Exact values, graphs and trig equations · Valeurs exactes, graphiques et équations trigonométriques
| English | Français |
|---|---|
| exact values/eɡˈzækt ˈvæljuːz/ | valeurs exactes |
| trig graphs/trɪɡ ɡræfz/ | graphiques trigonométriques |
| asymptotes/ˈæsɪmptəʊts/ | asymptotes |
| symmetry/ˈsɪmətri/ | symétrie |
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 · indéfinie |
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 · Graphiques trig & équations
(cos θ, sin θ)
As θ turns, sin and · et cos trace their waves — and repeat every 360°. · Alors que θ tourne, sin et cos tracent leurs ondes — et se répètent toutes les 360°.
What is the exact value of sin 30°? (as a decimal) · Quelle est la valeur exacte de sin 30° ? (en décimal)
sin 30° = 1/2 = 0.5. · sin 30° = 1/2 = 0,5.
What is the exact value of tan 45°? · Quelle est la valeur exacte de tan 45° ?
tan 45° = 1 (opposite = adjacent in a 45-45-90 triangle). · tan 45° = 1 (opposé = adjacent dans un triangle 45-45-90).
cos 60° = 1/2.
cos 60° = 1/2 is one of the key exact values to memorise. · cos 60° = 1/2 est l'une des valeurs exactes clés à mémoriser.
Trig graphs 三角函数图
- $y = \sin x$ and · et $y = \cos x$ are waves oscillating between $-1$ and · et $1$.
- $y = \tan x$ repeats every $180^{\circ}$ and has vertical asymptotes 渐近线 at $90^{\circ}$ and · et $270^{\circ}$.
Trig graphs are waves — sine and cosine oscillate between $-1$ and · et $1$, 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 · ou $x = 180 - 60 = 120^{\circ}$. Don't forget the second answer.
Solving trig equations
- Find the first · premier 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)? · Résolvez sin x = √3/2 pour 0–360°. Une réponse est 60°. Quelle est l'autre (degrés) ?
The sine wave is symmetric about 90°: 180 − 60 = 120°. · La courbe sin est symétrique par rapport à 90° : 180 − 60 = 120°.
Solve 2cos x + 1 = 0 for 0–360°. One answer is 120°. What is the other (degrees)? · Résolvez 2cos x + 1 = 0 pour 0–360°. Une réponse est 120°. Quelle est l'autre (degrés) ?
cos x = −1/2 gives x = 120° or 360 − 120 = 240°. · cos x = −1/2 donne x = 120° ou 360 − 120 = 240°.
If sin x = k has one answer x = 40°, the second answer is 180 − ______ = ______°. · Si sin x = k a une solution x = 40°, la seconde solution est 180 − ______ = ______°.
The second sine solution is 180° − 40° = 140°. · La seconde solution sin est 180° − 40° = 140°.
Worked examples
- $\sin x = \dfrac{\sqrt{3}}{2} \Rightarrow x = 60^{\circ}$ or · ou $120^{\circ}$.
- $2\cos x + 1 = 0 \Rightarrow \cos x = -\dfrac{1}{2} \Rightarrow x = 120^{\circ}$ or · ou $240^{\circ}$.
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°. · Trouver la plus petite solution de sin x = −1/2 pour 0° ≤ x < 360°.
Sine is negative in quadrants III and IV; the solutions are 210° and 330°. · Le sinus est négatif dans les quadrants III et IV ; les solutions sont 210° et 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 · et $1$
- a trig equation usually has two answers in $0^{\circ}$–$360^{\circ}$