Question

How do you find the exact values of cot, csc and sec for $ 30 $ degrees?

Answer 1

Hint: While solving this particular question we must convert the given trigonometric function into its corresponding simpler form that are $ \cot x = \dfrac{1}{{\tan x}} $ , $ \csc x = \dfrac{1}{{\sin x}} $ and $ \sec x = \dfrac{1}{{\cos x}} $ . Then we have to replace the value of x with $ 30 $ degrees .

Complete step by step solution:
We have to find the exact values of cot, csc and sec for $ 30 $ degrees ,
Let us find the exact value of cot first ,
We already know the relationship between tangent and cotangent that is ,
 $ \cot x = \dfrac{1}{{\tan x}} $
Therefore, we can write the given expression as ,
 $\Rightarrow \cot {30^ \circ } = \dfrac{1}{{\tan {{30}^ \circ }}} $
 $ {30^ \circ } $ is a special angle and we know that the tangent of $ {30^ \circ } $ is $ \dfrac{1}{{\sqrt 3 }} $ ,
Therefore, we will get the required result ,
 $\Rightarrow \cot {30^ \circ } = \dfrac{1}{{\dfrac{1}{{\sqrt 3 }}}} = \sqrt 3 $
Now, find the exact value of csc,
We already know the relationship between sine and cosecant that is ,
 $ \csc x = \dfrac{1}{{\sin x}} $
Therefore, we can write the given expression as ,
\[\csc {30^ \circ } = \dfrac{1}{{\sin {{30}^ \circ }}}\]
 $ {30^ \circ } $ is a special angle and we know that the sine of $ {30^ \circ } $ is $ \dfrac{1}{2} $ ,
Therefore, we will get the required result ,
 $\Rightarrow \csc {30^ \circ } = \dfrac{1}{2} = 2 $
Now, find the exact value of sec,
We already know the relationship between cosine and secant that is ,
 $ \sec x = \dfrac{1}{{\cos x}} $
Therefore, we can write the given expression as ,
\[\Rightarrow \sec {30^ \circ } = \dfrac{1}{{\cos {{30}^ \circ }}}\]
 $ {30^ \circ } $ is a special angle and we know that the cosine of $ {30^ \circ } $ is $ \dfrac{{\sqrt 3 }}{2} $ ,
Therefore, we will get the required result ,
 $\Rightarrow \sec {30^ \circ } = \dfrac{1}{{\dfrac{{\sqrt 3 }}{2}}} = \dfrac{2}{{\sqrt 3 }} $

Note: In order to solve and simplify the given expression we have to use the identities and express our given expression in the simplest form and thereby solve it. Questions similar in nature as that of above can be approached in a similar manner and we can solve it easily.