Question

Solve the following equation \[\sec \theta \cos 5\theta +1=0\], \[0\le \theta \le \dfrac{\pi }{2}\].

Answer 1

Hint: As we know that secant of any angle is the reciprocal of cosine of that angle. So we will substitute the value of secant in terms of cosine in the given equation.
Also, we will use the trigonometry identity to solve it further which is given as follows:
\[\operatorname{cosA}+\operatorname{cosB}=2cos\dfrac{A+B}{2}\cos \dfrac{A-B}{2}\].

Complete Step-by-step answer:
We have been given the equation \[\sec \theta \cos 5\theta +1=0\].
As we know that \[\sec \theta =\dfrac{1}{\cos \theta }\].
So by substituting the value of \[\sec \theta =\dfrac{1}{\cos \theta }\] in the above equation, we get as follows:
\[\dfrac{1}{\cos \theta }.\cos 5\theta +1=0\]
Taking \[\cos \theta \] as LCM, we get as follows:
\[\dfrac{\cos 5\theta +\cos \theta }{\cos \theta }=0\]
As we know that \[\operatorname{cosA}+cosB=2cos\dfrac{A+B}{2}\cos \dfrac{A-B}{2}\].
So, by using the formula in the above equation, we get as follows:
\[\begin{align}
  & \dfrac{2\cos \dfrac{5\theta +\theta }{2}.\cos \dfrac{5\theta -\theta }{2}}{\cos \theta }=0 \\
 & \dfrac{2\cos 3\theta \cos 2\theta }{\cos \theta }=0 \\
\end{align}\]
\[\Rightarrow \cos 3\theta =0\] and \[\cos 2\theta =0\].
As it is given that \[0\le \theta \le \dfrac{\pi }{2}\], we get as follows:
\[\begin{align}
  & \cos 3\theta =0 \\
 & 3\theta =\dfrac{\pi }{2},\dfrac{3\pi }{2} \\
 & \theta =\dfrac{\pi }{6},\dfrac{\pi }{2} \\
\end{align}\]
Also,
\[\begin{align}
  & \cos 2\theta =0 \\
 & 2\theta =\dfrac{\pi }{2} \\
 & \theta =\dfrac{\pi }{4} \\
\end{align}\]
Therefore, the solution for the equation are \[\theta =\dfrac{\pi }{4},\dfrac{\pi }{2},\dfrac{\pi }{6}\].

Note: We can also get the solution by drawing the graph of \[\cos 3\theta \] and \[\cos 3\theta \] for \[0\le \theta \le \dfrac{\pi }{2}\].
Also, be careful while choosing the value of ‘\[\theta \]’ from the equation \[\cos 3\theta =0\] and \[\cos 2\theta =0\] as we have been given the unit of \[\theta \], which is \[0\le \theta \le \dfrac{\pi }{2}\].
Also, that \[\cos \theta \ne 0\] as it makes the equation \[\infty =0\].
Take care of the sign while substituting the value of \[\cos 5\theta +\cos \theta \] by using the trigonometric identity.