三角函数恒等变形

基本性质

\sin^{2} α + \cos^{2} α = 1

$\sin^2\alpha+\cos^2\alpha=1$

\tan α = \frac{\sin α}{\cos α}

$\tan\alpha=\frac{\sin\alpha}{\cos\alpha}$

和差角公式

\sin (α \pm β) = \sin α \cos β \pm \cos α \sin β

$\sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta$

\cos (α \pm β) = \cos α \cos β \mp \sin α \sin β

$\cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta$

\tan (α \pm β) = \frac{\tan α \pm \tan β}{1 \mp \tan α \tan β}

$\tan(\alpha\pm\beta)=\frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta}$

二倍角公式

\sin 2 α = 2 \sin α \cos α

$\sin2\alpha=2\sin\alpha\cos\alpha$

\cos 2 α = \cos^{2} α - \sin^{2} α = 1 - 2 \sin^{2} α = 2 \cos^{2} α - 1

$\cos2\alpha=\cos^2\alpha-\sin^2\alpha=1-2\sin^2\alpha=2\cos^2\alpha-1$

\tan 2 α = \frac{2 \tan α}{1 - \tan^{2} α}

$\tan2\alpha=\frac{2\tan\alpha}{1-\tan^2\alpha}$

降幂公式

\sin α \cos α = \frac{1}{2} \sin 2 α

$\sin\alpha\cos\alpha=\frac12\sin2\alpha$

\sin^{2} α = \frac{1 - \cos 2 α}{2}

$\sin^2\alpha=\frac{1-\cos2\alpha}2$

\cos^{2} α = \frac{1 + \cos 2 α}{2}

$\cos^2\alpha=\frac{1+\cos2\alpha}2$

一点也不万能的公式

\sin α = \frac{2 \tan \frac{α}{2}}{1 + \tan^{2} \frac{α}{2}}

$\sin\alpha=\frac{2\tan\frac\alpha2}{1+\tan^2\frac\alpha2}$

\cos α = \frac{1 - \tan^{2} \frac{α}{2}}{1 + \tan^{2} \frac{α}{2}}

$\cos\alpha=\frac{1-\tan^2\frac\alpha2}{1+\tan^2\frac\alpha2}$

积化和差

\sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)]

$\sin\alpha\cos\beta=\frac12\left[\sin(\alpha+\beta)+\sin(\alpha-\beta)\right]$

\cos α \sin β = \frac{1}{2} [\sin (α + β) - \sin (α - β)]

$\cos\alpha\sin\beta=\frac12\left[\sin(\alpha+\beta)-\sin(\alpha-\beta)\right]$

\sin α \sin β = - \frac{1}{2} [\cos (α + β) - \cos (α - β)]

$\sin\alpha\sin\beta=-\frac12\left[\cos(\alpha+\beta)-\cos(\alpha-\beta)\right]$

\cos α \cos β = \frac{1}{2} [\cos (α + β) + \cos (α - β)]

$\cos\alpha\cos\beta=\frac12\left[\cos(\alpha+\beta)+\cos(\alpha-\beta)\right]$

\tan α \tan β = 1 - \frac{\tan α + \tan β}{\tan (α + β)} = \frac{\tan α - \tan β}{\tan (α - β)} - 1

$\tan\alpha\tan\beta=1-\frac{\tan\alpha+\tan\beta}{\tan(\alpha+\beta)}=\frac{\tan\alpha-\tan\beta}{\tan(\alpha-\beta)}-1$

和差化积

\sin α + \sin β = 2 \sin \frac{α + β}{2} \cos \frac{α - β}{2}

$\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}2\cos\frac{\alpha-\beta}2$

\sin α - \sin β = 2 \cos \frac{α + β}{2} \sin \frac{α - β}{2}

$\sin\alpha-\sin\beta=2\cos\frac{\alpha+\beta}2\sin\frac{\alpha-\beta}2$

\cos α + \cos β = 2 \cos \frac{α + β}{2} \cos \frac{α - β}{2}

$\cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}2\cos\frac{\alpha-\beta}2$

\cos α - \cos β = - 2 \sin \frac{α + β}{2} \sin \frac{α - β}{2}

$\cos\alpha-\cos\beta=-2\sin\frac{\alpha+\beta}2\sin\frac{\alpha-\beta}2$

\tan α \pm \tan β = \tan (α \pm β) (1 \mp \tan α \tan β)

$\tan\alpha\pm\tan\beta=\tan(\alpha\pm\beta)(1\mp\tan\alpha\tan \beta)$

杂题

\begin{aligned} \cos 75 ° \\ = & \cos (30 ° + 45 °) \\ = & \cos 30 ° \cos 45 ° - \sin 30 ° \sin 45 ° \\ = & \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2} - \frac{1}{2} \times \frac{\sqrt{2}}{2} \\ = & \frac{\sqrt{6} - \sqrt{2}}{2} \end{aligned}

$\begin{align*} &\cos75°\\ ={}&\cos(30°+45°)\\ ={}&\cos30°\cos45°-\sin30°\sin45°\\ ={}&\frac{\sqrt3}2\times\frac{\sqrt2}2-\frac12\times\frac{\sqrt2}2\\ ={}&\frac{\sqrt6-\sqrt2}2 \end{align*}$

\begin{aligned} \sin 105 ° \\ = & \sin (45 ° + 60 °) \\ = & \sin 45 ° \cos 60 ° + \cos 45 ° \sin 60 ° \\ = & \frac{\sqrt{2}}{2} \times \frac{1}{2} + \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} \\ = & \frac{\sqrt{6} + \sqrt{2}}{2} \end{aligned}

