Question

How do you solve $ {\sec ^2}x = 4 $ in the interval $ 0 \leqslant x \leqslant 2\pi $?

Answer 1

Hint: In the given question, we are required to find all the possible values of x that satisfy the given trigonometric equation $ {\sec ^2}x = 4 $ lying in the range $ 0 \leqslant x \leqslant 2\pi $ . For solving such types of questions where we have to solve trigonometric equations, we need to have basic knowledge of algebraic rules and identities as well as a strong grip on trigonometric formulae and identities.

Complete step by step solution:
We have to solve the given trigonometric equation $ {\sec ^2}x = 4 $ . We know that $ \sec \left( \theta \right) = \dfrac{1}{{\cos \left( \theta \right)}} $ .
So, converting $ \sec x $ into $ \cos x $ using the basic trigonometric formula, we get,
 $ \Rightarrow \dfrac{1}{{{{\cos }^2}x}} = 4 $
Now, Shifting the $ \cos x $ to right hand side of the equation and isolating the trigonometric ratio, we get,
 $ \Rightarrow \dfrac{1}{4} = {\cos ^2}x $
Taking square root both sides of the equation, we get,
 $ \Rightarrow \cos x = \pm \sqrt {\dfrac{1}{4}} $
 $ \Rightarrow \cos x = \pm \left( {\dfrac{1}{2}} \right) $
So, we have to find the values of x that satisfy either $ \cos x = \left( {\dfrac{1}{2}} \right) $ or $ \cos x = - \left( {\dfrac{1}{2}} \right) $ lying in the range $ 0 \leqslant x \leqslant 2\pi $.
We know that value of $ \cos \left( {\dfrac{\pi }{3}} \right) = \left( {\dfrac{1}{2}} \right) $ and $ \cos \left( {\dfrac{{5\pi }}{3}} \right) = \left( {\dfrac{1}{2}} \right) $ .
Also, value of $ \cos \left( {\dfrac{{2\pi }}{3}} \right) = - \left( {\dfrac{1}{2}} \right) $ and $ \cos \left( {\dfrac{{4\pi }}{3}} \right) = - \left( {\dfrac{1}{2}} \right) $ .
Hence, the solutions for the trigonometric equation $ {\sec ^2}x = 4 $ lying in the range $ 0 \leqslant x \leqslant 2\pi $ are: $ \dfrac{\pi }{3},\dfrac{{2\pi }}{3},\dfrac{{4\pi }}{3},\dfrac{{5\pi }}{3} $ .

Note: Such trigonometric equations can be solved by various methods by applying suitable trigonometric identities and formulae. The general solution of a given trigonometric solution may differ in form, but actually represents the correct solutions. The different forms of general equations are interconvertible into each other.