Exact values, graphs and trig equations
Introduced| English | Meaning |
|---|---|
| exact values/eɡˈzækt ˈvæljuːz/ | A value stated without rounding or approximation. |
| trig graphs/trɪɡ ɡræfz/ | trig graphs |
| asymptotes/ˈæsɪmptəʊts/ | Lines or curves that a graph approaches arbitrarily closely in a specified limiting direction. |
| symmetry/ˈsɪmətri/ | A property by which a shape or pattern matches itself under a transformation such as reflection or rotation. |
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)
Difficulty 2/5 · 140 coins + a small bonus on first completion today
sin 30° = 1/2 = 0.5.
What is the exact value of tan 45°?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
tan 45° = 1 (opposite = adjacent in a 45-45-90 triangle).
cos 60° = 1/2.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
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:
- For a positive sine value in $0^{\circ}\le x<360^{\circ}$, the second solution is $180^{\circ}-x$; for negative values use the lower-half graph.
- $\cos x = k$: 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)?
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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)?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
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 − ______ = ______°.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
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 $240^{\circ}$.
Negative trig values and tangent equations
- In $0^{\circ}\le x<360^{\circ}$, $\sin x=-1/2$ is negative below the horizontal axis: $x=210^{\circ},330^{\circ}$. A calculator value $-30^{\circ}$ must be replaced by its equivalent $330^{\circ}$ inside the required interval.
- Tangent repeats after $180^{\circ}$: $\tan x=1$ gives $x=45^{\circ},225^{\circ}$ in this interval. At $0$, $90$, $180$, $270$, $360^{\circ}$ check endpoints and undefined values rather than assuming every equation has exactly two solutions.
Find the smaller solution of sin x = −1/2 for 0° ≤ x < 360°.
Difficulty 2/5 · 140 coins + a small bonus on first completion today
Sine is negative in quadrants III and IV; the solutions are 210° and 330°.
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 $0^{\circ}$–$360^{\circ}$
Write from the other angle 150
- Solve sine of x equals 1/2 for 0 to 360 degrees.
- One solution is 30 degrees. Find the other.
Solve sine of x equals 1/2 for angles from 0 to 360 degrees. One answer is 30 degrees. What is the other?
180 minus 30 is 150.
What is the other solution?
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The second-quadrant partner of 30 is 150. Other options — 120 degrees (the lesson second solution); 30 degrees (the first solution repeated).