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.