Question

Find the magnitude of angle $ A $ , if $ \tan A - 2\cos A\tan A + 2\cos A - 1 = 0 $

Answer 1

Hint: The given equation is a trigonometric equation involving tangent and cosine functions, we will simplify the given equation by factorizing it and solve the equation for the value of $ A $ . After that we will get the value for the solution of the given equation for which it is true.
Magnitude of an angle is the amount by which an angle can be rotated to find the position of that angle. In our case the magnitude of our angle $ A $ is the value of $ A $ for which $ \tan A - 2\cos A\tan A + 2\cos A - 1 = 0 $ is satisfied. So, we have to find that.

Complete step-by-step answer:
The given trigonometric equation is: $ \tan A - 2\cos A\tan A + 2\cos A - 1 = 0 $
We will do algebraic manipulations here,
 $ \tan A - 2\cos A\tan A + 2\cos A - 1 = 0 $
Taking $ \tan A $ common from the first two terms we get:
 $ \tan A(1 - 2\cos A) + 2\cos A - 1 = 0 $
Now by taking $ - 1 $ common from the remaining terms we get:
 $ \tan A(1 - 2\cos A) + ( - 1)( - 2\cos A + 1) = 0 $
Reducing the above equation we get:
 $ \tan A(1 - 2\cos A) - 1(1 - 2\cos A) = 0 $
 $ \Rightarrow (\tan A - 1)(1 - 2\cos A) = 0 $
The above equation holds good if any of the following holds,
 $ \tan A - 1 = 0 $ or $ 1 - 2\cos A = 0 $
 $ \Rightarrow \tan A = 1 $ or $ 1 = 2\cos A $
 $ \Rightarrow \tan A = 1 $ or $ \cos A = \dfrac{1}{2} $
 $ \Rightarrow A = \dfrac{\pi }{4} $ or $ A = \dfrac{\pi }{3} $
So, the equation holds only for the values $ A = \dfrac{\pi }{3},\dfrac{\pi }{4} $
Therefore, the magnitude of angle $ A $ is $ \dfrac{\pi }{3} $ or $ \dfrac{\pi }{4} $ , if $ \tan A - 2\cos A\tan A + 2\cos A - 1 = 0 $
So, the correct answer is “ $ \dfrac{\pi }{3} $ or $ \dfrac{\pi }{4} $ ”.

Note: Since, we are asked about the magnitude of the angle so we need not find the general solution. If we were asked to find the general solution we would have found it specifically. The general solution of a trigonometric function is the set of all solutions of that function in the real number system, while the principal solution is the special case of general solution where the solution lies between $ [0,2\pi ] $