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1298cosx=2770sinx+325.8

Guest Oct 16, 2014

Best Answer 

 #2
avatar+19632 
+10

1298cosx=2770sinx+325.8

$$\begin{array}{rcl}
1298*cos(x) &=& 2770*sin(x)+325.8\\
\underbrace{1298}_{x_p}*\underbrace{cos(x)}_{n_x}
-\underbrace{2770}_{y_p}*\underbrace{sin(x)}_{n_y} &=& \underbrace{325.8}_d \qquad \text{line: } \vec{p}*\vec{n}=d \quad \vec{p}=(x_p,y_p)=(1298,-2770) \\
\tan{(\alpha)} &=& \frac{y_p}{x_p} = -\frac{2770}{1298}\\\\
cos(x-\alpha) &=& \frac{d}{ \sqrt{x_p^2+y_p^2} } \\\\
sin(x-\alpha) &=& \frac{ \sqrt{x_p^2+y_p^2-d^2}}{\sqrt{x_p^2+y_p^2} }\\\\
tan(x-\alpha) &=& \pm \sqrt{\frac{ x_p^2+y_p^2 }{d^2} -1 )} \\\\
x_{1,2} -\alpha &=& tan^{-1} { \left( \pm \sqrt{\frac{ x_p^2+y_p^2 }{d^2} -1 \right) } }\\\\
x_{1,2} &=& \alpha + tan^{-1} { \left( \pm \sqrt{\frac{ x_p^2+y_p^2 }{d^2} -1 \right) } }\\\\
x_{1,2} &=& tan^{-1} { \left( \frac{y_p}{x_p} \right) } + tan^{-1} { \left( \pm \sqrt{\frac{ x_p^2+y_p^2 }{d^2} -1 \right) } }\\\\
x_{1,2} &=& tan^{-1} { \left( \frac{-2770}{1298} \right) } + tan^{-1} { \left( \pm \sqrt{\frac{ (1298)^2+(-2770)^2 }{(325.8)^2} -1 \right) } }\\\\
x_{1,2} &=& ( -64.8925847874\ensurement{^{\circ}} \pm n*\pi) +(\pm 83.8861674476\ensurement{^{\circ}} \pm n*\pi)\\\\
x_1 &=& -64.8925847874\ensurement{^{\circ}} + 83.8861674476\ensurement{^{\circ}} \pm n*2\pi\\\\
x_2 &=& -64.8925847874\ensurement{^{\circ}} - 83.8861674476\ensurement{^{\circ}} \pm n*2\pi\\\\
x_1 &=& 18.9935826602\ensurement{^{\circ}} \pm n*2\pi\\\\
x_2 &=& -148.778752235\ensurement{^{\circ}} \pm n*2\pi
\end{array}$$

heureka  Oct 16, 2014
 #1
avatar+26750 
+5

See if the following graph helps

 

Intersecting curves

.

Alan  Oct 16, 2014
 #2
avatar+19632 
+10
Best Answer

1298cosx=2770sinx+325.8

$$\begin{array}{rcl}
1298*cos(x) &=& 2770*sin(x)+325.8\\
\underbrace{1298}_{x_p}*\underbrace{cos(x)}_{n_x}
-\underbrace{2770}_{y_p}*\underbrace{sin(x)}_{n_y} &=& \underbrace{325.8}_d \qquad \text{line: } \vec{p}*\vec{n}=d \quad \vec{p}=(x_p,y_p)=(1298,-2770) \\
\tan{(\alpha)} &=& \frac{y_p}{x_p} = -\frac{2770}{1298}\\\\
cos(x-\alpha) &=& \frac{d}{ \sqrt{x_p^2+y_p^2} } \\\\
sin(x-\alpha) &=& \frac{ \sqrt{x_p^2+y_p^2-d^2}}{\sqrt{x_p^2+y_p^2} }\\\\
tan(x-\alpha) &=& \pm \sqrt{\frac{ x_p^2+y_p^2 }{d^2} -1 )} \\\\
x_{1,2} -\alpha &=& tan^{-1} { \left( \pm \sqrt{\frac{ x_p^2+y_p^2 }{d^2} -1 \right) } }\\\\
x_{1,2} &=& \alpha + tan^{-1} { \left( \pm \sqrt{\frac{ x_p^2+y_p^2 }{d^2} -1 \right) } }\\\\
x_{1,2} &=& tan^{-1} { \left( \frac{y_p}{x_p} \right) } + tan^{-1} { \left( \pm \sqrt{\frac{ x_p^2+y_p^2 }{d^2} -1 \right) } }\\\\
x_{1,2} &=& tan^{-1} { \left( \frac{-2770}{1298} \right) } + tan^{-1} { \left( \pm \sqrt{\frac{ (1298)^2+(-2770)^2 }{(325.8)^2} -1 \right) } }\\\\
x_{1,2} &=& ( -64.8925847874\ensurement{^{\circ}} \pm n*\pi) +(\pm 83.8861674476\ensurement{^{\circ}} \pm n*\pi)\\\\
x_1 &=& -64.8925847874\ensurement{^{\circ}} + 83.8861674476\ensurement{^{\circ}} \pm n*2\pi\\\\
x_2 &=& -64.8925847874\ensurement{^{\circ}} - 83.8861674476\ensurement{^{\circ}} \pm n*2\pi\\\\
x_1 &=& 18.9935826602\ensurement{^{\circ}} \pm n*2\pi\\\\
x_2 &=& -148.778752235\ensurement{^{\circ}} \pm n*2\pi
\end{array}$$

heureka  Oct 16, 2014
 #3
avatar+92781 
0

Very impressive Heureka :)

Melody  Oct 16, 2014

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