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Magazine Chapter 11: Problem 22
a. On the same set of axes, sketch the graphs of \(y=\tan x\) and \(y=\cos \left(x+\frac{\pi}{2}\right)\) in the interval \(-\frac{\pi}{2} \leq x \leq \frac{3 \pi}{2}\) . b. How many points do the graphs of \(y=\tan x\) and \(y=\cos \left(x+\frac{\pi}{2}\right)\) have in common in the interval \(-\frac{\pi}{2} \leq x \leq \frac{3 \pi}{2} ?\)
Short Answer
Expert verified
The graphs intersect at 2 points: \((0,0)\) and \((\pi,0)\).
Step by step solution
01
Understand the Functions
We have two functions: \(y = \tan{x}\) and \(y = \cos\left(x + \frac{\pi}{2}\right)\). \(\tan x\) has vertical asymptotes where \(x\) is odd multiples of \(\frac{\pi}{2}\), and it's undefined there. \(\cos\left(x + \frac{\pi}{2}\right)\) is actually equivalent to \(-\sin x\).
02
Plot Key Points for y = tan x
Plot the tangents where \(x = 0, \pi/4, \pi/2, 3\pi/4, \pi, 5\pi/4, 3\pi/2\). Remember, the function is undefined at \(-\pi/2\), \(\pi/2\), and \(3\pi/2\), so these are vertical asymptotes.
03
Plot Key Points for y = -sin x
Rewrite \(\cos\left(x + \frac{\pi}{2}\right)\) to \(-\sin x\). Plot key points for \(-\sin x\) such as \((-\pi/2, -1)\), \( (0, 0) \), \((\pi/2, 1)\), \((\pi, 0)\), \((3\pi/2, -1)\).
04
Sketch Both Graphs on the Same Axes
On a graph ranging from \(-\pi/2\) to \(3\pi/2\), draw the periodic curve for \(\tan x\) with asymptotes at \(x = -\pi/2, \pi/2, 3\pi/2\). Then, overlay the curve of \(-\sin x\) over the same interval.
05
Identify Points of Intersection
By examining where the graphs intersect, find common x-values. The points of intersection typically occur where both \(\tan x = -\sin x\). In this case, they intersect at \(x = 0\), and \(x = \pi\).
06
Count Points of Intersection
There are two intersections: at \((0, 0)\) and \((\pi, 0)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Trigonometric Functions
Graphing trigonometric functions involves understanding the unique shapes and behaviors of these periodic functions. Each trigonometric function鈥攕ine, cosine, and tangent鈥攈as its own distinct shape:
- Sine and Cosine: These functions create wavelike patterns, each starting at a specific point on the y-axis and repeating every \(2\pi\) radians.
- Tangent: This graph has a distinguishable pattern marked by a central curve and several vertical lines, or asymptotes, that the function approaches but never touches.
- Periodic Nature: Trigonometric functions repeat their values at regular intervals. For sine and cosine, the interval is \(2\pi\), while tangent repeats every \(\pi\).
- Key Points: Identify values at crucial points such as \(0, \pi/2, \pi, 3\pi/2\) for cosine and sine; and odd multiples of \(\pi/2\) for tangent.
- Vertical Asymptotes: These are lines where the function is undefined due to division by zero鈥攖angent functions have them at odd multiples of \(\pi/2\).
Tangent Function
The tangent function, \(y = \tan x\), is unique among the basic trigonometric functions because it features vertical asymptotes in its graph. These asymptotes occur because the function is undefined where the cosine of \(x\) is zero, specifically at odd multiples of \(\pi/2\).
- Behavior: The tangent function increases or decreases without bound as it approaches these vertical asymptotes, creating distinct segments of the graph in between.
- Interval: The primary period of \(y = \tan x\) is \(\pi\), so it repeats itself every \(\pi\) units along the x-axis.
- Key Points: At \(x = 0\), \(\pi\), and other zero-cosine points, \(y = \tan x\) crosses through the x-axis, known as roots or zeros.
Cosine Function Transformations
Transformations of trigonometric functions, such as the cosine function, allow us to shift, stretch, or compress their graphs. The exercise focuses on the transformation \(y = \cos\left(x + \frac{\pi}{2}\right)\), which is equivalent to \(y = -\sin x\).
- Horizontal Shifts: Adding or subtracting from \(x\) inside the function's argument results in horizontal shifts. "Adding \(\frac{\pi}{2}\)" shifts the cosine graph to the left by \(\frac{\pi}{2}\).
- Equivalent Forms: By understanding identities, \(\cos\left(x + \frac{\pi}{2}\right)\) can be rewritten using the identity \(\cos(A + B) = \cos A \cos B - \sin A \sin B\), simplifying to \(-\sin x\).
- Wave Shape: The graph undergoes reflection due to the negative sign resulting in it mimicking the sine wave but inverted.
Intersection of Functions
When graphing functions together, finding their intersection points is crucial to understanding where they equate. In the context of this exercise, intersections between \(y = \tan x\) and \(-\sin x\) indicate where their outputs are equal for the same x-values.
- Intersection Points: These occur where both function values are the same (\(\tan x = -\sin x\)), revealing the solutions to equations graphically. In the given interval, intersections happen at specific x-values as these functions have differing periodicities.
- X-Values: For this exercise, key intersection points are at \(x = 0\) and \(x = \pi\), where both graphs we sketch coincide.
- Interpretation: The process of finding intersections visually can illuminate mathematical solutions, making complex equations easier to solve by seeing them. This is especially helpful where algebraic manipulation might be challenging.
