两角和与差的正弦、余弦和正切公式复习导学案(简单点)

两角和与差的正弦、余弦和正切公式复习导学案(简单点)

1.两角和与差的余弦、正弦、正切公式：$$\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta$$$$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$$$$\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta$$$$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$$$$\tan(\alpha-\beta)=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$$$$\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$$2.二倍角公式：$$\sin2\alpha=2\sin\alpha\cos\alpha$$$$\cos2\alpha=\cos^2\alpha-\sin^2\alpha=2\cos^2\alpha-1=1-2\sin^2\alpha$$$$\tan2\alpha=\frac{2\tan\alpha}{1-\tan^2\alpha}$$3.公式的逆用、变形等：$$(1)\tan(\alpha\pm\beta)=\frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta}$$$$(2)\cos^2\alpha+\sin^2\alpha=1,\sin^2\alpha=\frac{1-\cos2\alpha}{2},\cos^2\alpha=\frac{1+\cos2\alpha}{2}$$$$(3)1+\sin2\alpha=2\sin(\alpha+\frac{\pi}{4}),1-\sin2\alpha=2\sin(\alpha-\frac{\pi}{4}),\sin(\alpha\pm\beta)=2\sin\frac{\alpha\pm\beta}{2}\cos\frac{\alpha\mp\beta}{2}$$例1：(1)由题意可得：$$\tan(\alpha+\frac{\pi}{4})=\frac{7}{1}$$代入公式可得：$$\cos\alpha=\frac{1-\tan^2(\alpha+\frac{\pi}{4})}{1+\tan^2(\alpha+\frac{\pi}{4})}=-\frac{3}{4}$$$$\sin\alpha=\tan(\alpha+\frac{\pi}{4})\cos\alpha=\frac{7}{4}$$(2)由题意可得：$$\sin\alpha=\frac{3}{5}\cos2\alpha$$代入公式可得：$$\cos^2\alpha=1-\sin^2\alpha=\frac{16}{25}$$$$\cos2\alpha=2\cos^2\alpha-1=\frac{3}{5}$$$$\sin2\alpha=2\sin\alpha\cos\alpha=\frac{24}{25}$$所以：$$\tan2\alpha=\frac{\sin2\alpha}{\cos2\alpha}=\frac{8}{3}$$例2：$$\begin{aligned}\sin(65^\circ-x)\cos(x-20^\circ)+\cos(65^\circ-x)\cos(110^\circ-x)&=\frac{1}{2}[\sin(45^\circ-(x-20^\circ))+\sin(45^\circ+(x-20^\circ))]\\&+\frac{1}{2}[\cos(75^\circ-x)+\cos(75^\circ+x)]\\&=\frac{1}{2}[\cos(x-25^\circ)+\cos(155^\circ-x)]\\&=\cosx\cos25^\circ+\sinx\sin25^\circ-\cosx\cos25^\circ+\sinx\sin155^\circ\\&=\sinx(\sin25^\circ-\sin155^\circ)\\&=2\sinx\cos90^\circ\\&=\sin2x\end{aligned}$$注：第一段的公式使用了例2中的公式(3)。第二段使用了三角函数的和差化积公式和倍角公式。1.若tanα=2tan(π/5)，则10sin(α-π/5)等于()A.1B.2C.3D.42.已知α、β都是锐角，且cosα=5/5，sin(α+β)=3/5，则cosβ等于A.25/25B.25/5C.25/2555或5/25D.5/2555或253.已知cos(α-π/6)+sinα=47π/53，则sin(α+π/6)的值是________。4.sin20cos40+cos20sin40的值等于()A.1/4B.3/2C.13/2D.45.若tanα=3，tanβ=4/3，则tan(α-β)等于()A.-3B.3C.-1/3D.1/36.cos(π/5)cos(2π/5)的值等于()A.1B.1/4C.2D.47.已知<A<π，且cosA=3/25，那么sin2A等于()A.42/425B.7/125C.25/9D.25/48.cos85°+sin25°cos30°/cos25°等于()A.-3/2B.2C.1/2D.1/69.若θ∈[π/4，π/2]，sin2θ=37/8，则sinθ等于()A.3/5B.4/7C.4D.3/410.sin(π+α)=-32/5，α是第三象限角，则sin(π-α)等于()A.1/2B.-1/