Trigonometric functions

Description

The trigonometric functions $\sin$, $\cos$, and $\tan$ are defined for all angles, not just those in right-angled triangles.

Graphs:

$y = \sin x$: wave shape, period $360°$, range $[-1, 1]$, passes through $(0, 0)$ $y = \cos x$: wave shape, period $360°$, range $[-1, 1]$, passes through $(0, 1)$ $y = \tan x$: period $180°$, asymptotes at $x = 90°, 270°, \ldots$, no maximum or minimum

Key symmetry properties: $$\sin(180° - x) = \sin x$$ $$\cos(360° - x) = \cos x$$ $$\sin(-x) = -\sin x \quad (\sin \text{ is odd})$$ $$\cos(-x) = \cos x \quad (\cos \text{ is even})$$

CAST diagram — shows which functions are positive in each quadrant:

• All positive (0°–90°)
• Sine positive (90°–180°)
• Tangent positive (180°–270°)
• Cosine positive (270°–360°)

Used to find all solutions of trig equations in a given range.

Common error: only giving one solution to a trig equation — always check all quadrants in the given range.

Links

Exact trigonometric values Solving quadratics graphically

Make sure the calculator is in degree mode (D shown on screen) unless the question specifies radians.