Problem 47 - Precalculus: Functions and Graphs

Question

Refer to the graph of \(y=\tan x\) to find the exact values of \(x\) In the interval \((-\pi / 2,3 \pi / 2)\) that satisfy the equation. $$\tan x=1$$

Step-by-Step Solution

Answer

Solutions are at \( x = \frac{\pi}{4} \) and \( x = \frac{5\pi}{4} \).

Step 1: Identify where y=tan(x) equals 1

The tangent function, \( y = \tan x \), equals 1 at certain key angles where the sine and cosine functions have equal values, i.e., \( \frac{\sin x}{\cos x} = 1 \). This condition is satisfied at \( x = \frac{\pi}{4} + k\pi \), where \( k \) is an integer.

Step 2: Determine solutions within the interval

Identify the specific values of \( k \) such that \( x = \frac{\pi}{4} + k\pi \) falls within the interval \((-\frac{\pi}{2}, 3\pi/2)\). Substituting values for \( k \):- For \( k = 0 \), \( x = \frac{\pi}{4} \).- For \( k = 1 \), \( x = \frac{5\pi}{4} \).Examine if \( k = -1 \) and \( k = 2 \) fall within the interval, but they exceed the interval limits.

Key Concepts

Tangent Function

The tangent function, denoted as \( y = \tan x \), is one of the six fundamental trigonometric functions. It ties together the sine and cosine functions, as it can be expressed as the ratio of sine to cosine: \( \tan x = \frac{\sin x}{\cos x} \). This function has a periodicity, meaning it repeats its values over regular intervals. For tangent, the principal period is \( \pi \), or 180 degrees.Key characteristics of the tangent function include:

• Asymptotes: The tangent function has vertical asymptotes where \( \cos x = 0 \). These occur at \( x = \frac{\pi}{2} + k\pi \) for any integer \( k \). These are points where the function is undefined and the graph shoots off to infinity.
• Zeros: \( \tan x \) has zeros wherever \( \sin x = 0 \), which is at integer multiples of \( \pi \), such as \( 0, \pi, 2\pi, -\pi \), and so on.
• Range: The range of the tangent function is all real numbers, \((-\infty, \infty)\).

This makes the tangent function unique among the standard trigonometric functions, as both sine and cosine are bounded within -1 and 1.

Trigonometric Equations

Trigonometric equations involve finding the angle or angles that satisfy a given trigonometric identity. In our problem, the equation \( \tan x = 1 \) needs to be solved for specific \( x \) values.To tackle such problems:

• Identify the general solution: For \( \tan x = 1 \), the general solution where the tangent function equals 1 is at \( \frac{\pi}{4} + k\pi \), where \( k \) is any integer. This formula arises because tangent has a period of \( \pi \), meaning it repeats every \( \pi \) radians.
• Restrict the solution: Use the given interval, \((-\frac{\pi}{2}, 3\pi/2)\), to find specific \( x \) values by adjusting the integer \( k \) accordingly.

This process is crucial because tangent equations often have multiple solutions within a given range. Understanding how to manipulate \( k \) is essential to finding all solutions within specified limits.

Graphs of Trigonometric Functions

Graphs of trigonometric functions, like the tangent graph, provide visual insight into their behavior. The graph of \( y = \tan x \) is characterized by its periodic nature and repeating pattern. It helps to visualize how quickly or slowly the values change compared to other trigonometric functions.Important features to note when studying tangent graphs include:

• Periodic Intervals: The graph repeats every \( \pi \) units. Understanding the interval helps in predicting the behavior of the graph over different segments.
• Critical Points: Zero crossings occur at multiples of \( \pi \); asymptotes appear at positions \( \frac{\pi}{2} + k\pi \).
• Slope: Unlike sine and cosine which fluctuate smoothly between peaks, tangent increases or decreases sharply approaching infinity, before restabilizing across its period.

By interpreting these elements, one gains a deeper comprehension of how the function behaves over its domain, which is essential in solving problems, such as determining when \( \tan x = 1 \) within certain intervals.