Key points and features of the basic sine and cosine graphs

Basic sine and cosine graphs

A basic sine or cosine graph is built from five values spaced one quarter turn apart. Extend that pattern by repeating every full turn, then read vertical and horizontal features from the key points

1. Use the correct start pattern Begin with the five values for the stated function at consecutive quarter-turn positions \(\displaystyle \cos(0)=1,\quad \cos\left(\frac{\pi}{2}\right)=0,\quad \cos(\pi)=-1\)
2. Step by a quarter turn Move along the x-axis by one quarter turn for each successive key point \(\displaystyle 0,\quad \frac{\pi}{2},\quad \pi,\quad \frac{3\pi}{2},\quad 2\pi\)
3. Read the vertical features The maximum and minimum give the range, amplitude and equilibrium value \(\displaystyle -1\le y\le1,\quad A=\frac{1-(-1)}{2}=1,\quad y_{\rm eq}=0\)
4. Read the horizontal features One complete cycle covers four quarter-turn intervals, so the period is 2π \(\displaystyle T=4\times\frac{\pi}{2}=2\pi\)

Worked example

For the basic function below, list the five key points on the stated interval

1. \(\displaystyle \Delta x=\frac{\pi}{2}\)Successive key points are separated by one quarter turn
2. \(\displaystyle x=\frac{\pi}{2},\quad \pi,\quad \frac{3\pi}{2},\quad 2\pi,\quad \frac{5\pi}{2}\)Start at the left endpoint and add one quarter turn until the right endpoint is reached
3. \(\displaystyle y=0,\quad -1,\quad 0,\quad 1,\quad 0\)These are the five successive values of the basic cosine pattern from that starting position
4. \(\displaystyle \left(\frac{\pi}{2},0\right),\quad (\pi,-1),\quad \left(\frac{3\pi}{2},0\right),\quad (2\pi,1),\quad \left(\frac{5\pi}{2},0\right)\)Pair each x-value with its corresponding y-value to obtain the key points

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