Question

If \[\sin 2\theta = \cos \theta ,0 < \theta < \pi \], then the possible values of\[\theta \] are
A. \[{90^\circ },{60^\circ },{30^\circ }\]
B. \[{90^\circ },{150^\circ },{60^\circ }\]
C. \[{90^\circ },{45^\circ },{150^\circ }\]
D. \[{90^\circ },{30^\circ },{150^\circ }\]

Answer 1

Hints
Right-angle triangle lengths and angles are measured using trigonometry values of various ratios, including sine, cosine, tangent, secant, cotangent, and cosecant. When solving trigonometry problems, the values of the trigonometric functions for \[0^\circ ,{\rm{ }}30^\circ ,{\rm{ }}45^\circ ,{\rm{ }}60^\circ \], and \[90^\circ \] are frequently employed. Sine, Cosine, and Tangent are the three primary trigonometric ratios on which trigonometry values are based.
Formula used:
\[\cos \theta - \cos \left( {\frac{\pi }{2} - 2\theta } \right)\]
Complete step-by-step solution
We are given the equation \[\sin 2\theta = \cos \theta \] in the problem.
Therefore, we must discover broad answers or values of \[\theta \]
We know that the given equation in question is, \[\sin 2\theta = \cos \theta \]
Now solve for\[\cos \theta \]:
\[\cos \theta = \cos \left( {\frac{\pi }{2} - 2\theta } \right)\]
Now determine the value for\[\theta \]to find the general solution of\[\theta \]:
\[ \Rightarrow \theta = 2n\pi \pm \left( {\frac{\pi }{2} - 2\theta } \right)\]
The general solution for the given equation can be obtained as
\[ \Rightarrow \theta \pm 2\theta = 2n\pi \pm \frac{\pi }{2}\]
Now solve the above equation to make it less complicated:
\[3\theta = 2n\pi + \frac{\pi }{2}\]
In order to calculate the value for \[\theta \], now divide either sides of the equation by \[3\]:
\[\theta = \frac{1}{3}\left( {2n\pi + \frac{\pi }{2}} \right)\]
Simplify further by putting minus on both sides of the obtained equation:
\[ - \theta = 2n\pi - \frac{\pi }{2}\]
Now, on solving the equation we obtain the value of theta as
\[ \Rightarrow \theta = - \left( {2n\pi - \frac{\pi }{2}} \right)\]
Thus, the value of \[\theta \] lies between \[0\] and \[\pi \] are,
\[\frac{\pi }{6},\frac{\pi }{2},\frac{{5\pi }}{6}\]
In terms of degree, it can be written as
That is,\[{30^\circ },{90^\circ },{150^\circ }\]
Therefore, the possible values of \[\theta \] are \[{30^\circ },{90^\circ },{150^\circ }\]
Hence, the option D is correct.

Note
Students often forgot the trigonometry values. Simply learning the sin values first is a simple approach to retain the values. Remember the sin values from \[0^\circ \]to\[90^\circ \]; the cos values are simply the reverse of sin, i.e., the values of cos are from \[\sin 90^\circ \]to\[0^\circ \].