三角関数

The mathematics of cycles

• Tides, sound waves, alternating current, planetary orbits — all described by trigonometric 三角的 functions.
• The sine and cosine waves are the building blocks of periodic motion. Master them and you can model anything that repeats.

Graphs of sin, cos and tan

• $\sin$ and $\cos$ wave between $-1$ and $1$ and repeat every $360^\circ$ (or $2\pi$ radians).
• $\tan$ repeats every $180^\circ$ (or $\pi$ radians) and has vertical asymptotes 渐近线 at $90^\circ$ and $270^\circ$.

The unit circle: $\cos\theta$ is the $x$-coordinate, $\sin\theta$ is the $y$-coordinate. Since $x^2 + y^2 = 1$, we get $\cos^2\theta + \sin^2\theta \equiv 1$.

Exact values

• Learn these without a calculator:

A Ferris wheel: a point on the rim moves up and down like a sine curve

Key identities 恒等式

• The two fundamental identities: $\tan\theta \equiv \dfrac{\sin\theta}{\cos\theta}$ and $\sin^2\theta + \cos^2\theta \equiv 1$.
• These let you convert between functions and simplify expressions.

sin and cos wave between -1 and 1; tan breaks at 90 and 270 degrees

Solving trig equations

• To solve a trig equation: reduce it to one function (often a quadratic in $\sin$ or $\cos$), then find every solution in the given interval.
• Use the graph's symmetry: if $\sin\theta = k$ has solution $\theta = \alpha$, then $\theta = 180^\circ - \alpha$ is also a solution.
• Solve trigonometrical equations; use the inverse functions $\sin^{-1},\cos^{-1},\tan^{-1}$ (inverse trigonometric relations), which give principal values.

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