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sin(x+15 degree) = 3 cos( x-15 degree)

Guest Nov 25, 2015

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 #1
avatar+18715 
+15

sin(x+15 degree) = 3 cos( x-15 degree)

 

\(\small{ \boxed{~ \text{Formula: }\quad \begin{array}{lcl} \sin{ (x+y) } &=& \sin{(x)}\cdot \cos{(y)} + \sin{(y)} \cdot \cos{(x)} \\ \cos{ (x-y) } &=& \cos{(x)}\cdot \cos{(y)} + \sin{(x)} \cdot \sin{(y)} \\ \end{array} ~}\\ \begin{array}{rcl} \sin{ ( x + 15^{\circ} ) } &=& 3 \cos{ ( x - 15^{\circ} ) } \\ \sin{(x)}\cdot \cos{(15^{\circ})} + \sin{(15^{\circ})} \cdot \cos{(x)} &=& 3 \cdot [~ \cos{(x)}\cdot \cos{( 15^{\circ})} + \sin{(x)} \cdot \sin{( 15^{\circ})} ~] \\ \sin{(x)}\cdot \cos{(15^{\circ})} + \sin{(15^{\circ})} \cdot \cos{(x)} &=& 3 \cdot \cos{(x)}\cdot \cos{( 15^{\circ})} + 3 \cdot \sin{(x)} \cdot \sin{( 15^{\circ})} \quad | \quad : \cos{(x)} \quad x\ne 90^{\circ}\\ \tan{(x)}\cdot \cos{(15^{\circ})} + \sin{(15^{\circ})} &=& 3 \cdot \cos{( 15^{\circ})} + 3 \cdot \tan{(x)} \cdot \sin{( 15^{\circ})} \\ \tan{(x)}\cdot \cos{(15^{\circ})} - 3 \cdot \tan{(x)} \cdot \sin{( 15^{\circ})} &=& 3 \cdot \cos{( 15^{\circ})} - \sin{(15^{\circ})} \qquad | \qquad : \cos{(15^{\circ})} \\ \tan{(x)} - 3 \cdot \tan{(x)} \cdot \tan{( 15^{\circ})} &=& 3 - \tan{(15^{\circ})} \\ \tan{(x)}\cdot \left[~ 1 - 3 \cdot \tan{( 15^{\circ})} ~ \right] &=& 3 - \tan{(15^{\circ})} \\ \tan{(x)} &=& \frac{ 3 - \tan{(15^{\circ})} } { 1 - 3 \cdot \tan{( 15^{\circ})} } \\ \tan{(x)} &=& \frac{ 2.7320508076 } { 0.1961524227 } \\ \tan{(x)} &=& 13.9282032303 \\ \mathbf{ x } & \mathbf{=} & \mathbf{ 85.8933946491^{\circ} \pm k\cdot 180^{\circ} \qquad k \in Z } \end{array} }\)

 

laugh

heureka  Nov 25, 2015
edited by heureka  Nov 25, 2015
edited by heureka  Nov 25, 2015
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1+0 Answers

 #1
avatar+18715 
+15
Best Answer

sin(x+15 degree) = 3 cos( x-15 degree)

 

\(\small{ \boxed{~ \text{Formula: }\quad \begin{array}{lcl} \sin{ (x+y) } &=& \sin{(x)}\cdot \cos{(y)} + \sin{(y)} \cdot \cos{(x)} \\ \cos{ (x-y) } &=& \cos{(x)}\cdot \cos{(y)} + \sin{(x)} \cdot \sin{(y)} \\ \end{array} ~}\\ \begin{array}{rcl} \sin{ ( x + 15^{\circ} ) } &=& 3 \cos{ ( x - 15^{\circ} ) } \\ \sin{(x)}\cdot \cos{(15^{\circ})} + \sin{(15^{\circ})} \cdot \cos{(x)} &=& 3 \cdot [~ \cos{(x)}\cdot \cos{( 15^{\circ})} + \sin{(x)} \cdot \sin{( 15^{\circ})} ~] \\ \sin{(x)}\cdot \cos{(15^{\circ})} + \sin{(15^{\circ})} \cdot \cos{(x)} &=& 3 \cdot \cos{(x)}\cdot \cos{( 15^{\circ})} + 3 \cdot \sin{(x)} \cdot \sin{( 15^{\circ})} \quad | \quad : \cos{(x)} \quad x\ne 90^{\circ}\\ \tan{(x)}\cdot \cos{(15^{\circ})} + \sin{(15^{\circ})} &=& 3 \cdot \cos{( 15^{\circ})} + 3 \cdot \tan{(x)} \cdot \sin{( 15^{\circ})} \\ \tan{(x)}\cdot \cos{(15^{\circ})} - 3 \cdot \tan{(x)} \cdot \sin{( 15^{\circ})} &=& 3 \cdot \cos{( 15^{\circ})} - \sin{(15^{\circ})} \qquad | \qquad : \cos{(15^{\circ})} \\ \tan{(x)} - 3 \cdot \tan{(x)} \cdot \tan{( 15^{\circ})} &=& 3 - \tan{(15^{\circ})} \\ \tan{(x)}\cdot \left[~ 1 - 3 \cdot \tan{( 15^{\circ})} ~ \right] &=& 3 - \tan{(15^{\circ})} \\ \tan{(x)} &=& \frac{ 3 - \tan{(15^{\circ})} } { 1 - 3 \cdot \tan{( 15^{\circ})} } \\ \tan{(x)} &=& \frac{ 2.7320508076 } { 0.1961524227 } \\ \tan{(x)} &=& 13.9282032303 \\ \mathbf{ x } & \mathbf{=} & \mathbf{ 85.8933946491^{\circ} \pm k\cdot 180^{\circ} \qquad k \in Z } \end{array} }\)

 

laugh

heureka  Nov 25, 2015
edited by heureka  Nov 25, 2015
edited by heureka  Nov 25, 2015

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