Question

What is the exact value of $\sec 210$?

Answer 1

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.