Question

How do you evaluate \[\sec \left( {150^\circ } \right)\]?

Answer 1

Hint: Here, we will measure the given angle as a difference of two angles to convert it into acute angles using trigonometric identities. We will then simplify the expression and substitute the value of the secant of the obtained angle to find the required value.

Complete step-by-step solution:
First, we will simplify the given trigonometric ratio.
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
\[\sec \left( {150^\circ } \right) = \sec \left( {180^\circ - 30^\circ } \right)\]
The secant of an angle \[180^\circ - x\], is equal to the negative of the secant of angle \[x\], where \[x\] is an acute angle.
Therefore, we get
\[\sec \left( {150^\circ } \right) = \sec \left( {180^\circ - 30^\circ } \right) = - \sec 30^\circ \]
The secant of an angle measuring \[30^\circ \] is equal to \[\dfrac{2}{{\sqrt 3 }}\].
Substituting \[\sec 30^\circ = \dfrac{2}{{\sqrt 3 }}\] in the equation \[\sec \left( {150^\circ } \right) = - \sec 30^\circ \], we get
\[ \Rightarrow \sec \left( {150^\circ } \right) = - \dfrac{2}{{\sqrt 3 }}\]

Therefore, we get the value of the expression \[\sec 150^\circ \] as \[ - \dfrac{2}{{\sqrt 3 }}\].

Additional information:
Here, we can make a mistake if we convert \[\sec \left( {150^\circ } \right) = \sec \left( {180^\circ - 30^\circ } \right)\] to \[{\rm{cosec}}30^\circ \]. This is incorrect because \[180^\circ \] is an even multiple of \[90^\circ \]. If we rewrite \[\sec \left( {150^\circ } \right)\] as \[\sec \left( {90^\circ + 60^\circ } \right)\], then only it will become \[{\rm{cosec}}60^\circ \], which is equal to \[\dfrac{2}{{\sqrt 3 }}\]. Here, secant gets converted to cosecant because \[90^\circ \] is an odd multiple of \[90^\circ \].

Note:
We can simplify the value by converting it to cosine.
The secant of an angle is equal to the reciprocal of the cosine of that angle. This can be written as \[\sec x = \dfrac{1}{{\cos x}}\].
Therefore, we can write the given expression as
\[\sec 150^\circ = \dfrac{1}{{\cos 150^\circ }}\]
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
\[\cos \left( {150^\circ } \right) = \cos \left( {180^\circ - 30^\circ } \right) = - \cos 30^\circ \]
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
\[ \Rightarrow \cos \left( { - 210^\circ } \right) = \cos \left( {150^\circ } \right) = - \dfrac{{\sqrt 3 }}{2}\]
Substituting \[\cos \left( {150^\circ } \right) = - \dfrac{{\sqrt 3 }}{2}\] in the equation \[\sec 150^\circ = \dfrac{1}{{\cos 150^\circ }}\], we get
\[\begin{array}{l} \Rightarrow \sec 150^\circ = \dfrac{1}{{ - \dfrac{{\sqrt 3 }}{2}}}\\ \Rightarrow \sec 150^\circ = - \dfrac{2}{{\sqrt 3 }}\end{array}\]
Therefore, we get the value of the expression \[\sec 150^\circ \] as \[ - \dfrac{2}{{\sqrt 3 }}\].