Question

If $\sec 5A = {\text{cosec}}\left( {A - 30} \right)$, where $\angle A$ is acute angle, then $\angle A$=
A.35$^ \circ $
B.25$^ \circ $
C.20$^ \circ $
D.27$^ \circ $

Answer 1

Hint: We will use the trigonometric identity $\sec \theta = {\text{cosec}}\left( {{{90}^ \circ } - \theta } \right)$ to solve the question and to simplify the given equation for the value of $\angle A$ by equating it with $\sec 5A = {\text{cosec}}\left( {A - 30} \right)$.

Complete step-by-step answer:
We are given $\sec 5A = {\text{cosec}}\left( {A - 30} \right)$, where $\angle A$ is an acute angle.
Now, we have a trigonometric identity $\sec \theta = {\text{cosec}}\left( {{{90}^ \circ } - \theta } \right)$.
Using it, we can write $\sec 5A$as
$ \Rightarrow \sec 5A = {\text{cosec}}\left( {{{90}^ \circ } - 5A} \right)$
Substituting this value of $\sec 5A$ in the given equation, we get
$ \Rightarrow {\text{cosec}}\left( {{{90}^ \circ } - 5A} \right) = {\text{cosec}}\left( {A - {{30}^ \circ }} \right)$
$ \Rightarrow {90^ \circ } - 5A = A - {30^ \circ }$
$ \Rightarrow 6A = {120^ \circ }$
$ \Rightarrow A = \dfrac{{{{120}^ \circ }}}{6} = {20^ \circ }$
Therefore, $\angle A = {20^ \circ }$ , hence, option (C) is correct.

Additional Information: In mathematics, trigonometry is such a branch which is concerned with some defined and specific functions of angles and their applications and trigonometric identities are defined as equalities in trigonometry, involving trigonometric functions, which holds true for every value of the variables used and both of the sides of the equalities are defined.
The trigonometric identities are equations which are true for right angled triangles for each value of the variables. The term ratio can also be applied to trigonometric identities, since all trigonometric functions can be defined as ratios.

Note: In this question, you may go wrong while selecting the trigonometric identity to be applied in this question in order to solve it for the value of acute angle. You can also solve this question by converting secant and cosecant of the given angles in terms of cosine and sine of the angles respectively and then using the standard trigonometric identities for calculating the value of acute angle A.