}{c}\sin \left(-\frac{\pi }{2}\right)=-1;k=2n+1\\ \sin \left(\frac{\pi }{2}\right)=1;k=2n\end{array}\\ \tan \left(\frac{\pi }{2}+k\pi \right)\\ \cot \left(\frac{\pi }{2}+k\pi \right)=0\end{array}$