Problem 120 - Calculus of a Single Variable

Question

Proof Prove that for any real number \(y\) there exists \(x\) in \((-\pi / 2, \pi / 2)\) such that \(\tan x=y\) .

Step-by-Step Solution

Answer

The proof demonstrates that for any real number \(y\), there is a real number \(x\) within the interval \(-\pi/2 , \pi/2\) exclusive, such that \(\tan(x) = y\), based on the properties of the tangent function within this interval.

Step 1: Understanding the tangent function

The tangent function is defined as the ratio of the ordinate and abscissa of a point on the unit circle. This function is periodic with a period of \(\pi\). It is undefined for \(x = \pi/2 + n\pi\) where \(n\) is an integer because the denominator \( \cos x\) becomes zero, resulting in a division by zero.

Step 2: Range of the tangent function

Between \(-\pi/2\) and \(\pi/2\) exclusive, the denominator \(\cos x\) is always non-zero. Here, \(\tan x\) can take any real number value from \(-\infty\) to \(\infty\). This is because the value of \(\sin x/\cos x\) increases as \(x\) approaches \(\pi/2\) from the left side and decreases as \(x\) approaches \(-\pi/2\) from the right side.

Step 3: Proving the statement

Therefore, for every real number \(y\), we can find a number \(x\) in the interval \(-\pi/2, \pi/2\), such that \(y = \tan x\). This is because the range of the \(\tan\) function in this interval is \(-\infty, \infty\) or all real numbers. Hence, for any given \(y\), there exists such an \(x\) in the interval \(-\pi/2, \pi/2\).

Key Concepts

Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It serves as a powerful tool in trigonometry by allowing us to visualize the behavior of trigonometric functions. The coordinates of any point on the unit circle represent \( \cos \theta, \sin \theta \). For the tangent function, which is \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we observe this value as the slope of the line connecting the origin to the point on the circle.

• Unit circle helps in understanding trigonometric functions better.
• The coordinates of points on the circle are directly linked to angles and trigonometric values.
• Tangent is undefined when the line is vertical (when \( \cos \theta = 0 \)).

Visualizing the tangent function on the unit circle clarifies why it is undefined at certain points and how it behaves as angles change.

Periodic Function

A periodic function is a function that repeats its values in regular intervals or periods. The tangent function, \( \tan x \), is periodic with a period of \( \pi \). This means it repeats its values every \( \pi \) units.

• The concept of periodicity helps us understand how trigonometric functions behave over intervals.
• Tangent's periodicity is \( \pi \), meaning every \( \pi \) radians, the function looks the same.
• This periodic nature explains the repetitive pattern of the tangent waveform.

Understanding periodic functions is important for predicting the behavior of trigonometric functions like tangent over different angles.

Range of Tangent Function

The range of the tangent function provides insight into the values the function can take. Over the interval \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \), the tangent function spans all real numbers, from \( -\infty \) to \infty \). This is due to the fact that the tangent function increases or decreases without bound as it approaches the vertical asymptotes at \(-\frac{\pi}{2} \) and \( \frac{\pi}{2} \).

• The tangent function is unique in having a range of all real numbers within this specific interval.
• This vast range makes it possible for the tangent function to match any real number output, a key feature in solving equations like \( \tan x = y \).

Grasping the range of the tangent function helps explain why for any real number \( y \, there's an \( x \) such that \( \tan x = y \).

Inverse Trigonometric Functions

Inverse trigonometric functions help us determine angles if we know the value of the trigonometric function. For tangent, this inverse is denoted as \( \tan^{-1} y \) or \( \arctan y \), providing an angle whose tangent is \( y \).

• The inverse tangent function outputs angles in the principal range \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \).
• This means we can always find an angle for any real \( y \), solving the equation \( \tan x = y \).
• Understanding inverse trig functions is essential in various applications, from geometry to calculus.

Inverse trigonometric functions are powerful tools, enabling us to bridge the gap between trigonometric values and angles, particularly useful in solving equations involving real numbers.