Problem 157 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \(\tan x=\frac{\pi}{2}\) has no solution.

Step-by-Step Solution

Answer

False. The equation \(\tan x = \frac{\pi}{2}\) indeed has a solution. The fact that \(\frac{\pi}{2}\) is a real number should make us believe that there must be a corresponding value of 'x' where the tangent function equals \(\frac{\pi}{2}\). You can find it by taking the inverse tangent of \(\frac{\pi}{2}\).

Step 1: Understanding the Tangent Function

The tangent function, \(\tan x\), is defined as the ratio of the sine to the cosine of an angle. The function is periodic with a period of \(\pi\). It repeats its values every \(\pi\) units. It is undefined at odd multiples of \(\pi/2\). That's because these are the angles for which cos(x) = 0 and \(\tan x = \sin x / \cos x\)

Step 2: Reasoning the Statement

The statement to verify is whether or not the equation \(\tan x = \frac{\pi}{2}\) has a solution. \(\frac{\pi}{2}\) is a real number, which means that it is within the range of the tangent function. Therefore, there should be a real number solution for this equation where the tangent function equals \(\frac{\pi}{2}\).

Step 3: Finding the Solution

To find the exact solution for x, you need to use the inverse tangent function, \(\tan^{-1}\). So, apply inverse tangent to both sides of the equation: \(x = \tan^{-1}\left(\frac{\pi}{2}\right)\). This gives 'x' the value where \(\tan x = \frac{\pi}{2}\). Therefore, the statement 'The equation \(\tan x = \frac{\pi}{2}\) has no solution' is false.