三角関数
公式
\[
\begin{aligned}
& sin^2\theta+cos^2\theta=1 \\
& tan^2\theta+1=\frac{1}{cos^2\theta}\\
\end{aligned}
\]
正弦定理
\[
\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}=2R
\]
余弦定理
\[
a^2=b^2+c^2-2bc\cdot cosA
\]
加法定理
\[
\begin{aligned}
sin(\alpha\pm\beta) = sin\alpha\cdot cos\beta\pm cos\alpha\cdot sin\beta \\
cos(\alpha\pm\beta) = cos\alpha\cdot cos\beta\mp sin\alpha\cdot sin\beta \\
\end{aligned}
\]
倍角
\[
\begin{aligned}
sin2\theta &= 2sin\theta cos\theta \\
cos2\theta &= cos^2\theta-sin^2\theta \\
&= 2cos^2\theta -1 \\
&= 1-2sin^2\theta \\
tan2\theta &= \frac{2tan\theta}{1-tan^2\theta}
\end{aligned}
\]
\[
\begin{aligned}
sin3\theta &= 3sin\theta-4sin^3\theta \\
cos3\theta &= 4cos^3\theta - 3cos\theta \\
tan3\theta &= \frac{3tan\theta-tan^3\theta}{1-3tan^2\theta}
\end{aligned}
\]
半角
\[
\begin{aligned}
sin^2\frac{\theta}{2} &= \frac{1-cos\theta}{2} \\
cos^2\frac{\theta}{2} &= \frac{1+cos\theta}{2} \\
tan^2\frac{\theta}{2} &= \frac{1-cos\theta}{1+cos\theta}
\end{aligned}
\]
積和
\[
\begin{aligned}
sin\alpha\cdot cos\beta
&= \frac{1}{2}\lbrace sin(\alpha+\beta)+sin(\alpha-\beta)\rbrace \\
cos\alpha\cdot sin\beta
&= \frac{1}{2}\lbrace sin(\alpha+\beta)-sin(\alpha-\beta)\rbrace \\
cos\alpha\cdot cos\beta
&= \frac{1}{2}\lbrace cos(\alpha+\beta)+cos(\alpha-\beta)\rbrace \\
sin\alpha\cdot sin\beta
&= \frac{-1}{2}\lbrace cos(\alpha+\beta)-cos(\alpha-\beta)\rbrace \\
\end{aligned}
\]
和積
\[
\begin{aligned}
sin\alpha+sin\beta &= 2sin\frac{\alpha+\beta}{2}cos\frac{\alpha-\beta}{2} \\
sin\alpha-sin\beta &= 2cos\frac{\alpha+\beta}{2}sin\frac{\alpha-\beta}{2} \\
cos\alpha+cos\beta &= 2cos\frac{\alpha+\beta}{2}cos\frac{\alpha-\beta}{2} \\
cos\alpha-cos\beta &= -2sin\frac{\alpha+\beta}{2}sin\frac{\alpha-\beta}{2} \\
\end{aligned}
\]
逆数
\[
\begin{aligned}
\frac{1}{sinx} &= cosecx & セカント \\
\frac{1}{cosx} &= secx & コセカント \\
\frac{1}{tanx} &= cotx & コタンジェント \\
\end{aligned}
\]
逆関数
\(y=f(x)\)に対する\(x=f^{-1}(y)\)
角度\(\theta\)を求める
\(sin^{-1}\theta\)
\[
y_{sin}=sin(\theta)
\rightarrow
\theta = arcsin(y_{sin}) \\
\]
\(cos^{-1}\theta\)
\[
y_{cos}=cos(\theta)
\rightarrow
\theta = arccos(y_{cos})
\]
\(tan^{-1}\theta\)
\[
y_{tan}=tan(\theta)
\rightarrow
\theta = arctan(y_{tan})
\]
Reference
最終更新日:
August 14, 2023
作成日: August 14, 2023
作成日: August 14, 2023