Problem 21 - Algebra 2 and Trigonometry

Question

a. On the same set of axes, sketch the graphs of \(y=2 \sin x\) and \(y=\cos x\) in the interval \(0 \leq x \leq 2 \pi\) b. How many points do the graphs of \(y=2 \sin x\) and \(y=\cos x\) have in common in the interval \(0 \leq x \leq 2 \pi ?\)

Step-by-Step Solution

Answer

The graphs intersect at 2 points in the interval \(0 \leq x \leq 2\pi\).

Step 1: Understand the Functions

The first function is \( y = 2 \sin x \), which means the sine wave will have an amplitude of 2 rather than 1, and it will oscillate between -2 and 2.The second function is \( y = \cos x \), which will oscillate between -1 and 1 due to its amplitude of 1.Both functions are periodic with a period of \( 2\pi \), the same as the interval given.

Step 2: Plotting Key Points for y=2sin(x)

Identify key points for one period of \( y = 2 \sin x \):- At \( x = 0 \), \( y = 2 \sin 0 = 0 \)- At \( x = \pi/2 \), \( y = 2 \sin(\pi/2) = 2 \)- At \( x = \pi \), \( y = 2 \sin \pi = 0 \)- At \( x = 3\pi/2 \), \( y = 2 \sin(3\pi/2) = -2 \)- At \( x = 2\pi \), \( y = 2 \sin 2\pi = 0 \)Plot these points, and sketch the sine curve with a peak at 2 and a trough at -2 over the interval \( 0 \leq x \leq 2\pi \).

Step 3: Plotting Key Points for y=cos(x)

Identify key points for one period of \( y = \cos x \):- At \( x = 0 \), \( y = \cos 0 = 1 \)- At \( x = \pi/2 \), \( y = \cos(\pi/2) = 0 \)- At \( x = \pi \), \( y = \cos \pi = -1 \)- At \( x = 3\pi/2 \), \( y = \cos(3\pi/2) = 0 \)- At \( x = 2\pi \), \( y = \cos 2\pi = 1 \)Plot these points, and sketch the cosine curve oscillating between -1 and 1.

Step 4: Finding Intersection Points

The graphs intersect where \( 2\sin x = \cos x \). Using the identity \( \tan x = \frac{\sin x}{\cos x} \), solve: \[2 \sin x = \cos x \Rightarrow 2 \tan x = 1\]This simplifies to \( \tan x = \frac{1}{2} \).Find solutions of \( x \) in \( 0 \leq x \leq 2\pi \):- The principal solutions are \( x = \tan^{-1}(\frac{1}{2}) \).- Adding the period of tangent, \( \pi \), find the second point by calculating \( x = \pi + \tan^{-1}(\frac{1}{2}) \).These points provide the values of \( x \) where the graphs intersect in the given interval.

Key Concepts

Graphing Sine and Cosine

Understanding how to graph the sine and cosine functions is an essential skill in mathematics. These functions represent periodic behaviors that repeat their patterns over specific intervals.

For the sine function, such as the graph of \( y = 2 \sin x \), the curve displays oscillations with peaks and troughs. To properly sketch this graph, identify key points for one full cycle from \( 0 \) to \( 2\pi \):

• At \( x = 0 \), \( y = 0 \)
• At \( x = \pi/2 \), \( y = 2 \)
• At \( x = \pi \), \( y = 0 \)
• At \( x = 3\pi/2 \), \( y = -2 \)
• At \( x = 2\pi \), \( y = 0 \)

Plot these on your graph and connect them to form a smooth curve, peaking at \( y = 2 \) and dipping to \( y = -2 \).Next, graph the cosine function \( y = \cos x \), which is another period function oscillating between \( -1 \) and \( 1 \). Key points here include:

• At \( x = 0 \), \( y = 1 \)
• At \( x = \pi/2 \), \( y = 0 \)
• At \( x = \pi \), \( y = -1 \)
• At \( x = 3\pi/2 \), \( y = 0 \)
• At \( x = 2\pi \), \( y = 1 \)

With these points, sketch your cosine wave.Both graphs appear on the same set of axes for visual comparison, with sine reaching higher and lower due to its different amplitude.

Amplitude and Period

The amplitude and period are fundamental properties of trigonometric functions that define how these functions look when plotted.The **amplitude** of a function like sine or cosine describes the height from the center to the peak or from the center to the trough. For instance, in \( y = 2 \sin x \), the amplitude is 2:

• This indicates the graph will oscillate between +2 and -2 because it stretches twice as far from the midline compared to a standard sine wave, which has an amplitude of 1.
• So you see that the sine curve's highest point is at y = 2, and lowest is at y = -2.

For \( y = \cos x \), the amplitude is 1:

• This means it oscillates between +1 and -1, which is typical for standard cosine functions.

The **period** represents how long it takes for the function to repeat its entire cycle. Both \( y = 2 \sin x \) and \( y = \cos x \) have a period of \( 2\pi \), which means after \( 2\pi \) units along the x-axis, the waves begin their cycles again. The standard sine and cosine functions thus both complete one full period within the interval of \( 0 \leq x \leq 2\pi \).

Trigonometric Identities

Trigonometric identities are mathematical tools that simplify the manipulation and solving of trigonometric functions. In the problem, finding the intersection points of \( y = 2 \sin x \) and \( y = \cos x \) involves utilizing these identities.The equation \( 2 \sin x = \cos x \) can be transformed using the tangent identity:
For any angle \( x \), we know that \( \tan x = \frac{\sin x}{\cos x} \). This gives us:\[2 \sin x = \cos x \Rightarrow \frac{2 \sin x}{\cos x} = 1 \Rightarrow 2 \tan x = 1 \]This further simplifies to \( \tan x = \frac{1}{2} \). Finding the solutions for \( x \) within the interval \( 0 \leq x \leq 2\pi \) involves solving:

• The principal solution, \( x = \tan^{-1}(\frac{1}{2}) \).
• Since the tangent function has a period of \( \pi \), the next solution is \( x = \pi + \tan^{-1}(\frac{1}{2}) \).

Identifying these points reveals where the two graphs intersect within the given interval. Understanding and applying these identities effectively can ease the process of solving complex trigonometric equations.