Sketch the graph f(x)=sin(x)-cos(x)
there are lots of online graphing sites. However, this could help.
sinx - cosx
= √2 (sinx * 1/√2 - cosx * 1/√2)
= √2 sin(x - π/4)
confirming oobleck
https://www.wolframalpha.com/input/?i=plot+y+%3D+sinx+-+cosx%2C+y+%3D+%E2%88%9A2+sin%28x+-+%CF%80%2F4%29
Math is such fun when in self-quarantine.
To sketch the graph of the function f(x) = sin(x) - cos(x), we can follow a few steps:
Step 1: Determine the key properties of the function:
- Period: The period of f(x) is the same as the period of the individual sine and cosine functions, which is 2π.
- Amplitude: The amplitude of f(x) is determined by the sum of the maximum values of sine and cosine, which is √2.
Step 2: Find the intercepts:
To find the x-intercepts, we set f(x) = 0:
sin(x) - cos(x) = 0
Next, we can rewrite the equation as sin(x) = cos(x) and continue solving from there.
sin(x)/cos(x) = 1
Using the identity sin(x)/cos(x) = tan(x), we can rewrite the equation as:
tan(x) = 1
To find the x-values where tan(x) = 1, we can use the inverse tangent function (arctan):
x = arctan(1) + kπ
where k is an integer that allows for multiple solutions.
The solution for arctan(1) is π/4, so the x-intercepts occur at x = π/4 + kπ.
The y-intercept occurs when x = 0:
f(0) = sin(0) - cos(0) = 0 - 1 = -1
So, the y-intercept is -1.
Step 3: Sketch the graph:
Based on the properties we found in Step 1 and the intercepts we found in Step 2, we can now sketch the graph.
Start by plotting the x-intercepts at x = π/4 + kπ. Then plot the y-intercept at (0, -1). Connect the plotted points using a smooth curve that repeats every 2π and oscillates between the maximum and minimum values determined by the amplitude.
The graph will look like a wave that oscillates between two points above and below the x-axis, with a period of 2π.
Note: If you have access to graphing software or a graphing calculator, you can input the function f(x) = sin(x) - cos(x) to get a more accurate graph.