Question

Express it for $\tan x$ if we have given $a\cos x + b\sin x = c$ ?

Answer 1

Hint: We know that the tan function is the ratio of sin and cos function. So, to get the term of tan we will divide the whole equation by cos. we will take our tan term to one side to express the equation for tan. We can add or subtract some constant on both sides to make of tan term constant free.

Complete step by step answer:
We have given $a\cos x + b\sin x = c$ , we have to convert it to $\tan x$ .
We know that $\tan x$ is the ratio of sin and cos function.
So, to convert it for tan we will divide the whole equation by cos because cos will eliminate from $a\cos x$ term and $b\sin x$ will change to $b\tan x$
We divided the whole equation by cos, we get
$ \Rightarrow a + b\tan x = \dfrac{c}{{\cos x}}$
We know that $\sec x = \dfrac{1}{{\cos x}}$
So, R.H.S can be written in terms of sec as
$ \Rightarrow a + b\tan x = c\sec x$
We have subtracted a from both side
$ \Rightarrow b\tan x = c\sec x - a$
We will divide the whole equation by b, we get
$ \Rightarrow \tan x = \dfrac{{c\sec x - a}}{b}$
Hence, the equation $a\cos x + b\sin x = c$ when converted for tan is $\tan x = \dfrac{{c\sec x - a}}{b}$

Note:
 We can also solve this question by dividing the whole equation by sin and changing the $a\cos x$ in terms of cot then converting it in tan by reciprocating it. But this process will become lengthy and time consuming. So, it is very important to approach the right and easy method for solving these types of questions.