Question

Prove that $ \cos 510^\circ \cos 330^\circ + \sin 390^\circ \cos 120^\circ = - 1 $ .

Answer 1

Hint: We can write $ \cos 510^\circ $ as $ \cos \left( {360 + 150} \right) $ and using the properties of $ {\rm{cosine}} $ we obtain some value. $ \cos \left( {330^\circ } \right) $ can be written as $ \cos \left( {360 - 30} \right) $ using $ {\rm{cosine}} $ properties we get some value. $ \sin \left( {390^\circ } \right) $ can be written as $ \sin \left( {360 + 30} \right) $ using $ \sin $ properties we get some values and $ \cos 120^\circ $ can be written as $ \cos \left( {90 + 30} \right) $ using $ \cos $ properties we get some values.

Complete step-by-step answer:
The value of $ \cos 510^\circ \cos 330^\circ + \sin 390^\circ \cos 120^\circ = - 1 $ .
We have $ \cos 510^\circ $ that $ 510 $ can be written as $ 360 $ and $ 150 $ . Also, $ 150 $ can be written in terms of $ 180 $ and $ 30 $ since the angle for $ 510 $ is not known so the angles of the terms will be equal.
We know the property for the $ \cos $ is $ \cos \left( {2\pi + \theta } \right) = \cos \left( \theta \right) $ .
On substituting value in the above formula that is $ \theta $ as $ 150 $ , we get,
 $ \cos \left( {360 + 150} \right) $ which is $ \cos \left( {510} \right) $ . Now, let us use the property $ \cos \left( {2\pi + \theta } \right) = \cos \left( \theta \right) $ then we get,
 $ \cos \left( {510} \right) = \cos \left( {360 + 150} \right) $
Then we will take the property of $ \cos $ that is $ \cos \left( {\pi - \theta } \right) = - \cos \left( \theta \right) $ .
On substituting value of $ \pi $ as $ 180 $ and $ \theta $ as $ 30 $ .Then we get, $ \cos \left( {180 - 30} \right) $ which is $ \cos \left( {150} \right) $ . So,
  $ \cos \left( {150} \right) = \cos \left( {180 - 30} \right) $
Now, let us use the property $ \cos \left( {\pi - \theta } \right) = - \cos \left( \theta \right) $ then we get,
 $ \cos \left( {180 - 30} \right) = - \cos \left( {30} \right) $
To find the value for $ \cos \left( {330} \right) $ we use the following property for $ \cos $ , which is, $ \cos \left( {2\pi - \theta } \right) = \cos \left( \theta \right) $
On substitute the value of $ \pi $ and take $ \theta $ as $ 30 $ then we obtain ,
 $ \cos \left( {360 - 30} \right) $ which is equal to $ \cos \left( {330} \right) $ .
Hence, $ \cos \left( {330} \right) $ can be written as,
 $ \cos \left( {330} \right) = \cos \left( {360 - 30} \right) $
On using the property $ \cos \left( {2\pi - \theta } \right) = \cos \left( \theta \right) $ , we get,
 $ \cos \left( {360 - 30} \right) = \cos 30 $
To find the value for $ \sin \left( {390} \right) $ value,
We will use the property $ \sin \left( {2\pi + \theta } \right) = \sin \left( \theta \right) $ , then we get,
 $ \begin{array}{c}
\sin \left( {390} \right) = \sin \left( {360 + 30} \right)\\
 = \sin \left( {30} \right)
\end{array} $
To find the value for $ \cos \left( {120} \right) $ , we use the property that $ \cos \left( {\dfrac{\pi }{2} + 30} \right) = - \sin \left( {30} \right) $ , we get,
 $ \begin{array}{c}
\cos \left( {120} \right) = \cos \left( {90 + 30} \right)\\
 = - \sin \left( {30} \right)
\end{array} $
Putting all the values in the $ \cos 510^\circ \cos 330^\circ + \sin 390^\circ \cos 120^\circ $ , and also by applying property that is $ {\cos ^2}\theta + {\sin ^2}\theta = 1 $ , we obtain,
 $ \begin{array}{c}
 - \cos 30 \times \cos 30 + \sin \left( {30} \right) \times \left( { - \sin \left( {30} \right)} \right) = - {\cos ^2}30 - {\sin ^2}30\\
 = - \left( {{{\cos }^2}30 + {{\sin }^2}30} \right)\\
 = - 1
\end{array} $
Hence, $ \cos 510^\circ \cos 330^\circ + \sin 390^\circ \cos 120^\circ = - 1 $ .

Note: please be careful with the properties of the trigonometric function that is $ \sin $ , $ \cos $ properties. This problem can be proved by substituting the values which are known since, the value of $ \sin \left( {30} \right) $ and $ \cos \left( {30} \right) $ is known that is $ \sin \left( {30} \right) = \dfrac{1}{2} $ and $ \cos \left( {30} \right) = \dfrac{{\sqrt 3 }}{2} $ .