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It is given that a=2sin theta + cos theta b=2cos theta-sin theta, where theta is greater than or equal to 0 degrees and smaller than or equal to 360 degrees.

 

-Show that a squared +b squared is constant for all values if theta

 

-Given that 2a = 3b show that theta 4/7 and find the corresponding values of theta

Guest Nov 5, 2018
 #1
avatar+20164 
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It is given that a=2sin theta + cos theta b=2cos theta-sin theta,

where theta is greater than or equal to 0 degrees and smaller than or equal to 360 degrees.

 

-Show that a squared +b squared is constant for all values if theta

 

\(\begin{array}{|rcll|} \hline && \boxed{a= 2\sin(\theta) + \cos(\theta) \\ b= 2\cos(\theta) - \sin(\theta) } \\\\ \mathbf{a^2+b^2} &=& \Big(2\sin(\theta) + \cos(\theta)\Big)^2 + \Big(2\cos(\theta) - \sin(\theta)\Big)^2 \\\\ &=& 4\sin^2(\theta) + 4\sin(\theta)\cos(\theta) + \cos^2(\theta) \\ &+& 4\cos^2(\theta) - 4\sin(\theta)\cos(\theta) + \sin^2(\theta) \\\\ &=& 4\sin^2(\theta) + \cos^2(\theta)+ 4\cos^2(\theta) + \sin^2(\theta) \\\\ &=& 4\sin^2(\theta)+ 4\cos^2(\theta) + \cos^2(\theta) + \sin^2(\theta) \\\\ &=& 4\Big(\sin^2(\theta)+ \cos^2(\theta)\Big) + \Big(\sin^2(\theta) + \cos^2(\theta)\Big) \\\\ &=& \Big(\sin^2(\theta)+ \cos^2(\theta)\Big)(4+1) \\\\ &=& 5\Big(\sin^2(\theta)+ \cos^2(\theta)\Big) \quad | \quad \sin^2(\theta)+ \cos^2(\theta) = 1 \\\\ &=& 5 \cdot 1 \\\\ &\mathbf{=}& \mathbf{5} \\ \hline \end{array} \)

 

laugh

heureka  Nov 5, 2018
 #3
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Thank youuuuulaugh

Guest Nov 5, 2018
 #2
avatar+20164 
+3

-Given that 2a = 3b show that tan theta  = 4/7 and find the corresponding values of theta

 

\(\begin{array}{|lrcll|} \hline &&& \boxed{a= 2\sin(\theta) + \cos(\theta) \\ b= 2\cos(\theta) - \sin(\theta) } \\\\ &2a &=& 3b \\\\ &2\Big(2\sin(\theta) + \cos(\theta)\Big) &=& 3\Big( 2\cos(\theta) - \sin(\theta)\Big) \\\\ &4\sin(\theta) + 2\cos(\theta) &=& 6\cos(\theta) - 3\sin(\theta) \\\\ &4\sin(\theta) + 3\sin(\theta) &=& 6\cos(\theta) - 2\cos(\theta) \\\\ &7\sin(\theta) &=& 4\cos(\theta) \quad | \quad : \cos(\theta) \\\\ &7\cdot \dfrac{\sin(\theta)}{\cos(\theta)} &=& 4 \quad | \quad : 7 \\\\ &\dfrac{\sin(\theta)}{\cos(\theta)} &=& \dfrac{4}{7} \quad | \quad \dfrac{\sin(\theta)}{\cos(\theta)} = \tan(\theta) \\\\ &\tan(\theta) &=& \dfrac{4}{7} \\\\ & \theta &=& \arctan\left(\dfrac{4}{7}\right) + z\cdot 180^{\circ}, ~ z \in \mathbb{Z} \\\\ & \theta &=& 29.7448812969^{\circ} + z\cdot 180^{\circ} \\\\ z=0: & \mathbf{\theta} &\mathbf{=}& \mathbf{29.7448812969^{\circ}} \\\\ z=1: & \theta &=& 29.7448812969^{\circ}+ 180^{\circ} \\\\ & \mathbf{\theta} &\mathbf{=}& \mathbf{209.744881297^{\circ}} \\ \hline \end{array}\)

 

The corresponding values of theta are:  \(\mathbf{29.7448812969^{\circ}}\) and \(\mathbf{209.744881297^{\circ}}\)

 

laugh

heureka  Nov 5, 2018

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