$\begin{align*} &\sin105°\\ ={}&\sin(45°+60°)\\ ={}&\sin45°\cos60°+\cos45°\sin60°\\ ={}&\frac{\sqrt2}2\times\frac12+\frac{\sqrt2}2\times\frac{\sqrt3}2\\ ={}&\frac{\sqrt6+\sqrt2}2 \end{align*}$

\begin{aligned} \cos 61 ° \cos 16 ° + \sin 61 ° \sin 16 ° \\ = & \cos (61 ° - 16 °) \\ = & \cos 45 ° \\ = & \frac{\sqrt{2}}{2} \end{aligned}

$\begin{align*} &\cos61°\cos16°+\sin61°\sin16°\\ ={}&\cos(61°-16°)\\ ={}&\cos45°\\ ={}&\frac{\sqrt2}2 \end{align*}$

\begin{aligned} \sin 13 ° \cos 17 ° + \cos 13 ° \sin 17 ° \\ = & \sin (13 ° + 17 °) \\ = & \sin 30 ° \\ = & \frac{1}{2} \end{aligned}

$\begin{align*} &\sin13°\cos17°+\cos13°\sin17°\\ ={}&\sin(13°+17°)\\ ={}&\sin30°\\ ={}&\frac12 \end{align*}$

\begin{aligned} \sin 163 ° \sin 223 ° + \sin 253 ° \sin 313 ° \\ = & \sin 163 ° \sin 223 ° + \sin (163 ° + 90 °) \sin (223 ° + 90 °) \\ = & \sin 163 ° \sin 223 ° + \cos 163 ° \cos 223 ° \\ = & \cos (223 ° - 163 °) \\ = & \cos (- 60 °) \\ = & \cos 60 ° \\ = & \frac{1}{2} \end{aligned}

$\begin{align*} &\sin163°\sin223°+\sin253°\sin313°\\ ={}&\sin163°\sin223°+\sin(163°+90°)\sin(223°+90°)\\ ={}&\sin163°\sin223°+\cos163°\cos223°\\ ={}&\cos(223°-163°)\\ ={}&\cos(-60°)\\ ={}&\cos60°\\ ={}&\frac12 \end{align*}$

\begin{aligned} \cos 43 ° \cos 77 ° + \sin 43 ° \cos 167 ° \\ = & \cos 43 ° \cos 77 ° + \sin 43 ° \cos (77 ° + 90 °) \\ = & \cos 43 ° \cos 77 ° - \sin 43 ° \sin 77 ° \\ = & \cos (43 ° + 77 °) \\ = & \cos 120 ° \\ = & - \sin 30 ° \\ = & - \frac{1}{2} \end{aligned}

$\begin{align*} &\cos43°\cos77°+\sin43°\cos167°\\ ={}&\cos43°\cos77°+\sin43°\cos(77°+90°)\\ ={}&\cos43°\cos77°-\sin43°\sin77°\\ ={}&\cos(43°+77°)\\ ={}&\cos120°\\ ={}&-\sin30°\\ ={}&-\frac12\\ \end{align*}$

\begin{aligned} \frac{\sin 7 ° + \cos 15 ° \sin 8 °}{\cos 7 ° - \sin 15 ° \sin 8 °} \\ = & \frac{\sin (15 ° - 8 °) + \cos 15 ° \sin 8 °}{\cos (15 ° - 8 °) - \sin 15 ° \sin 8 °} \\ = & \frac{\sin 15 ° \cos 8 ° - \cos 15 ° \sin 8 ° + \cos 15 ° \sin 8 °}{\cos 15 ° \cos 8 ° + \sin 15 ° \sin 8 ° - \sin 15 ° \sin 8 °} \\ = & \frac{\sin 15 ° \cos 8 °}{\cos 15 ° \cos 8 °} \\ = & \tan 15 ° \\ = & \tan (45 ° - 30 °) \\ = & \frac{\tan 45 ° - \tan 30 °}{1 + \tan 45 ° \tan 30 °} \\ = & \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \times \frac{\sqrt{3}}{3}} \\ = & 2 - \sqrt{3} \end{aligned}

$\begin{align*} &\frac{\sin7°+\cos15°\sin8°}{\cos7°-\sin15°\sin8°}\\ ={}&\frac{\sin(15°-8°)+\cos15°\sin8°}{\cos(15°-8°)-\sin15°\sin8°}\\ ={}&\frac{\sin15°\cos8°-\cos15°\sin8°+\cos15°\sin8°}{\cos15°\cos8°+\sin15°\sin8°-\sin15°\sin8°}\\ ={}&\frac{\sin15°\cos8°}{\cos15°\cos8°}\\ ={}&\tan15°\\ ={}&\tan(45°-30°)\\ ={}&\frac{\tan45°-\tan30°}{1+\tan45°\tan30°}\\ ={}&\frac{1-\frac{\sqrt3}3}{1+1\times\frac{\sqrt3}3}\\ ={}&2-\sqrt3\\ \end{align*}$

\begin{aligned} \frac{1 + \tan 75 °}{1 - \tan 75 °} \\ = & \frac{\tan 45 ° + \tan 75 °}{1 - \tan 45 ° \tan 75 °} \\ = & \tan (45 ° + 75 °) \\ = & \tan 120 ° \\ = & - \tan 60 ° \\ = & - \sqrt{3} \end{aligned}

$\begin{align*} &\frac{1+\tan75°}{1-\tan75°}\\ ={}&\frac{\tan45°+\tan75°}{1-\tan45°\tan75°}\\ ={}&\tan(45°+75°)\\ ={}&\tan120°\\ ={}&-\tan60°\\ ={}&-\sqrt3\\ \end{align*}$

\begin{aligned} \frac{\cos 15 ° - \sin 15 °}{\cos 15 ° + \sin 15 °} \\ = & \frac{1 - \tan 15 °}{1 + \sin 15 °} \\ = & \frac{\tan 45 ° - \tan 15 °}{1 + \tan 45 ° \tan 15 °} \\ = & \tan 30 ° \\ = & \frac{\sqrt{3}}{3} \end{aligned}

