Find the length of the line segments whose endpoints have polar coordinate (7, 40 degres) and (15, 100 degrees).
Find the length of the line segments whose endpoints have polar coordinate (7, 40 degres) and (15, 100 degrees).
\(\text{Let $P_1 =\dbinom{x_1}{y_1}=\dbinom{7\cos(40^\circ)}{7\sin(40^\circ) } $} \\ \text{Let $P_2 =\dbinom{x_2}{y_2}=\dbinom{15\cos(100^\circ)}{15\sin(100^\circ)} $} \)
\(\begin{array}{|rcll|} \hline \mathbf{ \overline{P_1P_2} } &=& \mathbf{ \sqrt{ \left( x_2-x_1\right)^2 + \left(y_2-y_1 \right)^2 } } \\\\ &=& \sqrt{ \Big( 15\cos(100^\circ) - 7\cos(40^\circ)\Big)^2 + \Big( 15\sin(100^\circ) - 7\sin(40^\circ)\Big)^2 } \\ &=& \tiny{\sqrt{ 15^2\cos^2(100^\circ) -2*15*7\cos(100^\circ)\cos(40^\circ) + 7^2\cos^2(40^\circ) + 15^2\sin^2(100^\circ) -2*15*7\sin(100^\circ)\sin(40^\circ) + 7^2\sin^2(40^\circ) }} \\ &=& \tiny{\sqrt{ 7^2\Big(\underbrace{\sin^2(40^\circ)+\cos^2(40^\circ)}_{\small{=1}}\Big) + 15^2\Big(\underbrace{\sin^2(100^\circ)+\cos^2(100^\circ)}_{\small{=1}}\Big) -2*15*7\Big(\underbrace{\cos(100^\circ)\cos(40^\circ) + \sin(100^\circ)\sin(40^\circ)}_{\small{=\cos(100^\circ-40^\circ)}} \Big) }} \\ &=& \sqrt{ 7^2+ 15^2 -2*15*7\cos(100^\circ-40^\circ) } \\ &=& \sqrt{ 7^2+ 15^2 -2*15*7\cos(60^\circ) } \quad | \quad \cos(60^\circ)= \dfrac{1}{2} \\ &=& \sqrt{ 7^2+ 15^2 - 15*7 } \\ &=& \sqrt{ 169 } \\ &=& \mathbf{13} \\ \hline \end{array}\)
The length of the line is 13
Find the length of the line segments whose endpoints have polar coordinate (7, 40 degres) and (15, 100 degrees).
\(\text{Let $P_1 =\dbinom{x_1}{y_1}=\dbinom{7\cos(40^\circ)}{7\sin(40^\circ) } $} \\ \text{Let $P_2 =\dbinom{x_2}{y_2}=\dbinom{15\cos(100^\circ)}{15\sin(100^\circ)} $} \)
\(\begin{array}{|rcll|} \hline \mathbf{ \overline{P_1P_2} } &=& \mathbf{ \sqrt{ \left( x_2-x_1\right)^2 + \left(y_2-y_1 \right)^2 } } \\\\ &=& \sqrt{ \Big( 15\cos(100^\circ) - 7\cos(40^\circ)\Big)^2 + \Big( 15\sin(100^\circ) - 7\sin(40^\circ)\Big)^2 } \\ &=& \tiny{\sqrt{ 15^2\cos^2(100^\circ) -2*15*7\cos(100^\circ)\cos(40^\circ) + 7^2\cos^2(40^\circ) + 15^2\sin^2(100^\circ) -2*15*7\sin(100^\circ)\sin(40^\circ) + 7^2\sin^2(40^\circ) }} \\ &=& \tiny{\sqrt{ 7^2\Big(\underbrace{\sin^2(40^\circ)+\cos^2(40^\circ)}_{\small{=1}}\Big) + 15^2\Big(\underbrace{\sin^2(100^\circ)+\cos^2(100^\circ)}_{\small{=1}}\Big) -2*15*7\Big(\underbrace{\cos(100^\circ)\cos(40^\circ) + \sin(100^\circ)\sin(40^\circ)}_{\small{=\cos(100^\circ-40^\circ)}} \Big) }} \\ &=& \sqrt{ 7^2+ 15^2 -2*15*7\cos(100^\circ-40^\circ) } \\ &=& \sqrt{ 7^2+ 15^2 -2*15*7\cos(60^\circ) } \quad | \quad \cos(60^\circ)= \dfrac{1}{2} \\ &=& \sqrt{ 7^2+ 15^2 - 15*7 } \\ &=& \sqrt{ 169 } \\ &=& \mathbf{13} \\ \hline \end{array}\)
The length of the line is 13