B.4 Leyes de coseno y coseno
- Page ID
- 118163
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Ley del Coseno o Ley de Cosinos
La ley del coseno dice que, si un triángulo tiene lados de longitud\(a\text{,}\)\(b\) y\(c\) y el ángulo opuesto al lado de longitud\(c\) es\(\gamma\text{,}\) entonces
\ begin {align*} c^2 &= a^2+b^2 - 2ab\ cos\ gamma\ end {alinear*}
Observe que, cuando\(\gamma=\tfrac{\pi}{2}\text{,}\) esto se reduce a, (¡supise!) Teorema de Pitágoras\(c^2=a^2+b^2\text{.}\) Derivamos la ley del coseno.
Considera el triángulo de la izquierda. Ahora dibuja una línea perpendicular desde el lado de la longitud\(c\) hasta la esquina opuesta como se muestra. Esto demuestra que
\ begin {align*} c &= a\ cos\ beta + b\ cos\ alfa\\\ end {alinear*}
Multiplique esto por\(c\) para obtener una expresión para\(c^2\text{:}\)
\ begin {align*} c^2 &= ac\ cos\ beta + bc\ cos\ alfa\\\ end {alinear*}
Haciendo de manera similar para las otras esquinas da
\ begin {alinear*} a^2 &= ac\ cos\ beta + ab\ cos\ gamma\\ b^2 &= bc\ cos\ alfa + ab\ cos\ gamma\ end {align*}
Ahora combinando estos:
\ begin {alinear*} a^2+b^2-c^2 &= (bc-bc)\ cos\ alfa + (ac-ac)\ cos\ beta + 2ab\ cos\ gamma\ &= 2ab\ cos\ gamma\ end {alinear*}
según sea necesario.
Ley sinusoidal o ley de los senos
La ley sinusoidal dice que, si un triángulo tiene lados de longitud\(a, b\) y\(c\) y los ángulos opuestos esos lados son\(\alpha\text{,}\)\(\beta\) y\(\gamma\text{,}\) entonces
\ begin {align*}\ frac {a} {\ sin\ alpha} &=\ frac {b} {\ sin\ beta} =\ frac {c} {\ sin\ gamma}. \ end {alinear*}
Esta regla se entiende mejor calculando el área del triángulo usando la fórmula\(A = \frac{1}{2}ab\sin\theta\) del Apéndice A.10. Hacer esto de tres maneras da
\ begin {align*} 2A &= bc\ sin\ alfa\\ 2A &= ac\ sin\ beta\\ 2A &= ab\ sin\ gamma\ end {align*}
Dividiendo estas expresiones por\(abc\) da
\ begin {alinear*}\ frac {2A} {abc} &=\ frac {\ sin\ alfa} {a} =\ frac {\ sin\ beta} {b} =\ frac {\ sin\ gamma} {c}\ end {alinear*}
según sea necesario.