$\begin{align*} &\frac{\cos15°-\sin15°}{\cos15°+\sin15°}\\ ={}&\frac{1-\tan15°}{1+\sin15°}\\ ={}&\frac{\tan45°-\tan15°}{1+\tan45°\tan15°}\\ ={}&\tan30°\\ ={}&\frac{\sqrt3}3 \end{align*}$

\begin{aligned} \sin 15 ° \cos 15 ° \\ = & \frac{1}{2} \sin 30 ° \\ = & \frac{1}{2} \times \frac{1}{2} \\ = & \frac{1}{4} \end{aligned}

$\begin{align*} &\sin15°\cos15°\\ ={}&\frac12\sin30°\\ ={}&\frac12\times\frac12\\ ={}&\frac14\\ \end{align*}$

\begin{aligned} 1 - 2 \sin^{2} 75 ° \\ = & \cos (2 \times 75 °) \\ = & \cos 150 ° \\ = & - \cos 60 ° \\ = & - \frac{1}{2} \end{aligned}

$\begin{align*} &1-2\sin^275°\\ ={}&\cos(2\times75°)\\ ={}&\cos150°\\ ={}&-\cos60°\\ ={}&-\frac12\\ \end{align*}$

\begin{aligned} 1 - 2 \sin^{2} 75 ° \\ = & \cos (2 \times 75 °) \\ = & \cos 150 ° \\ = & - \cos 60 ° \\ = & - \frac{1}{2} \end{aligned}

$\begin{align*} &1-2\sin^275°\\ ={}&\cos(2\times75°)\\ ={}&\cos150°\\ ={}&-\cos60°\\ ={}&-\frac12\\ \end{align*}$

\begin{aligned} 1 - 2 \sin^{2} 75 ° \\ = & \cos (2 \times 75 °) \\ = & \cos 150 ° \\ = & - \cos 60 ° \\ = & - \frac{1}{2} \end{aligned}

$\begin{align*} &1-2\sin^275°\\ ={}&\cos(2\times75°)\\ ={}&\cos150°\\ ={}&-\cos60°\\ ={}&-\frac12\\ \end{align*}$

\begin{aligned} \frac{2 \tan 150 °}{1 - \tan^{2} 150 °} \\ = & \tan (2 \times 150 °) \\ = & \tan 300 ° \\ = & - \tan 60 ° \\ = & - \sqrt{3} \end{aligned}

$\begin{align*} &\frac{2\tan150°}{1-\tan^2150°}\\ ={}&\tan(2\times150°)\\ ={}&\tan300°\\ ={}&-\tan60°\\ ={}&-\sqrt3\\ \end{align*}$

\begin{aligned} \tan 15 ° + \cot 15 ° \\ = & \frac{\sin 15 °}{\cos 15 °} + \frac{\cos 15 °}{\sin 15 °} \\ = & \frac{\sin^{2} 15 ° + \cos^{2} 15 °}{\sin 15 ° \cos 15 °} \\ = & \frac{1}{\sin 15 ° \cos 15 °} \\ = & \frac{1}{\frac{1}{2} \sin 30 °} \\ = & \frac{1}{\frac{1}{2} \times \frac{1}{2}} \\ = & 4 \end{aligned}

$\begin{align*} &\tan15°+\cot15°\\ ={}&\frac{\sin15°}{\cos15°}+\frac{\cos15°}{\sin15°}\\ ={}&\frac{\sin^215°+\cos^215°}{\sin15°\cos15°}\\ ={}&\frac1{\sin15°\cos15°}\\ ={}&\frac1{\frac12\sin30°}\\ ={}&\frac1{\frac12\times\frac12}\\ ={}&4\\ \end{align*}$

\begin{aligned} \frac{2 \cos 10 ° - \sin 20 °}{\sin 70 °} \\ = & \frac{2 \cos (30 ° - 20 °) - \sin 20 °}{\cos 20 °} \\ = & \frac{2 (\cos 30 ° \cos 20 ° + \sin 30 ° \sin 20 °) - \sin 20 °}{\cos 20 °} \\ = & \frac{2 (\frac{\sqrt{3}}{2} \times \cos 20 ° + \frac{1}{2} \times \sin 20 °) - \sin 20 °}{\cos 20 °} \\ = & \frac{\sqrt{3} \cos 20 ° + \sin 20 ° - \sin 20 °}{\cos 20 °} \\ = & \sqrt{3} \end{aligned}

$\begin{align*} &\frac{2\cos10°-\sin20°}{\sin70°}\\ ={}&\frac{2\cos(30°-20°)-\sin20°}{\cos20°}\\ ={}&\frac{2(\cos30°\cos20°+\sin30°\sin20°)-\sin20°}{\cos20°}\\ ={}&\frac{2(\frac{\sqrt3}2\times\cos20°+\frac12\times\sin20°)-\sin20°}{\cos20°}\\ ={}&\frac{\sqrt3\cos20°+\sin20°-\sin20°}{\cos20°}\\ ={}&\sqrt3\\ \end{align*}$

\begin{aligned} \tan 20 ° + 4 \sin 20 ° \\ = & \frac{\sin 20 °}{\cos 20 °} + \frac{4 \sin 20 ° \cos 20 °}{\cos 20 °} \\ = & \frac{\sin 20 ° + 2 \sin 40 °}{\cos 20 °} \\ = & \frac{\sin 20 ° + 2 \sin (60 ° - 20 °)}{\cos 20 °} \\ = & \frac{\sin 20 ° + 2 \sin 60 ° \cos 20 ° - 2 \cos 60 ° \sin 20 °}{\cos 20 °} \\ = & \frac{\sin 20 ° + \sqrt{3} \cos 20 ° - \sin 20 °}{\cos 20 °} \\ = & \sqrt{3} \end{aligned}

