Exact values, graphs and trig equations
| English | Chinese | Pinyin |
|---|---|---|
| exact values | 精确值 | jīng què zhí |
| trig graphs | 三角函数图 | sān jiǎo hán shù tú |
| asymptotes | 渐近线 | jiàn jìn xiàn |
| symmetry | 对称 | duì chèn |
The values you must know by heart
- Some trig values are so important that you need them without a calculator. They come from two special right triangles: the $45$-$45$-$90$ and the $30$-$60$-$90$.
The exact values 精确值 table
| $x$ | $0^{\circ}$ | $30^{\circ}$ | $45^{\circ}$ | $60^{\circ}$ | $90^{\circ}$ |
|---|---|---|---|---|---|
| $\sin x$ | $0$ | $\dfrac{1}{2}$ | $\dfrac{\sqrt{2}}{2}$ | $\dfrac{\sqrt{3}}{2}$ | $1$ |
| $\cos x$ | $1$ | $\dfrac{\sqrt{3}}{2}$ | $\dfrac{\sqrt{2}}{2}$ | $\dfrac{1}{2}$ | $0$ |
| $\tan x$ | $0$ | $\dfrac{1}{\sqrt{3}}$ | $1$ | $\sqrt{3}$ | undefined |
Memory trick for sine: $0, 1, 2, 3, 4$ under the root and over $2$: $\;\dfrac{\sqrt{0}}{2}, \dfrac{\sqrt{1}}{2}, \dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{3}}{2}, \dfrac{\sqrt{4}}{2}$. Cosine is the reverse.
A Ferris wheel: a point on the rim traces a sine curve
Trig graphs & equations
(cos θ, sin θ)
As θ turns, sin and cos trace their waves — and repeat every 360°.
What is the exact value of sin 30°? (as a decimal)
sin 30° = 1/2 = 0.5.
What is the exact value of tan 45°?
tan 45° = 1 (opposite = adjacent in a 45-45-90 triangle).
cos 60° = 1/2.
cos 60° = 1/2 is one of the key exact values to memorise.
Trig graphs 三角函数图
- $y = \sin x$ and $y = \cos x$ are waves oscillating between $-1$ and $1$.
- $y = \tan x$ repeats every $180^{\circ}$ and has vertical asymptotes 渐近线 at $90^{\circ}$ and $270^{\circ}$.
Trig graphs are waves — sine and cosine oscillate between $-1$ and $1$, tangent repeats every $180^{\circ}$.
Two solutions. A trig equation usually has two answers in $0^{\circ}$–$360^{\circ}$. For $\sin x = \dfrac{\sqrt{3}}{2}$: $x = 60^{\circ}$ or $x = 180 - 60 = 120^{\circ}$. Don't forget the second answer.
Solving trig equations
- Find the first solution using the inverse function.
- Use the graph's symmetry 对称 to find the second:
- $\sin x = k$: second solution is $180^{\circ} - x$.
- $\cos x = k$: second solution is $360^{\circ} - x$.
The two special triangles give the exact values: the $45^\circ$ triangle and the $30^\circ$-$60^\circ$ triangle
Solve sin x = √3/2 for 0–360°. One answer is 60°. What is the other (degrees)?
The sine wave is symmetric about 90°: 180 − 60 = 120°.
Solve 2cos x + 1 = 0 for 0–360°. One answer is 120°. What is the other (degrees)?
cos x = −1/2 gives x = 120° or 360 − 120 = 240°.
If sin x = k has one answer x = 40°, the second answer is 180 − ______ = ______°.
The second sine solution is 180° − 40° = 140°.
Worked examples
- $\sin x = \dfrac{\sqrt{3}}{2} \Rightarrow x = 60^{\circ}$ or $120^{\circ}$.
- $2\cos x + 1 = 0 \Rightarrow \cos x = -\dfrac{1}{2} \Rightarrow x = 120^{\circ}$ or $240^{\circ}$.
You've got it
- know the exact values: $\sin 30^{\circ} = \dfrac{1}{2}$, $\cos 60^{\circ} = \dfrac{1}{2}$, $\tan 45^{\circ} = 1$
- sine and cosine waves run between $-1$ and $1$
- a trig equation usually has two answers in $0^{\circ}$–$360^{\circ}$