Trigonometric Equations and Inequalities · 三角方程与不等式
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| period/ˈpɪərɪəd/ | 周期 | zhōu qī |
| inverse/ɪnˈvɜːs/ | 反函数 | fǎn hán shù |
| sine/saɪn/ | 正弦 | zhèng xián |
When does the wave hit a target?
- "At what times is the tide exactly $3$ m?" becomes a trig equation.
- Because the wave repeats, the answer is not one time but a whole rhythm of times.
- Solving means finding every angle where the function equals a target value.
- The unit circle and the period together deliver them all.
波什么时候击中目标?
- "潮汐在什么时刻恰好是 $3$ 米?"就变成一个三角方程。
- 因为波重复,答案不是一个时刻,而是一整串有节律的时刻。
- 求解意味着找出函数等于某个目标值的每一个角。
- 单位圆和周期一起,把它们全都交出来。
Finding the first solution
- To solve $\sin x = 0.5$, first find one angle with the inverse 反函数: $\arcsin(0.5) = \tfrac{\pi}{6}$.
- That gives a single "starter" solution inside the restricted range.
- For cosine or tangent, use $\arccos$ or $\arctan$ the same way.
- This first angle is the seed from which all others grow.
找第一个解
- 要解 $\sin x = 0.5$,先用反函数(inverse)找一个角:$\arcsin(0.5) = \tfrac{\pi}{6}$。
- 这给出受限范围内一个"起始"解。
- 对余弦或正切,用 $\arccos$ 或 $\arctan$ 同样处理。
- 这第一个角是其余所有解生长出来的种子。
To find a first solution of $\sin x = 0.5$, you use… · 要找 $\sin x = 0.5$ 的第一个解,你用……
The inverse · 反函数 gives one angle: $\arcsin(0.5) = 30°$ ($\tfrac{\pi}{6}$). Then use symmetry for the rest. · 反函数给出一个角:$\arcsin(0.5) = 30°$($\tfrac{\pi}{6}$)。然后用对称性找其余的。
Then find them all
- The sine 正弦 wave meets a horizontal target line once every cycle, forever.
- So a trig equation usually has infinitely many solutions.
- Use the symmetry of the graph to find the second solution within one period.
- Then add whole multiples of the period 周期 to generate the endless rest.
然后找出全部
- 正弦(sine)波每个循环都与一条水平目标线相交一次,永远如此。
- 所以一个三角方程通常有无穷多个解。
- 利用图像的对称性,在一个周期内找到第二个解。
- 然后加上周期(period)的整数倍,生成无尽的其余解。
Every crossing is a solution · 每一个交点都是一个解
y = sin x
The sine wave meets a horizontal target line again and again, so a trig equation has endlessly many solutions. · 正弦波一次次地与一条水平目标线相交,所以三角方程有无穷多个解。
Over all real numbers, $\sin x = 0.5$ has… · 在全体实数上,$\sin x = 0.5$ 有……
The wave meets the line $y = 0.5$ once every cycle, forever — infinitely many solutions. · 波每个循环都与直线 $y = 0.5$ 相交一次,永远如此——无穷多个解。
To list every solution, add whole multiples of the ____ to a first solution. · 要列出每一个解,在第一个解上加上____的整数倍。
Because sine repeats, adding any multiple of its period · 周期 ($2\pi$) gives another solution. · 因为正弦重复,加上它的周期($2\pi$)的任意倍数就得到另一个解。
Select all · 所有 true statements about trig equations. · 选出关于三角方程的所有正确说法。
Sine is capped at $1$, so $\sin x = 5$ has no solution. The other three are correct. · 正弦上限是 $1$,所以 $\sin x = 5$ 没有解。其余三条正确。
Inequalities
- A trig inequality like $\sin x > 0.5$ asks for the intervals where the wave is above a line.
- Solve the matching equation first to find the boundary angles.
- Then read off which arcs of the wave satisfy the inequality.
- The answer is a repeating set of intervals, one per period.
不等式
- 像 $\sin x > 0.5$ 这样的三角不等式,求的是波在一条线上方的那些区间。
- 先解对应的方程,找到边界角。
- 然后读出波的哪些弧段满足不等式。
- 答案是一组重复的区间,每个周期一组。
The equation · 方程 $\sin x = 2$ has no solution. · 方程 $\sin x = 2$ 没有解。
Sine never exceeds $1$, so $\sin x = 2$ can never happen — no solution exists. · 正弦从不超过 $1$,所以 $\sin x = 2$ 永远不可能——没有解。
When there is no solution
- Sine and cosine never leave $[-1, 1]$, so $\sin x = 2$ has no solution at all.
- Always check that the target value is reachable before solving.
- A tangent target, by contrast, is always reachable, since tangent is unbounded.
- Knowing each function's range saves you from chasing impossible answers.
当没有解时
- 正弦和余弦从不离开 $[-1, 1]$,所以 $\sin x = 2$ 根本没有解。
- 求解之前,永远要检查目标值是否可达。
- 相比之下,正切的目标总是可达的,因为正切是无界的。
- 知道每个函数的值域,能让你不去追逐不可能的答案。
Do not report just the calculator's one answer. $\arcsin(0.5)$ gives $\tfrac{\pi}{6}$, but $\sin x = 0.5$ also holds at $\tfrac{5\pi}{6}$ and at every $+2\pi$ beyond both. A trig equation almost always has more solutions than the inverse alone reveals.
不要只报告计算器给的那一个答案。$\arcsin(0.5)$ 给出 $\tfrac{\pi}{6}$,但 $\sin x = 0.5$ 在 $\tfrac{5\pi}{6}$ 处也成立,并在两者之外每 $+2\pi$ 处都成立。三角方程几乎总是比单靠反函数揭示的解要多。
Solve $\cos x = 0$ for all $x$.
- One solution: $\arccos(0) = \tfrac{\pi}{2}$.
- Cosine is also $0$ at $\tfrac{3\pi}{2}$ — that is $\tfrac{\pi}{2} + \pi$.
- So the full solution is $x = \tfrac{\pi}{2} + k\pi$ for every integer $k$.
对所有 $x$ 求解 $\cos x = 0$。
- 一个解:$\arccos(0) = \tfrac{\pi}{2}$。
- 余弦在 $\tfrac{3\pi}{2}$ 处也是 $0$——也就是 $\tfrac{\pi}{2} + \pi$。
- 所以完整的解是 $x = \tfrac{\pi}{2} + k\pi$,对每个整数 $k$。
To solve a trig equation, use the inverse for a first angle, then the graph's symmetry and the period to list every solution — often infinitely many. Check the target is within the function's range first (sine and cosine cannot exceed $1$). Inequalities give repeating intervals.
要解一个三角方程,用反函数找第一个角,再用图像的对称性和周期列出每一个解——通常是无穷多个。先检查目标在函数的值域内(正弦和余弦不能超过 $1$)。不等式给出重复的区间。