$\begin{align*} &\tan20°+4\sin20°\\ ={}&\frac{\sin20°}{\cos20°}+\frac{4\sin20°\cos20°}{\cos20°}\\ ={}&\frac{\sin20°+2\sin40°}{\cos20°}\\ ={}&\frac{\sin20°+2\sin(60°-20°)}{\cos20°}\\ ={}&\frac{\sin20°+2\sin60°\cos20°-2\cos60°\sin20°}{\cos20°}\\ ={}&\frac{\sin20°+\sqrt3\cos20°-\sin20°}{\cos20°}\\ ={}&\sqrt3 \end{align*}\\$

\begin{aligned} \tan 72 ° - \tan 42 ° - \frac{\sqrt{3}}{3} \tan 72 ° \tan 42 ° \\ = & \tan (72 ° - 42 °) (1 + \tan 72 ° \tan 42 °) - \tan 30 ° \tan 72 ° \tan 42 ° \\ = & \tan 30 ° (1 + \tan 72 ° \tan 42 °) - \tan 30 ° (\tan 72 ° \tan 42 °) \\ = & \tan 30 ° \\ = & \frac{\sqrt{3}}{3} \end{aligned}

$\begin{align*} &\tan72°-\tan42°-\frac{\sqrt3}3\tan72°\tan42°\\ ={}&\tan(72°-42°)(1+\tan72°\tan42°)-\tan30°\tan72°\tan42°\\ ={}&\tan30°(1+\tan72°\tan42°)-\tan30°(\tan72°\tan42°)\\ ={}&\tan30°\\ ={}&\frac{\sqrt3}3\\ \end{align*}$

\begin{aligned} \frac{\tan 20 ° + \tan 40 ° + \tan 120 °}{\tan 20 ° \tan 40 °} \\ = & \frac{\tan 60 ° - \tan 20 ° \tan 40 ° \tan 60 ° - \tan 60 °}{\tan 20 ° \tan 40 °} \\ = & - \tan 60 ° \\ = & - \sqrt{3} \end{aligned}

$\begin{align*} &\frac{\tan20°+\tan40°+\tan120°}{\tan20°\tan40°}\\ ={}&\frac{\tan60°-\tan20°\tan40°\tan60°-\tan60°}{\tan20°\tan40°}\\ ={}&-\tan60°\\ ={}&-\sqrt3\\ \end{align*}$

\begin{aligned} \tan (30 ° - α) + \tan (30 ° + α) + \sqrt{3} \tan (30 ° - α) \tan (30 ° + α) \\ = & \tan (30 ° - α + 30 ° + α) [1 - \tan (30 ° + α) \tan (30 ° - α)] + \sqrt{3} \tan (30 ° - α) \tan (30 ° + α) \\ = & \sqrt{3} [1 - \tan (30 ° + α) \tan (30 ° - α)] + \sqrt{3} \tan (30 ° - α) \tan (30 ° + α) \\ = & \sqrt{3} - \sqrt{3} \tan (30 ° + α) \tan (30 ° - α) + \sqrt{3} \tan (30 ° - α) \tan (30 ° + α) \\ = & \sqrt{3} \end{aligned}

$\begin{align*} &\tan(30°-\alpha)+\tan(30°+\alpha)+\sqrt3\tan(30°-\alpha)\tan(30°+\alpha)\\ ={}&\tan(30°-\alpha+30°+\alpha)\left[1-\tan(30°+\alpha)\tan(30°-\alpha)\right]+\sqrt3\tan(30°-\alpha)\tan(30°+\alpha)\\ ={}&\sqrt3\left[1-\tan(30°+\alpha)\tan(30°-\alpha)\right]+\sqrt3\tan(30°-\alpha)\tan(30°+\alpha)\\ ={}&\sqrt3-\sqrt3\tan(30°+\alpha)\tan(30°-\alpha)+\sqrt3\tan(30°-\alpha)\tan(30°+\alpha)\\ ={}&\sqrt3\\ \end{align*}$

\begin{aligned} \tan 10 ° (3 + \tan 30 ° \tan 40 ° + \tan 40 ° \tan 50 ° + \tan 50 ° \tan 60 °) \\ = & \tan 10 ° [(1 + \tan 30 ° \tan 40 °) + (1 + \tan 40 ° \tan 50 °) + (1 + \tan 50 ° \tan 60 °)] \\ = & \tan 10 ° [(1 + \frac{\tan 40 ° - \tan 30 °}{\tan 10 °} - 1) + (1 + \frac{\tan 50 ° - \tan 40 °}{\tan 10 °} - 1) + (1 + \frac{\tan 60 ° - \tan 50 °}{\tan 10 °} - 1)] \\ = & \tan 10 ° (\frac{\tan 40 ° - \tan 30 °}{\tan 10 °} + \frac{\tan 50 ° - \tan 40 °}{\tan 10 °} + \frac{\tan 60 ° - \tan 50 °}{\tan 10 °}) \\ = & \tan 40 ° - \tan 30 ° + \tan 50 ° - \tan 40 ° + \tan 60 ° - \tan 50 ° \\ = & \tan 60 ° - \tan 30 ° \\ = & \tan 60 ° - \tan 30 ° \\ = & \frac{2 \sqrt{3}}{3} \end{aligned}

