Skip to main content
Mathematics LibreTexts

B.4 Cosine and Sine Laws

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Cosine Law or Law of Cosines

    The cosine law says that, if a triangle has sides of length \(a\text{,}\) \(b\) and \(c\) and the angle opposite the side of length \(c\) is \(\gamma\text{,}\) then

    \begin{align*} c^2 &= a^2+b^2 - 2ab\cos\gamma \end{align*}

    Observe that, when \(\gamma=\tfrac{\pi}{2}\text{,}\) this reduces to, (surpise!) Pythagoras' theorem \(c^2=a^2+b^2\text{.}\) Let's derive the cosine law.


    Consider the triangle on the left. Now draw a perpendicular line from the side of length \(c\) to the opposite corner as shown. This demonstrates that

    \begin{align*} c &= a \cos \beta + b \cos \alpha\\ \end{align*}

    Multiply this by \(c\) to get an expression for \(c^2\text{:}\)

    \begin{align*} c^2 &= ac \cos \beta + bc \cos \alpha\\ \end{align*}

    Doing similarly for the other corners gives

    \begin{align*} a^2 &= ac \cos \beta + ab \cos \gamma\\ b^2 &= bc \cos \alpha + ab \cos \gamma \end{align*}

    Now combining these:

    \begin{align*} a^2+b^2-c^2 &= (bc-bc) \cos \alpha + (ac-ac)\cos\beta + 2ab \cos \gamma\\ &= 2ab\cos \gamma \end{align*}

    as required.

    Sine Law or Law of Sines

    The sine law says that, if a triangle has sides of length \(a, b\) and \(c\) and the angles opposite those sides are \(\alpha\text{,}\) \(\beta\) and \(\gamma\text{,}\) then

    \begin{align*} \frac{a}{\sin \alpha} &= \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}. \end{align*}


    This rule is best understood by computing the area of the triangle using the formula \(A = \frac{1}{2}ab\sin\theta\) of Appendix A.10. Doing this three ways gives

    \begin{align*} 2A &= bc \sin \alpha\\ 2A &= ac \sin \beta\\ 2A &= ab \sin \gamma \end{align*}

    Dividing these expressions by \(abc\) gives

    \begin{align*} \frac{2A}{abc} &= \frac{\sin \alpha}{a} = \frac{\sin\beta}{b} = \frac{\sin \gamma}{c} \end{align*}

    as required.

    This page titled B.4 Cosine and Sine Laws is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Joel Feldman, Andrew Rechnitzer and Elyse Yeager via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.