Solution:

Given,

$2 sin² \theta - cos \theta = 1$

To find:

 $\theta = ?$

Now,

$2 sin² \theta - cos \theta = 1$

or, $2 sin² \theta = 1 +cos \theta$

or, $sin² \theta = \dfrac{1 +cos \theta}{2}$

or, $sin² \theta = cos²\frac{\theta}{2}$

or, $sin \theta = cos\frac{ \theta}{2}$

Either,


$sin \theta = cos\frac{ \theta}{2}$

or, $2 sin\frac{\theta}{2}cos\frac{ \theta}{2} = cos\frac{ \theta}{2} $

or, $2 sin\frac{\theta}{2} = 1$

or, $sin\frac{\theta}{2} = \dfrac{1}{2}$ --- (i)

or, $sin\frac{\theta}{2} = sin30°$

or, $\frac{\theta}{2} = 30°$

$\therefore, \theta = 60°$


Or,

$sin \theta = cos\frac{ \theta}{2}$

or, $sin \theta = sin(90° + \frac{ \theta}{2})$

or, $\theta = \frac{180°+\theta}{2}$

or, $2\theta = 180° +\theta$

$\therefore, \theta = 180°$


Or,

Solving (i)

$sin\frac{\theta}{2} = \dfrac{1}{2}$

or, $sin\frac{\theta}{2} = sin150°$

or, $\frac{\theta}{2} = 150°$

$\therefore, \theta = 300°$


So,


The value of $\theta$ in the given trigonometric equations could be 60°, 180° or 300°.



Question: Solve the following trigonometric equation: 2 sin² 'theta' - cos 'theta' = 1 | SciPiPupil


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