Question

The number of values of \[\theta \] in satisfying the equation \[2{\sin ^2}\theta = 4 + 3\]\[\cos \theta \] are
A. \[0\]
B. \[1\]
C. \[2\]
D. \[3\]

Answer 1

Hint
In this case, we have been given the equation that \[2{\sin ^2}\theta = 4 + 3\]\[\cos \theta \] _{where we have to find the number of values of \theta satisfying the given equation. For that we have to restructure the equation according to the formulas to make it less complicated to solve. Then, we have to use trigonometry's trivial identity to solve it further to obtain the desired solution.}
Formula used:
\[{\sin ^2}\theta = 1 - {\cos ^2}\theta \;\]
Complete step-by-step solution
The given equation is
\[2{\sin ^2}\theta = 4 + 3\cos \theta \]
This can also be written as,
\[2 - 2{\cos ^2}\theta = 4 + 3\cos \theta \]
By solving the equation, it becomes
\[ = > 2{\cos ^2}\theta + 3\cos \theta + 2\]
Which eventually becomes,
\[ = > 0\]
With the trigonometric formulae, the equation can be written as
\[\cos \theta = \frac{{ - 3 \pm \sqrt {9 - 16} }}{4}\]
 This is an imaginary equation, hence there is no solution (A).

Note
When the value beneath the radical component of the quadratic formula is negative, imaginary numbers (and complex roots) appear in relation to quadratic equations. In this case, the set of real numbers contains no roots (or zeros) for the equation. Complex conjugate pairs contain the complex roots of a quadratic equation with real coefficients.
Imaginary numbers are those that produce a negative square root when multiplied by themselves. In other terms, imaginary numbers are those that have no known value and are represented by the square root of negative numbers. They can also range from one real number double solution where the parabola only hits the x-axis once.