Question

How do you find the x intercept of $y = {\sec ^4}\left( {\dfrac{{\pi x}}{8}} \right) - 4$ ?

Answer 1

Hint : We are given with a function of trigonometry such that the x intercept of the function is to be found. We will put the y value equals to zero. Then we will get the function in the form of ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ .
Then we will write the function in two separate brackets equating to zero. On proceeding we will find the value for which the function has x intercepts.

Complete step by step solution:
Given that,
$y = {\sec ^4}\left( {\dfrac{{\pi x}}{8}} \right) - 4$
Putting y equals to zero because we have to find the x intercept,
${\sec ^4}\left( {\dfrac{{\pi x}}{8}} \right) - 4 = 0$
Now this is of the form,
${\left( {{{\sec }^2}\left( {\dfrac{{\pi x}}{8}} \right)} \right)^2} - {2^2} = 0$
This is same as ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$
Now on expanding the function so given
$\left( {{{\sec }^2}\left( {\dfrac{{\pi x}}{8}} \right) - 2} \right)\left( {{{\sec }^2}\left( {\dfrac{{\pi x}}{8}} \right) + 2} \right) = 0$
Now we will equate them one by one to zero,
$\left( {{{\sec }^2}\left( {\dfrac{{\pi x}}{8}} \right) + 2} \right) = 0$
Taking 2 on LHS we get,
$\left( {{{\sec }^2}\left( {\dfrac{{\pi x}}{8}} \right)} \right) = - 2$
This is not the acceptable case.
So let’s put the second bracket to zero.
$\left( {{{\sec }^2}\left( {\dfrac{{\pi x}}{8}} \right) - 2} \right) = 0$
Taking 2 on LHS we get,
$\left( {{{\sec }^2}\left( {\dfrac{{\pi x}}{8}} \right)} \right) = 2$
Taking the square root on both sides,
$\sec \left( {\dfrac{{\pi x}}{8}} \right) = \pm \sqrt 2$
We know that $\sec \dfrac{\pi }{4} = \sqrt 2$
Whereas sec function is positive in the first and fourth quadrant. and it is negative in the second and third quadrant. So the value of $\dfrac{{\pi x}}{8} = \dfrac{\pi }{4},\dfrac{{3\pi }}{4},\dfrac{{5\pi }}{4}....$
In order to find the value of x we can write from above angles as ,
$x = 2,6,10....$
Now we need to form an equation that defines the value of x for n different values. So after observing the values we can write or express it as,
$x = 4n - 2,$ because for the values of $n = 1,2,3...$ we get the values of x intercepts mentioned above.
This is the complete answer.
So, the correct answer is “ $x = 4n - 2,$ for $n = 1,2,3...$ ”.

Note : Here note that the x intercept of the given function is the value of the angle so obtained for different values of the x. also note that the root so obtained is either positive or negative depending on the value of x. The value that is not acceptable because the value on LHS does not exist for any value of sec.