We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
249
3
avatar

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

 Nov 5, 2018
 #1
avatar+22558 
+11

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

 Nov 5, 2018
 #3
avatar
+1

Thank youuuuulaugh

Guest Nov 5, 2018
 #2
avatar+22558 
+10

-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

 Nov 5, 2018

11 Online Users

avatar
avatar