Question

What is the exact value of \[\sec 210\]?

Answer 1

Hint: In this type of question we have to use the concepts of trigonometry. We know that the secant is the reciprocal of cosine so we can write this secant expression as a reciprocal of cosine. So we have given \[\sec 210\] writing it as a reciprocal of \[\cos 210\] and then substituting the value of \[\cos 210\] will give us the required solution. Also as we have to find the exact value some rules of indices are also useful here. In this question we use \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\] and expressing \[\sqrt{x}\] as \[{{x}^{\dfrac{1}{2}}}\].

Complete step by step solution:
Now, we have to find out the exact value of the trigonometric expression \[\sec 210\].
As we know that the secant is the reciprocal of cosine by writing \[\sec 210\] as a reciprocal of \[\cos 210\].
\[\Rightarrow \sec 210=\dfrac{1}{\cos 210}................\text{e}{{\text{q}}^{\text{n}}}\left( 1 \right)\]
Now, we have to find the value of \[\cos 210\]
\[\Rightarrow \cos 210=\cos \left( 180+30 \right)\]
By using the formula, \[\cos \left( 180+\theta \right)=-\cos \theta \] we can write,
\[\Rightarrow \cos 210=-\cos 30\]
Now, according to the trigonometric ratio values we know that the value of \[\cos 30\] is equal to \[\dfrac{\sqrt{3}}{2}\].
\[\Rightarrow \cos 210=-\dfrac{\sqrt{3}}{2}\]
Hence, \[\text{e}{{\text{q}}^{\text{n}}}\left( 1 \right)\] becomes,
\[\Rightarrow \sec 210=\dfrac{1}{\left( -\dfrac{\sqrt{3}}{2} \right)}\]
\[\Rightarrow \sec 210=-\dfrac{2}{\sqrt{3}}\]
Now as we have to find the exact value of \[\sec 210\] we multiply numerator as well as denominator by \[\sqrt{3}\].
\[\Rightarrow \sec 210=-\dfrac{2\times \sqrt{3}}{\sqrt{3}\times \sqrt{3}}\]
By combining and simplifying the denominator,
\[\Rightarrow \sec 210=-\dfrac{2\sqrt{3}}{{{\left( \sqrt{3} \right)}^{2}}}\]
Now as we know that \[\sqrt{3}\] can also be expressed as \[{{3}^{\dfrac{1}{2}}}\]
\[\Rightarrow \sec 210=-\dfrac{2\sqrt{3}}{{{\left( {{3}^{\dfrac{1}{2}}} \right)}^{2}}}\]
By using \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\] we can write,
\[\Rightarrow \sec 210=-\dfrac{2\sqrt{3}}{3}\]
Hence, the exact value of \[\sec 210\] is \[-\dfrac{2\sqrt{3}}{3}\].

Note: In this question students have to note that we have to find the exact value of \[\sec 210\], so after obtaining value of \[\sec 210\] from \[\cos 210\] multiplying numerator and denominator by \[\sqrt{3}\] is a must. Also students have to take care about the value of \[\cos 30\] if it is not known then it is hard to solve this question.