Question

Solve \[{\tan ^2}3\theta = 3\]

Answer 1

Hint: Here we are asked to solve the given equation which contains trigonometric functions. Using the basic definition of trigonometric conversions to convert to the simplest forms possible. Preferably convert them into terms of single or double angles so that the computation is feasible for us to handle. After these conversions we can be able to simplify the equation easily. Keep in mind that this is not a polynomial equation to solve for any unknown variable; it contains a trigonometric function thus we have to solve it for the value of theta.

Complete answer:
It is given that \[{\tan ^2}3\theta = 3\] We aim to solve this equation for the value of theta. Let us first expand the given function using trigonometric identities.
We know \[{\tan ^2}\theta = {\sec ^2}\theta - 1\]
So,
\[{\sec ^2}3\theta - 1 - 3 = 0\]
\[\Rightarrow {\sec ^2}3\theta = 4\]

Also,
\[{\sec ^2}\theta = \dfrac{1}{{{{\cos }^2}\theta }}\]
Then,
\[\dfrac{1}{{{{\cos }^2}3\theta }} = 4\]
\[\Rightarrow {\cos ^2}3\theta = \dfrac{1}{4}\]
\[\Rightarrow {\cos ^2}3\theta = \dfrac{1}{{2.2}}\]
\[\Rightarrow 2{\cos ^2}3\theta = \dfrac{1}{2}\]
From known properties, \[2{\cos ^2}\theta = 1 + \cos 2\theta \]
Then,
\[1 + \cos 6\theta = \dfrac{1}{2}\]
\[\Rightarrow \cos 6\theta = - \dfrac{1}{2}\]
Using the angle values from the table we get,

\[\cos {0^ \circ }\]	\[\cos {30^ \circ }\]	\[\cos {45^ \circ }\]	\[\cos {60^ \circ }\]	\[\cos {90^ \circ }\]
\[1\]	\[\dfrac{{\sqrt 3 }}{2}\]	\[\dfrac{1}{{\sqrt 2 }}\]	\[\dfrac{1}{2}\]	\[0\]

So,
\[ \Rightarrow \cos 6\theta = - \cos \dfrac{\pi }{3}\]
Now again we write it as,
\[\Rightarrow \cos 6\theta = \cos (\pi - \dfrac{\pi }{3})\]
\[\Rightarrow \cos 6\theta = \cos (\dfrac{{2\pi }}{3})\]
Now we know that if \[\cos x = \cos y\] then \[x = 2n\pi \pm y\] where\[n = 0, \pm 1, \pm 2\]
We get,
\[6\theta = 2n\pi \pm \dfrac{{2\pi }}{3}\]
\[\theta = \dfrac{{n\pi }}{3} \pm \dfrac{\pi }{9}\]
Therefore, \[\theta = \dfrac{\pi }{3}(n \pm \dfrac{1}{3})\] where \[n = 0, \pm 1, \pm 2\]

Note:
To solve these types of problem we must need to know all the basic trigonometric identities because trigonometric identities are the key to solve this problem. It plays a major role in simplifying or breaking the given equation into a simple form which makes the solving easier. We must give more attention to using the correct trigonometric identity because some identities may make the simpler form into a complex form. Another important thing that we need to know is the trigonometric ratio table here we are solving the given function for the value of theta it can only be found by using the trigonometric ratio table.