Question

How do you evaluate $\cos \left( { - 210^\circ } \right)$.

Answer 1

Here, we will convert the given angle measures into acute angles using trigonometric identities. Then, we will express the obtained angle as a difference of two angles. We will then simplify the expression and substitute the value of the cosine of the obtained angle to find the required value.

Complete step-by-step answer:
First, we will simplify the given trigonometric ratios.
We can rewrite the given angle as a positive angle.
Rewriting the term of the expression, we get
$\cos \left( { - 210^\circ } \right) = \cos \left( { - 360^\circ + 150^\circ } \right)$
The cosine of an angle $- 360^\circ + x$, is equal to the cosine of angle $x$.
Therefore, we get
$\Rightarrow \cos \left( { - 210^\circ } \right) = \cos \left( {150^\circ } \right)$
We can rewrite the angle as the sum or difference of a multiple of $90^\circ$ or $180^\circ$, and an acute angle.
Rewriting the term of the expression, we get
$\Rightarrow \cos \left( {150^\circ } \right) = \cos \left( {180^\circ - 30^\circ } \right)$
The cosine of an angle $180^\circ - x$, is equal to the negative of the cosine of angle $x$, where $x$ is an acute angle.
Therefore, we get
$\Rightarrow \cos \left( {150^\circ } \right) = \cos \left( {180^\circ - 30^\circ } \right) = - \cos 30^\circ$
The cosine of an angle measuring $30^\circ$ is equal to $\dfrac{{\sqrt 3 }}{2}$.
Substituting $\cos 30^\circ = \dfrac{{\sqrt 3 }}{2}$ in the equation $\cos \left( {150^\circ } \right) = - \cos 30^\circ$, we get
$\Rightarrow \cos \left( {150^\circ } \right) = - \dfrac{{\sqrt 3 }}{2}$
Therefore, we get
$\Rightarrow \cos \left( { - 210^\circ } \right) = \cos \left( {150^\circ } \right) = - \dfrac{{\sqrt 3 }}{2}$
Therefore, we get the value of the expression $\cos \left( { - 210^\circ } \right)$ as $- \dfrac{{\sqrt 3 }}{2}$.

Note: A common mistake is to convert $\cos \left( {150^\circ } \right) = \cos \left( {180^\circ - 30^\circ } \right)$ to $\sin 30^\circ$. This is incorrect because $180^\circ$ is an even multiple of $90^\circ$. If we rewrite $\cos 150^\circ$ as $\cos \left( {90^\circ + 60^\circ } \right)$, then only it will become $\sin 60^\circ$, which is equal to $\dfrac{{\sqrt 3 }}{2}$. Here, cosine gets converted to sine because $90^\circ$ is an odd multiple of $90^\circ$.
The cosine of any negative angle $- x$ is equal to the cosine of the positive angle $x$.
Therefore, we can write the given expression as
$\cos \left( { - 210^\circ } \right) = \cos 210^\circ$
Writing 210 as sum of 180 and 30, we get
$\cos 210^\circ = \cos \left( {180^\circ + 30^\circ } \right)$
The cosine of an angle $180^\circ + x$, is equal to the negative of the cosine of angle $x$, where $x$ is an acute angle.
Thus, we get
$\cos 210^\circ = - \cos 30^\circ = - \dfrac{{\sqrt 3 }}{2}$