Question

Express the trigonometric ratios $ \sin {\rm{ A, sec A, tan A}} $ in terms of $ \cot {\rm{ A}} $ .

Answer 1

Hint: Remember some basic formulae to solve this question easily. Keep in mind $ \sin {\rm{ A = }}\dfrac{1}{{\cos ec{\rm{ A}}}} $ , $ \tan {\rm{ A = }}\dfrac{{\sin {\rm{ A}}}}{{\cos {\rm{ A}}}} $ and $ \cot {\rm{ A = }}\dfrac{{\cos {\rm{ A}}}}{{\sin {\rm{ A}}}} $
If there is an equation of the form $ {x^2} = {a^2} $ , then after taking square roots on both sides, we get two values of $ x $ , they are $ + a $ and \[ - a\].

Complete step-by-step answer:
We know that $ \tan {\rm{ A = }}\dfrac{{\sin {\rm{ A}}}}{{\cos {\rm{ A}}}}......\left( 1 \right) $
 $ \cot {\rm{ A = }}\dfrac{{\cos {\rm{ A}}}}{{\sin {\rm{ A}}}}......\left( 2 \right) $
From equations (1) and (2), we can conclude that $ \tan {\rm{ A = }}\dfrac{1}{{{\rm{cot A}}}} $
We know that $ 1 + {\cot ^2}A = {\rm{cose}}{{\rm{c}}^2}A $
On rearranging the terms, we get $ {\rm{cose}}{{\rm{c}}^2}A = 1 + {\cot ^2}A $
On taking square roots on both sides of the equation we get,
 $\Rightarrow {\rm{cosec A = }} \pm \sqrt {1 + {{\cot }^2}A} ......\left( 3 \right) $
We know that, $ \sin {\rm{ A = }}\dfrac{1}{{\cos ec{\rm{ A}}}} $
Substitute the value of $ {\rm{cosec A = }} \pm \sqrt {1 + {{\cot }^2}A} $ taken from equation (3) in the equation $ \sin {\rm{ A = }}\dfrac{1}{{\cos ec{\rm{ A}}}} $ to find $ \sin A $ in terms of $ \cot A $
 $\Rightarrow {\rm{sin A = }}\dfrac{1}{{ \pm \sqrt {1 + {{\cot }^2}A} }} $
It is known that $ {\sec ^2}A = 1 + {\tan ^2}A $
Taking square roots on both sides of the equation, we get, $ \sec A = \pm \sqrt {1 + {{\tan }^2}A} $
We know that, $ \tan {\rm{ A = }}\dfrac{1}{{\cot {\rm{ A}}}} $
Substitute $ \tan {\rm{ A = }}\dfrac{1}{{\cot {\rm{ A}}}} $ in the equation $ \sec A = \pm \sqrt {1 + {{\tan }^2}A} $ to find $ \sec A $ in terms of $ \cot A $
 $ \sec A = \pm \sqrt {1 + {{\left( {\dfrac{1}{{\cot {\rm{ A}}}}} \right)}^2}} $

Additional information:
Trigonometric ratios can be defined only in a right-angled triangle.
As per the basic definition of sin of an angle, it states that $ \sin {\rm{ A}} $ is the ratio of the opposite side of angle $ A $ and the longest side i.e. hypotenuse of the right-angled triangle.
As per the basic definition of cos of an angle, it states that $ \cos {\rm{ A}} $ is the ratio of the adjacent side of angle $ A $ and the longest side i.e. hypotenuse of the right-angled triangle.
The inverse of the $ \sin {\rm{ A}} $ is $ \cos ec{\rm{ A}} $ and the inverse of $ \cos {\rm{ A}} $ is $ {\rm{sec A}} $ .
As per the basic definition of tan of an angle, it states that $ \tan {\rm{ A}} $ is the ratio of the opposite side of angle $ A $ to the adjacent side of the angle $ A $
As per the basic definition of tan of an angle, it states that $ \tan {\rm{ A}} $ is the ratio of the adjacent side of angle $ A $ to the opposite side of the angle $ A $

Note: In this type of question, students make mistakes. They need to take note that $ {\cot ^2}A $ cannot be equal to $ 1 + \cos e{c^2}A $ . Also, $ {\tan ^2}A $ cannot be equal to $ 1 + {\sec ^2}A $ .
In addition to this, they need to make sure that the basic definitions are utilized properly while solving this type of question.