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Law of Sines and Cosines

Used for oblique triangles (triangles with no right angles).

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Law of Sines

Law of Cosines

$\begin{equation*} \frac{sin(A)}{a} = \frac{sin(B)}{b} = \frac{sin(C)}{c} \end{equation*}$

use for: AAS, ASA, SSA

$\begin{equation*} a^{2} = b^{2} + c^{2} - 2 \cdot b \cdot c \cdot cos(A) \end{equation*}$

$\begin{equation*} b^{2} = a^{2} + c^{2} - 2 \cdot a \cdot c \cdot cos(B) \end{equation*}$

$\begin{equation*} c^{2} = a^{2} + b^{2} - 2 \cdot a \cdot b \cdot cos(C) \end{equation*}$

use for: SSS or SAS

Example

A ski lift rises at a 28 $^\circ$ angle during the first 41 feet up a mountain to achieve a height of 20 feet, which is maintained during ride up the mountain. Determine the length of the cable needed for this initial rise.

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2 slides & non-included angle

$\hookrightarrow$ SSA

$\hookrightarrow$ Law of Sines

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$\begin{equation*} \frac{sin(A)}{a} = \frac{sin(B)}{b} \end{equation*}$

$\begin{equation*} \frac{sin(28^\circ)}{\text{20 ft}} = \frac{sin(B)}{\text{41 ft}} \end{equation*}$

$\begin{equation*} \angle B = 74.24^\circ \end{equation*}$

$\begin{equation*} \angle A + \angle B + \angle C = 180^\circ \end{equation*}$

$\begin{equation*} 28^\circ + 74.24^\circ + \angle C = 180^\circ \end{equation*}$

$\begin{equation*} \angle C = 77.76^\circ \end{equation*}$

* * *

$\begin{equation*} \frac{sin(A)}{a} = \frac{sin(C)}{c} \end{equation*}$

$\begin{equation*} \frac{sin(77.76^\circ)}{c} = \frac{sin(28^\circ)}{\text{20 ft}} \end{equation*}$

$\begin{equation*} c = 41.63 \text{ ft} = x \end{equation*}$