Bài tập nâng cao · Bài 2
Bài tập nâng cao Giá trị lượng giác của góc lượng giác · Bài 2
Rút gọn các biểu thức sau:
a) $A=\cos\left( \frac{\pi}{2}+x \right)+\cos\left( 2\pi-x \right)+\cos\left( 3\pi+x \right)$
b) $B=2\cos x-3\cos\left( \pi-x \right)+5\sin\left( \frac{7\pi}{2}-x \right)+\cot\left( \frac{3\pi}{2}-x \right)$
c) $C=2\sin\left( \frac{\pi}{2}+x \right)+\sin\left( 5\pi-x \right)+\sin\left( \frac{3\pi}{2}+x \right)+\cos\left( \frac{\pi}{2}+x \right)$
d) $D=\cos\left( 5\pi-x \right)-\sin\left( \frac{3\pi}{2}+x \right)+\tan\left( \frac{3\pi}{2}-x \right)+\cot\left( 3\pi-x \right)$
a) $A=\cos\left( \frac{\pi}{2}+x \right)+\cos\left( 2\pi-x \right)+\cos\left( 3\pi+x \right)$
b) $B=2\cos x-3\cos\left( \pi-x \right)+5\sin\left( \frac{7\pi}{2}-x \right)+\cot\left( \frac{3\pi}{2}-x \right)$
c) $C=2\sin\left( \frac{\pi}{2}+x \right)+\sin\left( 5\pi-x \right)+\sin\left( \frac{3\pi}{2}+x \right)+\cos\left( \frac{\pi}{2}+x \right)$
d) $D=\cos\left( 5\pi-x \right)-\sin\left( \frac{3\pi}{2}+x \right)+\tan\left( \frac{3\pi}{2}-x \right)+\cot\left( 3\pi-x \right)$
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Lời giải
a)
$\begin{array}{l} A=\cos\left( \frac{\pi}{2}+x \right)+\cos\left( 2\pi-x \right)+\cos\left( 3\pi+x \right) \\ \,\,\,\,\,\,=-\sin x+\cos\left( -x \right)-\cos x \\ \,\,\,\,\,\,=-\sin x+\cos x-\cos x \\ \,\,\,\,\,\,=-\sin x \end{array}$
b)
$\begin{array}{l} B=2\cos x-3\cos\left( \pi-x \right)+5\sin\left( \frac{7\pi}{2}-x \right)+\cot\left( \frac{3\pi}{2}-x \right) \\ \,\,\,\,=2\cos x+3\cos x+5\sin\left( 3\pi+\frac{\pi}{2}-x \right)+\cot\left( \pi+\frac{\pi}{2}-x \right) \\ \,\,\,\,=5\cos x-5\sin\left( \frac{\pi}{2}-x \right)+\cot\left( \frac{\pi}{2}-x \right) \\ \,\,\,\,=5\cos x-5\cos x+\tan x \\ \,\,\,\,=\tan x \end{array}$
c)
$\begin{array}{l} C=2\sin\left( \frac{\pi}{2}+x \right)+\sin\left( 5\pi-x \right)+\sin\left( \frac{3\pi}{2}+x \right)+\cos\left( \frac{\pi}{2}+x \right) \\ \,\,\,\,\,=2\cos x+\sin\left( 4\pi+\pi-x \right)+\sin\left( \pi+\frac{\pi}{2}+x \right)-\sin x \end{array}$
$\begin{array}{l} \,\,\,\,\,=2\cos x+\sin\left( \pi-x \right)-\sin\left( \frac{\pi}{2}+x \right)-\sin x \\ \,\,\,\,\,=2\cos x+\sin x-\cos x-\sin x \\ \,\,\,\,\,=\cos x \end{array}$
d)
$\begin{array}{l} D=\cos\left( 5\pi-x \right)-\sin\left( \frac{3\pi}{2}+x \right)+\tan\left( \frac{3\pi}{2}-x \right)+\cot\left( 3\pi-x \right) \\ \,\,\,\,=\cos\left( 4\pi+\pi-x \right)-\sin\left( \pi+\frac{\pi}{2}+x \right)+\tan\left( \pi+\frac{\pi}{2}-x \right)+\cot\left( 2\pi+\pi-x \right) \\ \,\,\,\,=\cos\left( \pi-x \right)+\sin\left( \frac{\pi}{2}+x \right)+\tan\left( \frac{\pi}{2}-x \right)+\cot\left( \pi-x \right) \\ \,\,\,\,=-\cos x+\cos x+\cot x-\cot x=0 \end{array}$
$\begin{array}{l} A=\cos\left( \frac{\pi}{2}+x \right)+\cos\left( 2\pi-x \right)+\cos\left( 3\pi+x \right) \\ \,\,\,\,\,\,=-\sin x+\cos\left( -x \right)-\cos x \\ \,\,\,\,\,\,=-\sin x+\cos x-\cos x \\ \,\,\,\,\,\,=-\sin x \end{array}$
b)
$\begin{array}{l} B=2\cos x-3\cos\left( \pi-x \right)+5\sin\left( \frac{7\pi}{2}-x \right)+\cot\left( \frac{3\pi}{2}-x \right) \\ \,\,\,\,=2\cos x+3\cos x+5\sin\left( 3\pi+\frac{\pi}{2}-x \right)+\cot\left( \pi+\frac{\pi}{2}-x \right) \\ \,\,\,\,=5\cos x-5\sin\left( \frac{\pi}{2}-x \right)+\cot\left( \frac{\pi}{2}-x \right) \\ \,\,\,\,=5\cos x-5\cos x+\tan x \\ \,\,\,\,=\tan x \end{array}$
c)
$\begin{array}{l} C=2\sin\left( \frac{\pi}{2}+x \right)+\sin\left( 5\pi-x \right)+\sin\left( \frac{3\pi}{2}+x \right)+\cos\left( \frac{\pi}{2}+x \right) \\ \,\,\,\,\,=2\cos x+\sin\left( 4\pi+\pi-x \right)+\sin\left( \pi+\frac{\pi}{2}+x \right)-\sin x \end{array}$
$\begin{array}{l} \,\,\,\,\,=2\cos x+\sin\left( \pi-x \right)-\sin\left( \frac{\pi}{2}+x \right)-\sin x \\ \,\,\,\,\,=2\cos x+\sin x-\cos x-\sin x \\ \,\,\,\,\,=\cos x \end{array}$
d)
$\begin{array}{l} D=\cos\left( 5\pi-x \right)-\sin\left( \frac{3\pi}{2}+x \right)+\tan\left( \frac{3\pi}{2}-x \right)+\cot\left( 3\pi-x \right) \\ \,\,\,\,=\cos\left( 4\pi+\pi-x \right)-\sin\left( \pi+\frac{\pi}{2}+x \right)+\tan\left( \pi+\frac{\pi}{2}-x \right)+\cot\left( 2\pi+\pi-x \right) \\ \,\,\,\,=\cos\left( \pi-x \right)+\sin\left( \frac{\pi}{2}+x \right)+\tan\left( \frac{\pi}{2}-x \right)+\cot\left( \pi-x \right) \\ \,\,\,\,=-\cos x+\cos x+\cot x-\cot x=0 \end{array}$