$\begin{align*} &\tan10°(3+\tan30°\tan40°+\tan40°\tan50°+\tan50°\tan60°)\\ ={}&\tan10°\left[(1+\tan30°\tan40°)+(1+\tan40°\tan50°)+(1+\tan50°\tan60°)\right]\\ ={}&\tan10°\left[(1+\frac{\tan40°-\tan30°}{\tan10°}-1)+(1+\frac{\tan50°-\tan40°}{\tan10°}-1)+(1+\frac{\tan60°-\tan50°}{\tan10°}-1)\right]\\ ={}&\tan10°(\frac{\tan40°-\tan30°}{\tan10°}+\frac{\tan50°-\tan40°}{\tan10°}+\frac{\tan60°-\tan50°}{\tan10°})\\ ={}&\tan40°-\tan30°+\tan50°-\tan40°+\tan60°-\tan50°\\ ={}&\tan60°-\tan30°\\ ={}&\tan60°-\tan30°\\ ={}&\frac{2\sqrt3}3\\ \end{align*}$

\begin{aligned} \frac{\sin 50 ° (1 + \sqrt{3} \tan 10 °) - \cos 20 °}{\cos 80 ° \sqrt{1 - \cos 20 °}} \\ = & \frac{\sin 50 ° (\frac{\cos 10 °}{\cos 10 °} + \frac{\sqrt{3} \sin 10 °}{\cos 10 °}) - \cos 20 °}{\cos 80 ° \sqrt{1 - \cos 20 °}} \\ = & \frac{\frac{\sin 50 °}{\cos 10 °} \times (\sqrt{3} \sin 10 ° + \cos 10 °) - \cos 20 °}{\cos 80 ° \sqrt{1 - \cos 20 °}} \\ = & \frac{\frac{\sin 50 °}{\cos 10 °} \times 2 (\frac{\sqrt{3}}{2} \sin 10 ° + \frac{1}{2} \cos 10 °) - \cos 20 °}{\cos 80 ° \sqrt{1 - \cos 20 °}} \\ = & \frac{\frac{\sin 50 °}{\cos 10 °} \times 2 (\cos 30 ° \sin 10 ° + \sin 30 ° \cos 10 °) - \cos 20 °}{\cos 80 ° \sqrt{1 - \cos 20 °}} \\ = & \frac{\frac{\sin 50 °}{\cos 10 °} \times 2 \sin (30 ° + 10 °) - \cos 20 °}{\cos 80 ° \sqrt{1 - \cos 20 °}} \\ = & \frac{\frac{\cos 40 °}{\sin 80 °} \times 2 \sin 40 ° - \cos 20 °}{\cos 80 ° \sqrt{1 - \cos 20 °}} \\ = & \frac{1 - \cos 20 °}{\cos 80 ° \sqrt{1 - \cos 20 °}} \\ = & \frac{1 - (1 - 2 \sin^{2} 10 °)}{\cos 80 ° \sqrt{1 - (1 - 2 \sin^{2} 10 °)}} \\ = & \frac{2 \sin^{2} 10 °}{\cos 80 ° \sqrt{2 \sin^{2} 10 °}} \\ = & \frac{2 \sin^{2} 10 °}{\sqrt{2} \sin^{2} 10 °} \\ = & \sqrt{2} \end{aligned}

$\begin{align*} &\frac{\sin50°(1+\sqrt3\tan10°)-\cos20°}{\cos80°\sqrt{1-\cos20°}}\\ ={}&\frac{\sin50°(\frac{\cos10°}{\cos10°}+\frac{\sqrt3\sin10°}{\cos10°})-\cos20°}{\cos80°\sqrt{1-\cos20°}}\\ ={}&\frac{\frac{\sin50°}{\cos10°}\times(\sqrt3\sin10°+\cos10°)-\cos20°}{\cos80°\sqrt{1-\cos20°}}\\ ={}&\frac{\frac{\sin50°}{\cos10°}\times2(\frac{\sqrt3}2\sin10°+\frac12\cos10°)-\cos20°}{\cos80°\sqrt{1-\cos20°}}\\ ={}&\frac{\frac{\sin50°}{\cos10°}\times2(\cos30°\sin10°+\sin30°\cos10°)-\cos20°}{\cos80°\sqrt{1-\cos20°}}\\ ={}&\frac{\frac{\sin50°}{\cos10°}\times2\sin(30°+10°)-\cos20°}{\cos80°\sqrt{1-\cos20°}}\\ ={}&\frac{\frac{\cos40°}{\sin80°}\times2\sin40°-\cos20°}{\cos80°\sqrt{1-\cos20°}}\\ ={}&\frac{1-\cos20°}{\cos80°\sqrt{1-\cos20°}}\\ ={}&\frac{1-(1-2\sin^210°)}{\cos80°\sqrt{1-(1-2\sin^210°)}}\\ ={}&\frac{2\sin^210°}{\cos80°\sqrt{2\sin^210°}}\\ ={}&\frac{2\sin^210°}{\sqrt2\sin^210°}\\ ={}&\sqrt2\\ \end{align*}$

\begin{aligned} (\tan 10 ° - \sqrt{3}) \times \frac{\cos 10 °}{\sin 50 °} \\ = & (\tan 10 ° - \tan 60 °) \times \frac{\cos 10 °}{\sin 50 °} \\ = & [\tan (10 ° - 60 °)] (1 + \tan 10 ° \tan 60 °) \times \frac{\cos 10 °}{\sin 50 °} \\ = & - \tan 50 ° (1 + \tan 10 ° \tan 60 °) \times \frac{\cos 10 °}{\sin 50 °} \\ = & - \tan 50 ° (1 + \sqrt{3} \tan 10 °) \times \frac{\cos 10 °}{\sin 50 °} \\ = & - 2 \tan 50 ° (\frac{1}{2} \times \frac{\cos 10 °}{\cos 10 °} + \frac{\sqrt{3}}{2} \times \frac{\sin 10}{\cos 10}) \times \frac{\cos 10 °}{\sin 50 °} \\ = & - 2 \tan 50 ° (\cos 60 ° \cos 10 ° + \sin 60 ° \sin 10) \times \frac{1}{\sin 50 °} \\ = & - 2 \tan 50 ° (\cos 60 ° - 10 °) \times \frac{1}{\sin 50 °} \\ = & - 2 \tan 50 ° (\cos 60 ° - 10 °) \times \frac{1}{\sin 50 °} \\ = & - 2 \end{aligned}

