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例1 求满足下列各式的锐角$\alpha$:
(1)$2 \sin \alpha -\sqrt{3}=0$; (2)$\cos ( \alpha -25^{ \circ })=\dfrac{\sqrt{2}}{2}$;
(3)$\tan ^{2} \alpha -2\sqrt{3}\tan \alpha +3=0$; (4)$2 \cos ^{2} \alpha -5 \cos \alpha +2=0$.
(1)$2 \sin \alpha -\sqrt{3}=0$; (2)$\cos ( \alpha -25^{ \circ })=\dfrac{\sqrt{2}}{2}$;
(3)$\tan ^{2} \alpha -2\sqrt{3}\tan \alpha +3=0$; (4)$2 \cos ^{2} \alpha -5 \cos \alpha +2=0$.
答案
解：$ 2sina=\sqrt{3}$
$sina=\frac {\sqrt{3}}{2}$
所以a=60°
解：a-25°=45°
a=70°
解：$ (tana-\sqrt{3})²= 0$
$tana=\sqrt{3}$
α=60°
解： (2cosa- 1)(cosa-2)= 0
$cosα=\frac {1}{2}$或cosa = 2(舍去)
所以a=60°
$sina=\frac {\sqrt{3}}{2}$
所以a=60°
解：a-25°=45°
a=70°
解：$ (tana-\sqrt{3})²= 0$
$tana=\sqrt{3}$
α=60°
解： (2cosa- 1)(cosa-2)= 0
$cosα=\frac {1}{2}$或cosa = 2(舍去)
所以a=60°
