Graph f(x) = sin(3x)/sin(x) - cos(3x)/cos(x) and make a conjecture. Prove your conjecture.
http://www.wolframalpha.com/input/?i=plot+sin%283x%29%2Fsin%28x%29+-+cos%283x%29%2Fcos%28x%29+
mmmhhhh?
let y = sin(3x)/sin(x) - cos(3x)/cos(x)
= (sin(3x)cos(x) - cos(3x)sinx)/(sinxcosx)
= sin(3x - x)/(sinxcos)
= sin(2x)/sinxcosx
= 2sinxcosx/(sinxcosx)
= 2
so it is a straight line y = 2 as shown in the graph
To graph the function f(x) = sin(3x)/sin(x) - cos(3x)/cos(x), we can start by analyzing each term separately.
The first term sin(3x)/sin(x) can be simplified using the trigonometric identity sin(2θ) = 2sin(θ)cos(θ):
sin(3x)/sin(x) = (sin(2x)cos(x) + cos(2x)sin(x)) / sin(x)
= (2sin(x)cos(x)cos(x) + (1 - 2sin^2(x))sin(x)) / sin(x)
= 2cos^2(x) + sin(x) - 2sin^3(x)/sin(x)
= 2cos^2(x) + sin(x) - 2sin^2(x)
The second term cos(3x)/cos(x) can be simplified using the trigonometric identity cos(2θ) = cos^2(θ) - sin^2(θ):
cos(3x)/cos(x) = (2cos^2(x) - 1 - 2sin^2(x)) / cos(x)
= 2cos^2(x) - 1 - 2sin^2(x)/cos(x)
Combining both terms, we have:
f(x) = (2cos^2(x) + sin(x) - 2sin^2(x)) - (2cos^2(x) - 1 - 2sin^2(x))/cos(x)
= 2cos^2(x) + sin(x) - 2sin^2(x) - 2cos^2(x) + 1 + 2sin^2(x))/cos(x)
= 1 + sin(x) / cos(x)
Now, to graph the function f(x), firstly observe that f(x) is undefined when cos(x) = 0 since division by 0 is undefined. Therefore, the vertical asymptotes of the graph occur at x = π/2, 3π/2, 5π/2, etc.
Let's examine some key points on the graph:
1. When x = 0,
f(0) = 1 + sin(0)/cos(0) = 1
So, we have a point (0, 1) on the graph.
2. When x = π/4,
f(π/4) = 1 + sin(π/4)/cos(π/4) = 1 + 1/sqrt(2) = (sqrt(2) + 1) / sqrt(2)
So, we have a point (π/4, (sqrt(2) + 1) / sqrt(2)) on the graph.
3. When x = π/2,
f(π/2) = undefined
4. When x = π,
f(π) = 1 + sin(π)/cos(π) = 1 - 1/(-1) = 2
So, we have a point (π, 2) on the graph.
5. When x = 3π/2,
f(3π/2) = undefined
From the above analysis and using the properties of the sinusoidal and cosinusoidal functions, a conjecture can be made: The function f(x) = sin(3x)/sin(x) - cos(3x)/cos(x) is a periodic function with vertical asymptotes at each odd multiple of π/2, and it oscillates between the values of 1 and 2 as x increases.
To graph the function f(x) = sin(3x)/sin(x) - cos(3x)/cos(x), we can follow these steps:
Step 1: Determine the x-intercepts.
To find the x-intercepts, we need to solve the equation sin(3x)/sin(x) - cos(3x)/cos(x) = 0. This means that either sin(3x)/sin(x) = 0 or cos(3x)/cos(x) = 0.
For sin(3x)/sin(x) = 0, sin(3x) must be equal to 0. The solutions to this equation are x = 0, π/3, 2π/3, and so on. These values represent the x-intercepts.
For cos(3x)/cos(x) = 0, cos(3x) must be equal to 0. The solutions to this equation are x = π/6, π/2, 5π/6, and so on. However, we need to exclude the values where sin(x) = 0, as dividing by zero is undefined. So, we remove x = π/2 and x = 5π/2 from our list of x-intercepts.
Step 2: Determine the y-intercept.
To find the y-intercept, we evaluate f(x) when x = 0. Substituting x = 0 into the function, we get f(0) = sin(0)/sin(0) - cos(0)/cos(0). Since sin(0) = 0 and cos(0) = 1, we have f(0) = 0/0 - 1/1 = -1.
Therefore, the y-intercept is at (0, -1).
Step 3: Determine the symmetry of the graph.
Since the function f(x) contains both sin(3x) and cos(3x), we can examine their properties to determine the symmetry.
sin(3x) is an odd function, meaning sin(-x) = -sin(x). This tells us that sin(3(-x)) = sin(-3x) = -sin(3x). Therefore, sin(3x) is an odd function.
cos(3x) is an even function, meaning cos(-x) = cos(x). Therefore, cos(3x) is also an even function.
Since addition and subtraction of odd and even functions preserve the symmetry, we can conclude that f(x) is an even function. This means the graph is symmetric with respect to the y-axis.
Step 4: Plotting the graph and making a conjecture.
Using the x-intercepts, y-intercept, symmetry, and the shape of sine and cosine functions, we can plot the graph of f(x) over a desired interval.
After plotting the graph, we can observe its behavior and make a conjecture. However, without a specific interval or additional information provided, it is difficult to make an accurate conjecture.
To prove the conjecture or analyze the behavior of the function further, you may need to use calculus techniques such as finding the first derivative to identify critical points and extreme values, and the second derivative to determine concavity and inflection points.