$\begin{align*} &(\tan10°-\sqrt3)\times\frac{\cos10°}{\sin50°}\\ ={}&(\tan10°-\tan60°)\times\frac{\cos10°}{\sin50°}\\ ={}&\left[\tan(10°-60°)\right](1+\tan10°\tan60°)\times\frac{\cos10°}{\sin50°}\\ ={}&-\tan50°(1+\tan10°\tan60°)\times\frac{\cos10°}{\sin50°}\\ ={}&-\tan50°(1+\sqrt3\tan10°)\times\frac{\cos10°}{\sin50°}\\ ={}&-2\tan50°(\frac12\times\frac{\cos10°}{\cos10°}+\frac{\sqrt3}2\times\frac{\sin10}{\cos10})\times\frac{\cos10°}{\sin50°}\\ ={}&-2\tan50°(\cos60°\cos10°+\sin60°\sin10)\times\frac1{\sin50°}\\ ={}&-2\tan50°(\cos60°-10°)\times\frac1{\sin50°}\\ ={}&-2\tan50°(\cos60°-10°)\times\frac1{\sin50°}\\ ={}&-2 \end{align*}$

\begin{aligned} \sqrt{\sin^{2} 80 °} [2 \sin 50 ° + \sin 10 ° (1 + \sqrt{3} \tan 10 °)] \\ = & \sqrt{\sin^{2} 80 °} [2 \sin 50 ° + 2 \sin 10 ° (\frac{1}{2} + \frac{\sqrt{3}}{2} \tan 10 °)] \\ = & 2 \sqrt{\sin^{2} 80 °} [\sin 50 ° + \frac{\sin 10 °}{\cos 10 °} \times (\cos 60 ° \cos 10 ° + \sin 60 ° \sin 10 °)] \\ = & 2 \sqrt{\sin^{2} 80 °} [\sin 50 ° + \frac{\sin 10 °}{\cos 10 °} \times (\cos 60 ° - 10 °)] \\ = & 2 \cos 10 ° (\sin 50 ° + \tan 10 ° \cos 50 °) \\ = & 2 (\sin 50 ° \cos 10 ° + \cos 50 ° \sin 10 °) \\ = & 2 \sin (50 ° + 10 °) \\ = & \sqrt{3} \end{aligned}

$\begin{align*} &\sqrt{\sin^280°}\left[2\sin50°+\sin10°(1+\sqrt3\tan10°)\right]\\ ={}&\sqrt{\sin^280°}\left[2\sin50°+2\sin10°(\frac12+\frac{\sqrt3}2\tan10°)\right]\\ ={}&2\sqrt{\sin^280°}\left[\sin50°+\frac{\sin10°}{\cos10°}\times(\cos60°\cos10°+\sin60°\sin10°)\right]\\ ={}&2\sqrt{\sin^280°}\left[\sin50°+\frac{\sin10°}{\cos10°}\times(\cos60°-10°)\right]\\ ={}&2\cos10°(\sin50°+\tan10°\cos50°)\\ ={}&2(\sin50°\cos10°+\cos50°\sin10°)\\ ={}&2\sin(50°+10°)\\ ={}&\sqrt3\\ \end{align*}$

\begin{aligned} \sqrt{1 + \cos 20 °} [2 \sin 50 ° + \sin 10 ° (1 + \sqrt{3} \tan 10 °)] \\ = & \sqrt{1 + \cos 20 °} [2 \sin 50 ° + 2 \sin 10 ° (\frac{1}{2} + \frac{\sqrt{3}}{2} \tan 10 °)] \\ = & 2 \sqrt{1 + \cos 20 °} [\sin 50 ° + \frac{\sin 10 °}{\cos 10 °} \times (\cos 60 ° \cos 10 ° + \sin 60 ° \sin 10 °)] \\ = & 2 \sqrt{1 + \cos 20 °} [\sin 50 ° + \frac{\sin 10 °}{\cos 10 °} \times (\cos 60 ° - 10 °)] \\ = & 2 \sqrt{1 + \cos 20 °} (\sin 50 ° + \tan 10 ° \cos 50 °) \\ = & 2 \sqrt{1 + 2 \cos^{2} 10 ° - 1} (\sin 50 ° + \tan 10 ° \cos 50 °) \\ = & 2 \sqrt{1 + 2 \cos^{2} 10 ° - 1} (\sin 50 ° + \tan 10 ° \cos 50 °) \\ = & 2 \sqrt{2} \cos 10 ° (\sin 50 ° + \tan 10 ° \cos 50 °) \\ = & 2 \sqrt{2} \sin 60 ° \\ = & \sqrt{6} \end{aligned}

