Question

Find the value of \[\sec {210^o}\].
A. \[\dfrac{2}{{\sqrt 3 }}\]
B. \[ - \dfrac{1}{{\sqrt 2 }}\]
C. \[ - \dfrac{2}{{\sqrt 3 }}\]
D. \[ - 2\]

Answer 1

Hint: First we know that the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. Trigonometric ratios are Sine, Cosine, Tangent, Cotangent, Secant and Cosecant. We know that \[\sec x = \dfrac{1}{{\cos x}}\] for any angle \[x\]. Then express the given angle as a sum/difference of any two standard angles. Then using the trigonometric identities, we reduce it. Finally simplifying we get the solution.

Complete step by step answer:
Given \[\sec {210^o}\]---------(1)
Since \[\sec x = \dfrac{1}{{\cos x}}\], then the expression (1) can be written as
\[\sec {210^o} = \dfrac{1}{{\cos {{210}^o}}}\]--(2)
Since \[{210^o}\]is not a standard angle. So, we can write \[{210^o} = {180^o} + {30^o}\], then the equation (2) becomes
\[\sec {210^o} = \dfrac{1}{{\cos \left( {{{180}^o} + {{30}^o}} \right)}}\]
\[ \Rightarrow \sec {210^o} = \dfrac{1}{{\cos \left( {\pi + \dfrac{\pi }{6}} \right)}}\]--(3)
Using the trigonometric identity (8), we get
\[\sec {210^o} = - \dfrac{1}{{\cos \left( {\dfrac{\pi }{6}} \right)}}\]
\[ \Rightarrow \sec {210^o} = - \dfrac{1}{{\left( {\dfrac{{\sqrt 3 }}{2}} \right)}}\]
\[ \therefore \] \[\sec {210^o} = - \dfrac{2}{{\sqrt 3 }}\].
Hence the exact value of \[\cos {30^o} = - \dfrac{2}{{\sqrt 3 }} = - \dfrac{{2\sqrt 3 }}{3}\].

Hence the correct option is C.

Additional information: The first trigonometric table was apparently compiled by Hipparchus known as "the father of trigonometry". Trigonometry used in oceanography in calculating the height of tides in oceans. Trigonometry can also be used to roof a house, to make the roof inclined and the height of the roof in buildings etc. It is used in the naval and aviation industries.

Note: The standard angles of trigonometric ratios are \[{0^0}\], \[{30^0}\], \[{45^0}\], \[{60^0}\] and \[{90^0}\]. In this question learners have to note that we have to find the exact value of \[\sec {210^o}\], so after obtaining the value of \[\sec {210^o}\] multiplying numerator and denominator by \[\sqrt 3 \]is a must. Also note that the learners have to take care about the value of \[\cos {30^o}\]if it is not known then it is difficult to solve this question.