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 $\cos \left( {30} \right) = \dfrac{{\sqrt 3 }}{2}$ .