$\begin{align*} &\sqrt{1+\cos20°}\left[2\sin50°+\sin10°(1+\sqrt3\tan10°)\right]\\ ={}&\sqrt{1+\cos20°}\left[2\sin50°+2\sin10°(\frac12+\frac{\sqrt3}2\tan10°)\right]\\ ={}&2\sqrt{1+\cos20°}\left[\sin50°+\frac{\sin10°}{\cos10°}\times(\cos60°\cos10°+\sin60°\sin10°)\right]\\ ={}&2\sqrt{1+\cos20°}\left[\sin50°+\frac{\sin10°}{\cos10°}\times(\cos60°-10°)\right]\\ ={}&2\sqrt{1+\cos20°}(\sin50°+\tan10°\cos50°)\\ ={}&2\sqrt{1+2\cos^210°-1}(\sin50°+\tan10°\cos50°)\\ ={}&2\sqrt{1+2\cos^210°-1}(\sin50°+\tan10°\cos50°)\\ ={}&2\sqrt2\cos10°(\sin50°+\tan10°\cos50°)\\ ={}&2\sqrt2\sin60°\\ ={}&\sqrt6\\ \end{align*}$

\begin{aligned} 4 \cos 50 ° - \tan 40 ° \\ = & 4 \sin 40 ° - \frac{\sin 40 °}{\cos 40 °} \\ = & \frac{4 \sin 40 ° \cos 40 ° - \sin 40 °}{\cos 40 °} \\ = & \frac{2 \sin 80 ° - \sin 40 °}{\cos 40 °} \\ = & \frac{2 \cos 10 ° - \sin (10 ° + 30 °)}{\cos 40 °} \\ = & \frac{2 \cos 10 ° - (\sin 10 ° \cos 30 ° + \cos 10 ° \sin 30 °)}{\cos 40 °} \\ = & \frac{2 \cos 10 ° - (\frac{\sqrt{3}}{2} \sin 10 ° + \frac{1}{2} \cos 10 °)}{\cos 40 °} \\ = & \frac{\frac{3}{2} \cos 10 ° - \frac{\sqrt{3}}{2} \sin 10 °}{\cos 40 °} \\ = & \sqrt{3} \times \frac{\frac{\sqrt{3}}{2} \cos 10 ° - \frac{1}{2} \sin 10 °}{\cos 40 °} \\ = & \sqrt{3} \times \frac{\cos 30 ° \cos 10 ° - \sin 30 ° \sin 10 °}{\cos 40 °} \\ = & \sqrt{3} \times \frac{\cos (30 ° + 10 °)}{\cos 40 °} \\ = & \sqrt{3} \end{aligned}

$\begin{align*} &4\cos50°-\tan40°\\ ={}&4\sin40°-\frac{\sin40°}{\cos40°}\\ ={}&\frac{4\sin40°\cos40°-\sin40°}{\cos40°}\\ ={}&\frac{2\sin80°-\sin40°}{\cos40°}\\ ={}&\frac{2\cos10°-\sin(10°+30°)}{\cos40°}\\ ={}&\frac{2\cos10°-(\sin10°\cos30°+\cos10°\sin30°)}{\cos40°}\\ ={}&\frac{2\cos10°-(\frac{\sqrt3}2\sin10°+\frac12\cos10°)}{\cos40°}\\ ={}&\frac{\frac32\cos10°-\frac{\sqrt3}2\sin10°}{\cos40°}\\ ={}&\sqrt3\times\frac{\frac{\sqrt3}2\cos10°-\frac12\sin10°}{\cos40°}\\ ={}&\sqrt3\times\frac{\cos30°\cos10°-\sin30°\sin10°}{\cos40°}\\ ={}&\sqrt3\times\frac{\cos(30°+10°)}{\cos40°}\\ ={}&\sqrt3\\ \end{align*}\\$

\begin{aligned} \frac{\sqrt{3} \tan 12 ° - 3}{4 \cos^{2} 12 ° \sin 12 ° - 2 \sin 12 °} \\ = & \sqrt{3} \times \frac{\tan 12 ° - \tan 60 °}{4 \cos^{2} 12 ° \sin 12 ° - 2 \sin 12 °} \\ = & \sqrt{3} \times \frac{\frac{\sin 12 ° \cos 60 ° - \cos 12 ° \sin 60 °}{\cos 12 ° \cos 60 °}}{4 \cos^{2} 12 ° \sin 12 ° - 2 \sin 12 °} \\ = & \sqrt{3} \times \frac{\frac{\sin (12 ° - 60 °)}{\frac{1}{2} \cos 12 °}}{4 \cos^{2} 12 ° \sin 12 ° - 2 \sin 12 °} \\ = & \sqrt{3} \times \frac{\frac{- 2 \sin 48 °}{\cos 12 °}}{4 \cos^{2} 12 ° \sin 12 ° - 2 \sin 12 °} \\ = & \sqrt{3} \times \frac{- 2 \sin 48 °}{2 \sin 12 ° \cos 12 ° (2 \cos^{2} 12 ° - 1)} \\ = & \sqrt{3} \times \frac{- 2 \sin 48 °}{\sin 24 ° \cos 24 °} \\ = & \sqrt{3} \times \frac{- 2 \sin 48 °}{\frac{1}{2} \sin 48 °} \\ = & - 4 \sqrt{3} \end{aligned}

