Question

How do you solve: $ {\csc ^4}x - 4{\csc ^2}x = 0? $

Answer 1

Hint: Here we will simplify the given expression by taking the common factors and equate to zero to get the values of the angles. Also use the inverse trigonometric functions and trigonometric value table for the reference values in the measure of “pi” terms.

Complete step by step solution:
Take the given expression: $ {\csc ^4}x - 4{\csc ^2}x = 0 $
Take the common factor common from the above expression –
 $ {\csc ^2}x({\csc ^2}x - 4) = 0 $
From the above expression, here we have two cases –
 $ {\csc ^2}x = 0 $ and $ {\csc ^2}x - 4 = 0 $
 $ {\csc ^2}x = 0 $
Take square root on both the sides of the equation –
 $ \csc x = 0 $
Cosec is the inverse of the sine.
 $ \dfrac{1}{{\sin x}} = 0 $
Cross multiply the above expression; the denominator is multiplied with the numerator of the opposite side.
 $ \dfrac{1}{0} = \sin x $
The above expression can be re-written as:
 $ \sin x = \infty $
The above value does not exist.
 $ {\csc ^2}x - 4 = 0 $
Make the cosecant the subject and move other terms on the opposite side of the equation. Sign of the terms also changes when moved to the opposite side. Negative terms change to the opposite side and vice-versa.
 $ {\csc ^2}x = 4 $
Take square root on both the sides of the equation –
\[\sqrt {{{\csc }^2}x} = \sqrt {{2^2}} \]
Square and square root cancel each other.
\[\csc x = \pm 2\]
Cosecant is the inverse of the sine function.
 $ \sin x = \pm \dfrac{1}{2} $
Here we have two cases –
 $ x = {\sin ^{ - 1}}\left( {\dfrac{1}{2}} \right) $ and $ x = {\sin ^{ - 1}}\left( { - \dfrac{1}{2}} \right) $
By referring the trigonometric table –
 $ x = \dfrac{\pi }{6} $ or $ x = - \dfrac{\pi }{6} $
By referring All STC rule, primary values are repeated at the integer multiples of $ \pi $
 $ x = \dfrac{\pi }{6} + n\pi $ and $ x = - \dfrac{\pi }{6} + n\pi ,n \in Z $

Note: Go through the trigonometric table having different angles of measures and remember the correlation between the six trigonometric functions. Remember the All STC rule, which is also recognized as ASTC rule in the geometry which states that all the trigonometric ratios in the first quadrant ( $ 0^\circ \;{\text{to 90}}^\circ $ ) are positive, sine and cosec are positive in the second quadrant ( $ 90^\circ {\text{ to 180}}^\circ $ ), tangent and cot ant are positive in the third quadrant ( $ 180^\circ \;{\text{to 270}}^\circ $ ) and sine and cosec are positive in the fourth quadrant ( $ 270^\circ {\text{ to 360}}^\circ $ ).