Question

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

Answer 1

Hint: 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
\[\cos 210^\circ = - \cos 30^\circ = - \dfrac{{\sqrt 3 }}{2}\]