$\begin{align*} &\frac{\sqrt3\tan12°-3}{4\cos^212°\sin12°-2\sin12°}\\ ={}&\sqrt3\times\frac{\tan12°-\tan60°}{4\cos^212°\sin12°-2\sin12°}\\ ={}&\sqrt3\times\frac{\frac{\sin12°\cos60°-\cos12°\sin60°}{\cos12°\cos60°}}{4\cos^212°\sin12°-2\sin12°}\\ ={}&\sqrt3\times\frac{\frac{\sin(12°-60°)}{\frac12\cos12°}}{4\cos^212°\sin12°-2\sin12°}\\ ={}&\sqrt3\times\frac{\frac{-2\sin48°}{\cos12°}}{4\cos^212°\sin12°-2\sin12°}\\ ={}&\sqrt3\times\frac{-2\sin48°}{2\sin12°\cos12°(2\cos^212°-1)}\\ ={}&\sqrt3\times\frac{-2\sin48°}{\sin24°\cos24°}\\ ={}&\sqrt3\times\frac{-2\sin48°}{\frac12\sin48°}\\ ={}&-4\sqrt3\\ \end{align*}$

\begin{aligned} \frac{1}{\sin 10 °} - \frac{\sqrt{3}}{\cos 10 °} \\ = & \frac{\cos 10 ° - \sqrt{3} \sin 10 °}{\sin 10 ° \cos 10 °} \\ = & 2 \times \frac{\frac{1}{2} \cos 10 ° - \frac{\sqrt{3}}{2} \sin 10 °}{\sin 10 ° \cos 10 °} \\ = & 2 \times \frac{\sin 30 ° \cos 10 ° - \cos 30 ° \sin 10 °}{\sin 10 ° \cos 10 °} \\ = & 2 \times \frac{\sin (30 ° - 10 °)}{\frac{1}{2} \sin 20 °} \\ = & 4 \end{aligned}

$\begin{align*} &\frac1{\sin10°}-\frac{\sqrt3}{\cos10°}\\ ={}&\frac{\cos10°-\sqrt3\sin10°}{\sin10°\cos10°}\\ ={}&2\times\frac{\frac12\cos10°-\frac{\sqrt3}2\sin10°}{\sin10°\cos10°}\\ ={}&2\times\frac{\sin30°\cos10°-\cos30°\sin10°}{\sin10°\cos10°}\\ ={}&2\times\frac{\sin(30°-10°)}{\frac12\sin20°}\\ ={}&4\\ \end{align*}$

\begin{aligned} \cos^{4} x - 2 \sin x \cos x - \sin^{4} x \\ = & (\cos^{2} x + \sin^{2} x) (\cos^{2} x - \sin^{2} x) - 2 \sin x \cos x \\ = & \cos 2 x - \sin 2 x \\ = & \sqrt{2} (\frac{\sqrt{2}}{2} \cos 2 x - \frac{\sqrt{2}}{2} \sin 2 x) \\ = & \sqrt{2} (\sin \frac{π}{4} \cos 2 x - \cos \frac{π}{4} \sin 2 x) \\ = & \sqrt{2} (\sin \frac{π}{4} \cos 2 x - \cos \frac{π}{4} \sin 2 x) \\ = & \sqrt{2} \sin (\frac{π}{4} - 2 x) \\ = & \sqrt{2} \sin (2 x - \frac{3 π}{4}) \end{aligned}

$\begin{align*} &\cos^4x-2\sin x\cos x-\sin^4x\\ ={}&(\cos^2x+\sin^2x)(\cos^2x-\sin^2x)-2\sin x\cos x\\ ={}&\cos2x-\sin2x\\ ={}&\sqrt2(\frac{\sqrt2}2\cos2x-\frac{\sqrt2}2\sin2x)\\ ={}&\sqrt2(\sin\frac\pi4\cos2x-\cos\frac\pi4\sin2x)\\ ={}&\sqrt2(\sin\frac\pi4\cos2x-\cos\frac\pi4\sin2x)\\ ={}&\sqrt2\sin(\frac\pi4-2x)\\ ={}&\sqrt2\sin(2x-\frac{3\pi}4)\\ \end{align*}$

\begin{aligned} \sin (x + \frac{π}{6}) + \sin (x - \frac{π}{6}) + \cos x \\ = & \sin (x + \frac{π}{6}) + \sin x \cos \frac{π}{6} - \cos x \sin \frac{π}{6} + \cos x \\ = & \sin (x + \frac{π}{6}) + \sin x \cos \frac{π}{6} + \cos x \sin \frac{π}{6} \\ = & \sin (x + \frac{π}{6}) + \sin (x + \frac{π}{6}) \\ = & 2 \sin (x + \frac{π}{6}) \end{aligned}

$\begin{align*} &\sin(x+\frac\pi6)+\sin(x-\frac\pi6)+\cos x\\ ={}&\sin(x+\frac\pi6)+\sin x\cos\frac\pi6-\cos x\sin\frac\pi6+\cos x\\ ={}&\sin(x+\frac\pi6)+\sin x\cos\frac\pi6+\cos x\sin\frac\pi6\\ ={}&\sin(x+\frac\pi6)+\sin(x+\frac\pi6)\\ ={}&2\sin(x+\frac\pi6)\\ \end{align*}$

\begin{aligned} a \sin x + b \cos x \\ = & \sqrt{a^{2} + b^{2}} (\frac{a}{\sqrt{a^{2} + b^{2}}} \sin x + \frac{b}{\sqrt{a^{2} + b^{2}}} \cos x) \\ = & \sqrt{a^{2} + b^{2}} (\sin x \cos α + \cos x \sin α) \\ = & \sqrt{a^{2} + b^{2}} \sin (x + α) \end{aligned}

$\begin{align*} &a\sin x+b\cos x\\ ={}&\sqrt{a^2+b^2}(\frac{a}{\sqrt{a^2+b^2}}\sin x+\frac{b}{\sqrt{a^2+b^2}}\cos x)\\ ={}&\sqrt{a^2+b^2}(\sin x\cos\alpha+\cos x\sin\alpha)\\ ={}&\sqrt{a^2+b^2}\sin(x+\alpha) \end{align*}$

posted @ 2023-01-03 22:39 蒟酱阅读(346) 评论(0) 编辑收藏举报

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