三角函数

特别为了某人写的23333,可能会有点问题23333

\(\theta\)	0°	30°	45°	60°	90°	120°	135°	150°	180°
弧度	0	\(\frac{\pi}{6}\)	\(\frac{\pi}{4}\)	\(\frac{\pi}{3}\)	\(\frac{\pi}{2}\)	\(\frac{2\pi}{3}\)	\(\frac{3\pi}{4}\)	\(\frac{5\pi}{6}\)	\(\pi\)
\(\sin{\theta}\)	0	\(\frac{1}{2}\)	\(\frac{\sqrt{2}}{2}\)	\(\frac{\sqrt{3}}{2}\)	1	\(\frac{\sqrt{3}}{2}\)	\(\frac{\sqrt{2}}{2}\)	\(\frac{1}{2}\)	0
\(\cos{\theta}\)	1	\(\frac{\sqrt{3}}{2}\)	\(\frac{\sqrt{2}}{2}\)	\(\frac{1}{2}\)	0	-\(\frac{1}{2}\)	-\(\frac{\sqrt{2}}{2}\)	-\(\frac{\sqrt{3}}{2}\)	-1
\(\tan{\theta}\)	0	\(\frac{\sqrt{3}}{3}\)	1	\(\sqrt{3}\)	无	-\(\sqrt{3}\)	-1	-\(\frac{\sqrt{3}}{3}\)	0

\(\theta\)	210°	225°	240°	270°	300°	315°	330°	360°	0°
弧度	\(\frac{7\pi}{6}\)	\(\frac{5\pi}{4}\)	\(\frac{4\pi}{3}\)	\(\frac{3\pi}{2}\)	\(\frac{5\pi}{3}\)	\(\frac{7\pi}{4}\)	\(\frac{11\pi}{6}\)	0	0
\(\sin{\theta}\)	-\(\frac{1}{2}\)	-\(\frac{\sqrt{2}}{2}\)	-\(\frac{\sqrt{3}}{2}\)	-1	-\(\frac{\sqrt{3}}{2}\)	-\(\frac{\sqrt{2}}{2}\)	-\(\frac{1}{2}\)	0	0
\(\cos{\theta}\)	-\(\frac{\sqrt{3}}{2}\)	-\(\frac{\sqrt{2}}{2}\)	-\(\frac{1}{2}\)	0	\(\frac{1}{2}\)	\(\frac{\sqrt{2}}{2}\)	\(\frac{\sqrt{3}}{2}\)	1	1
\(\tan{\theta}\)	\(\frac{\sqrt{3}}{3}\)	1	\(\sqrt{3}\)	无	-\(\sqrt{3}\)	-1	-\(\frac{\sqrt{3}}{3}\)	0	0

基本公式:

\(\sin^2{\alpha}+\cos^2{\alpha}=1\)

\(\frac{\sin{\alpha}}{\cos{\alpha}}=\tan{\alpha}\)

诱导公式:

\(\sin{(\alpha+k\times2\pi)}=\sin{\alpha} \quad \cos{(\alpha+k\times2\pi)}=\cos{\alpha} \quad \tan{(\alpha+k\times2\pi)}=\tan{\alpha} \quad (k \in Z)\)

\(\sin{(\alpha+\pi)}=-\sin{\alpha} \quad \cos{(\alpha+\pi)}=-\cos{\alpha} \quad \tan{(\alpha+\pi)}=\tan{\alpha}\)

\(\sin{(-\alpha)}=-\sin{\alpha} \quad \cos{(-\alpha)}=\cos{\alpha} \quad \tan{(-\alpha)}=-\tan{\alpha}\)

\(\sin{(\pi-\alpha)}=\sin{\alpha} \quad \cos{(\pi-\alpha)}=-\cos{\alpha} \quad \tan{(\pi-\alpha)}=-\tan{\alpha}\)

$\sin{(\frac{\pi}{2}-\alpha)}=\cos{\alpha} \quad \cos{(\frac{\pi}{2}-\alpha)}=\sin{\alpha} $

$\sin{(\frac{\pi}{2}+\alpha)}=\cos{\alpha} \quad \cos{(\frac{\pi}{2}+\alpha)}=-\sin{\alpha} $

