Key points and features of the basic sine and cosine graphs

For the basic function below, which five-point pattern should be used first?

Vary the integer n and inspect the points at angle nπ

Which statement holds for all settings?

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. 1Use 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. 2Step 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. 3Read 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. 4Read the horizontal features One complete cycle covers four quarter-turn intervals, so the period is 2π \(\displaystyle T=4\times\frac{\pi}{2}=2\pi\)

Original setup

\(\displaystyle x=3\pi/2\)

Find the quarter-turn index of the given x-value

Original setup

\(\displaystyle x=3\pi/2\)

Use the sine pattern to find the y-value at the given x-value

Start pattern swapped. One of these lines is right. Which one?

From individual features to the whole drawing

Use the existing feature work to construct the complete graph yourself before checking the feedback

1. 1Prepare the complete picture Label axes and units, exact key points, all allowed branches and endpoints. Use radians for the angle axis of the circular-function graphs
2. 2Check the connections and domain Join smooth waves, retain filled/open endpoints State the restricted domain/range separately from the period of the full function
3. 3Check your drawing Full drawings and explanations are ungraded. Compare your work with the feedback; short entries check individual values only

Use unit-circle heights to draw a full sine wave

The input axis now measures an angle in radians. Mark (−π,0),(−π/2,−1),(0,0),(π/2,1),(π,0), then join a smooth sine curve, not straight chords. Both interval endpoints are included. The range is [−1,1], with minimum at −π/2 and maximum at π/2. The zeros are −π,0,π. The full real sine function has period 2π, no asymptotes and origin symmetry. For cosine, use the horizontal unit-circle coordinates instead: cos0=1, not 0.

Worked graph

\(\displaystyle s(x)=\sin x,\quad -\pi\le x\le\pi\)

How many x-axis intercepts does the restricted worked sine graph have, counting included endpoints?

Your complete circular drawing

On separate axes, draw the functions over the restricted interval shown. Label quarter-turn points, every x- and y-intercept, all maxima/minima and included/excluded endpoints. State each exact domain and range. Explain why an excluded endpoint does not necessarily remove that output from the range. State the period of each full real function and whether either has an asymptote

Review all points, endpoints and branches

For sine, the quarter-turn outputs are −1,0,1,0,−1; the first endpoint (−π/2,−1) is filled and the last (3π/2,−1) is open. Its included zeros are 0 and π, minimum −1 occurs at the left endpoint and maximum 1 at π/2, so range [−1,1]. For cosine, the outputs are 0,1,0,−1,0 with the same endpoint conventions. Zeros are −π/2 and π/2; the right endpoint zero is excluded. Its maximum is (0,1), minimum (π,−1), range [−1,1]. Both domains are [−π/2,3π/2). The full functions have period 2π and no asymptotes. The two y-intercepts are (0,0) and (0,1). The excluded sine endpoint has output −1, which also occurs at the included left endpoint. The excluded cosine endpoint has output 0, which also occurs at included points