Bài 1.6 trang 15

Đề bài

Không dùng máy tính cầm tay, tính:

a) ${\sin }^2\frac{\pi }{4}+\cos \left(-\frac{\pi }{2}\right);$

b) ${\tan }^2\left({30}^0\right)-{\cot }^2\left({240}^0\right);$

c) ${\sin }^3\frac{\pi }{2}-\cos 5\pi ;$

d) $\tan \frac{11\pi }{3}-\cot \left(-\frac{21\pi }{4}\right)$.

Lời giải chi tiết

a)

$\begin{array}{c}{\sin }^2\frac{\pi }{4}+\cos \left(-\frac{\pi }{2}\right)\\ ={\sin }^2\frac{\pi }{4}+\cos \frac{\pi }{2}\\ ={\left(\frac{\sqrt{2}}{2}\right)}^2+0=\frac{1}{2}\end{array}$

b)

$\begin{array}{c}{\tan }^2\left({30}^0\right)-{\cot }^2\left({240}^0\right)\\ ={\left(\frac{1}{\sqrt{3}}\right)}^2-\cot \left({60}^0\right)\\ =\frac{1}{3}-\frac{\sqrt{3}}{3}=\frac{1-\sqrt{3}}{3}\end{array}$

c)

$\begin{array}{c}{\sin }^3\frac{\pi }{2}-\cos 5\pi \\ =1^3-\cos \left(\pi +4\pi \right)\\ =1-\cos \pi \\ =1-\left(-1\right)=2\end{array}$

d)

$\begin{array}{c}\tan \frac{11\pi }{3}-\cot \left(-\frac{21\pi }{4}\right)\\ =\tan \left(\frac{2}{3}\pi +3\pi \right)-\cot \left(-\frac{\pi }{4}-5\pi \right)\\ =\tan \left(\frac{2\pi }{3}\right)-\cot \left(-\frac{\pi }{4}\right)\\ =-\sqrt{3}+\cot \left(\frac{\pi }{4}\right)\\ =-\sqrt{3}+1\end{array}$