Trigonometry
Introduced| English | Meaning |
|---|---|
| trigonometric/ˌtrɪɡənəʊˈmetrɪk/ | Relating to functions such as sine, cosine and tangent and their relationships with angles. |
| asymptotes/ˈæsɪmptəʊts/ | Lines or curves that a graph approaches arbitrarily closely in a specified limiting direction. |
| identities/aɪˈdentɪtiz/ | Equations that are true for every permitted value of their variables. |
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 $\cos^2\theta + \sin^2\theta \equiv 1$.
The unit circle
(cos θ, sin θ)
Spin the angle θ around the circle: the horizontal leg is cos θ and the vertical leg is sin θ.
Exact values
- Learn these without a calculator:
| $\theta$ | $0^\circ$ | $30^\circ$ | $45^\circ$ | $60^\circ$ | $90^\circ$ |
|---|---|---|---|---|---|
| $\sin$ | $0$ | $\tfrac12$ | $\tfrac{1}{\sqrt2}$ | $\tfrac{\sqrt3}{2}$ | $1$ |
| $\cos$ | $1$ | $\tfrac{\sqrt3}{2}$ | $\tfrac{1}{\sqrt2}$ | $\tfrac12$ | $0$ |
| $\tan$ | $0$ | $\tfrac{1}{\sqrt3}$ | $1$ | $\sqrt3$ | undefined |
Memory trick. The sine values follow the pattern $\dfrac{\sqrt{0}}{2}, \dfrac{\sqrt{1}}{2}, \dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{3}}{2}, \dfrac{\sqrt{4}}{2}$ for $0°, 30°, 45°, 60°, 90°$. Cosine is the reverse.
A Ferris wheel: a point on the rim moves up and down like a sine curve
Sine & cosine graphs
Drag the amplitude, period and shifts of y = a·sin(bx + c) + d and watch the curve change against the base wave.
What is the exact value of sin 30°?
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sin 30° = 1/2 = 0.5.
What is the value of tan 45°?
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tan 45° = 1.
What is the exact value of cos 60°?
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cos 60° = 1/2 = 0.5.
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.
$\sin^2\theta + \cos^2\theta \equiv 1$, not $(\sin\theta + \cos\theta)^2 = 1$. The identity is about the sum of the squares, not the square of the sum. This is a common exam mistake.
sin and cos wave between -1 and 1; tan breaks at 90 and 270 degrees
For any angle θ, what does sin²θ + cos²θ equal?
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This is the identity sin²θ + cos²θ ≡ 1.
Which identity is correct?
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tan θ ≡ sin θ / cos θ; and sin²θ + cos²θ ≡ 1.
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.
If sin θ = 0.5 and 0° ≤ θ ≤ 360°, then θ = 30° is the only solution.
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sin θ = 0.5 also has solution θ = 180° − 30° = 150° in the given range.
You've got it
- $\sin, \cos$ stay in $[-1, 1]$, repeat every $360^\circ$; know the $30/45/60$ exact values
- identities: $\tan\theta \equiv \dfrac{\sin\theta}{\cos\theta}$, $\sin^2\theta + \cos^2\theta \equiv 1$
- solve by reducing to one trig function, then finding all solutions in the interval