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Chapter 2: Problem 6

Draw the graphs of $$ f(x)=\sin x-\cos x $$

Short Answer

Expert verified
The graph of \(f(x)=\sin x-\cos x\) is a sinusoidal function that passes through the origin and looks different from standard sine and cosine graphs due to the subtraction operation. It oscillates between -\(\sqrt{2}\) and \(\sqrt{2}\) with a period of \(2\pi\).

Step by step solution

01

Understand the function

The function \(f(x)=\sin x-\cos x\) is a combination of two simple trigonometric functions, the sine and the cosine function. These functions have a period of \(2\pi\) and their graphical representations are waves that oscillate between -1 and 1.
02

Plot basic sine and cosine function

Start by plotting the basic functions \(\sin x\) and \(\cos x\) on the same set of axes. For values of \(x\) from \(0\) to \(2\pi\), do the following: For every angle, calculate the sine and cosine values. Plot \(x\) as the angle, and the y-value as the result of the sine or cosine of that angle.
03

Subtract the cosine function from sine function

To get the graph of \(f(x)=\sin x-\cos x\), subtract the y-values of the cosine function from the y-values of the sine function at each corresponding x-value. Plot these points.
04

Draw the graph

Connect the points from step 3 to draw the graph of \(f(x)=\sin x-\cos x\). It will still be a wave-like curve but it looks different from basic sine and cosine curves. Unlike the simple sine and cosine functions, this function will pass through the origin (0,0).

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