E3.1, E3.5, E7.4 - Trigonometric graphs (Extended)

Sine and cosine graphs

The graph $y=\sin x$ records the vertical coordinate of this rotating endpoint. The graph $y=\cos x$ records its horizontal coordinate. Angles in this lesson are in degrees.

Worked example 1: sketching sine

Question. Describe a sketch of $y=\sin x$ for $0^\circ\le x\le360^\circ$, stating its zeros, maximum and minimum.

Plan. Place the five quarter-turn points, then join them with the smooth shape traced by the rotating arm's height.

The curve starts at $(0^\circ,0)$, rises to $(90^\circ,1)$, falls through $(180^\circ,0)$ to $(270^\circ,-1)$, then rises to $(360^\circ,0)$. It flattens at its maximum and minimum; it has no sharp corners.

Answer. Zeros: $0^\circ,180^\circ,360^\circ$. Maximum: 1 at $90^\circ$. Minimum: $-1$ at $270^\circ$.

Check. The endpoint cannot be higher than 1 or lower than $-1$, because the arm has length 1. Its height is positive during the upper half-turn and negative during the lower half-turn, matching the sketch.

Worked example 2: sketching cosine

Question. Describe a sketch of $y=\cos x$ on the same interval, stating its zeros, maximum and minimum.

Plan. Track the horizontal position at the five quarter turns.

The curve starts at $(0^\circ,1)$, falls through $(90^\circ,0)$ to $(180^\circ,-1)$, then rises through $(270^\circ,0)$ to $(360^\circ,1)$. Join these points smoothly, flattening at the top and bottom.

Answer. Zeros: $90^\circ,270^\circ$. Maximum: 1 at both $0^\circ$ and $360^\circ$. Minimum: $-1$ at $180^\circ$.

Check. The arm points furthest left at $180^\circ$, so its horizontal coordinate must be $-1$. Reflection of the graph in $x=180^\circ$ leaves it unchanged: equal angles either side give equal horizontal positions.

Watch out

Do not join the quarter-turn points by straight segments. A trigonometric graph is smooth. Include both endpoints when the interval uses $\le$.

Vertical asymptote

A vertical asymptote is a vertical line approached by a graph as its values grow without bound in magnitude. For $y=\tan x$ on one full turn, the asymptotes are $x=90^\circ$ and $x=270^\circ$. The dashed lines show these positions; they are not parts of the graph.

Worked example 3: the first tangent branch

Question. Sketch the part of $y=\tan x$ for $0^\circ\le x<90^\circ$, showing $x=45^\circ$ and the asymptote.

Plan. Use the exact values at 0 and 45 degrees, then show how the branch approaches its break.

$\tan0^\circ=0$ gives $(0^\circ,0)$, and $\tan45^\circ=1$ gives $(45^\circ,1)$. Draw an increasing curve through them, becoming steeper and rising without bound as $x$ approaches $90^\circ$ from the left. Draw $x=90^\circ$ dashed.

Answer. One increasing branch through $(0^\circ,0)$ and $(45^\circ,1)$, approaching the vertical asymptote $x=90^\circ$.

Check. The values at 30 and 60 degrees are $1/\sqrt3$ and $\sqrt3$: they lie below and above 1, respectively, as the rising curve requires.