The graphs of $\sin x$, $\cos x$, and $\tan x$ are periodic — they repeat the same pattern over and over. Understanding their shapes lets you find multiple solutions to trig equations.
Sine ($y = \sin x$):
- Starts at the origin $(0°, 0)$
- Oscillates between $-1$ and $+1$
- Period of 360° — crosses the x-axis at 0°, 180°, 360°, ...
- Reaches its peak of 1 at 90° and its trough of $-1$ at 270°
Cosine ($y = \cos x$):
- Starts at $(0°, 1)$ — i.e. $\cos 0° = 1$
- Also oscillates between $-1$ and $+1$, period of 360°
- Identical shape to $\sin x$, just shifted 90° to the left
Tangent ($y = \tan x$):
- Period of 180° — repeats twice as often
- Has asymptotes (vertical lines it never crosses) at 90°, 270°, etc. — these are angles where tan is undefined
- No maximum or minimum: it goes from $-\infty$ to $+\infty$ between each asymptote
Why this matters: If your calculator gives $\sin x = 0.5 \Rightarrow x = 30°$, the graph shows there's also a solution at $x = 150°$ in the range $0°$–$360°$. Sketch the graph to find all solutions in a given range.