口诀:奇变偶不变,符号看象限

加减法:

\(\sin{(\alpha+\beta)}=\sin{\alpha}\times\cos{\beta}+\sin{\beta}\times\cos{\alpha}\quad\sin{(\alpha-\beta)}=\sin{\alpha}\times\cos{\beta}-\sin{\beta}\times\cos{\alpha}\)

\(\cos{(\alpha+\beta)}=\cos{\alpha}\times\cos{\beta}-\sin{\beta}\times\sin{\alpha}\quad\cos{(\alpha-\beta)}=\cos{\alpha}\times\cos{\beta}+\sin{\beta}\times\sin{\alpha}\)

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

二倍角公式:

\(\sin{(2\times\alpha)}=2\times\sin{\alpha}\times\cos{\alpha}\)

\(\cos{(2\times\alpha)}=\cos^2{\alpha}-\sin^2{\alpha}=2\times\cos^2{\alpha}-1=1-2\times\sin^2{\alpha}\)

\(\tan{(2\times\alpha)}=\frac{2\times\tan{\alpha}}{1-\tan^2{\alpha}}\)

\(\tan{\frac{\alpha}{2}}=\frac{\sin{\alpha}}{1+\cos{\alpha}}=\frac{1-\cos{\alpha}}{\sin{\alpha}}\)

\(\sin^2{\alpha}=\frac{1-\cos{(2\times\alpha)}}{2}\quad\cos^2{\alpha}=\frac{\sin{(2\times\alpha)}-1}{2}\)

\((\sin{\alpha}+\cos{\alpha})^2=1+\sin{2\times\alpha}\)

和差化积:

\(\sin{\alpha}-\sin{\beta}=2\times\cos{\frac{\alpha+\beta}{2}}\times\sin{\frac{\alpha-\beta}{2}}\quad\sin{\alpha}+\sin{\beta}=2\times\cos{\frac{\alpha-\beta}{2}}\times\sin{\frac{\alpha+\beta}{2}}\)
\(\cos{\alpha}-\cos{\beta}=2\times\cos{\frac{\alpha+\beta}{2}}\times\cos{\frac{\alpha-\beta}{2}}\quad\cos{\alpha}+\cos{\beta}=2\times\sin{\frac{\alpha-\beta}{2}}\times\sin{\frac{\alpha+\beta}{2}}\)

积化和差:

\(\sin{\alpha}\times\cos{\beta}=\frac{\sin{(\alpha+\beta)}+\sin{(\alpha-\beta)}}{2}\quad\sin{\alpha}\times\sin{\beta}=\frac{\cos{(\alpha+\beta)}-\cos{(\alpha-\beta)}}{2}\)

\(\cos{\alpha}\times\sin{\beta}=\frac{\sin{(\alpha+\beta)}-\sin{(\alpha-\beta)}}{2}\quad\cos{\alpha}\times\cos{\beta}=\frac{\cos{(\alpha+\beta)}+\cos{(\alpha-\beta)}}{2}\)

在任意三角形ABC中,定义角A对边为a,角B对边为b,角C对边为c,则有:
1.\(\sin{(A+B)}=\sin{C}\quad\cos{(A+B)}=-\cos{C}\)

2.正弦定理:

\(\frac{a}{\sin{A}}=\frac{b}{\sin{B}}=\frac{c}{\sin{C}}=2R\),其中R为该三角形外接圆的半径

3.余弦定理:

\(c^2=a^2+b^2-2 \times a \times b \times \cos{C} \quad a^2=b^2+c^2-2 \times b \times c \times \cos{A} \quad b^2=a^2+c^2-2 \times a \times c \times \cos{B}\)

对于三角函数: \(f(x)=A\sin{(\omega x+\varphi)}\)

\(A\): 振幅

$ \omega $: 三角函数在y轴方向的压缩程度,当 $ \omega > 1$ 时,表示被压缩, \(\omega < 1\) 时表示拉伸.

\(\omega x+\varphi\) : 三角函数的相位

\(\varphi\) : 三角函数的初相

求周期\(T\): \(T= \frac{2 \pi}{\omega}\)

求频率\(f\): \(f=\frac{1}{T}\)