Trigonometric Equations and Inequalities
Introduced| English | Meaning |
|---|---|
| period/ˈpɪərɪəd/ | The time taken for one complete oscillation or cycle. |
| inverse/ɪnˈvɜːs/ | A function that reverses another function's mapping on a suitable one-to-one domain. |
| sine/saɪn/ | For an acute angle in a right-angled triangle, the opposite side divided by the hypotenuse. |
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.
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.
To find a first solution of $\sin x = 0.5$, you use…
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The inverse gives one angle: $\arcsin(0.5) = 30°$ ($\tfrac{\pi}{6}$). Then use symmetry for the rest.
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.
Every crossing is a solution
y = sin x
The sine wave meets a horizontal target line again and again, so a trig equation has endlessly many solutions.
Over all real numbers, $\sin x = 0.5$ has…
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The wave meets the line $y = 0.5$ once every cycle, forever — infinitely many solutions.
To list every solution, add whole multiples of the ____ to a first solution.
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Because sine repeats, adding any multiple of its period ($2\pi$) gives another solution.
Select all true statements about trig equations.
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Sine is capped at $1$, so $\sin x = 5$ has no solution. The other three are correct.
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.
The equation $\sin x = 2$ has exactly one solution in $[0, 2\pi)$.
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False. Sine never exceeds $1$, so $\sin x = 2$ has no solution at all.
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.
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.
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 $k$.
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.
Write from two solutions
- Solve sine of x equals 0 on 0 less than or equal to x less than 2 pi.
- Use the period and the symmetry.
How many solutions does sine of x equals 0 have on the interval from 0 inclusive to 2 pi exclusive?
The sine wave crosses zero twice in that half-open interval.
How many solutions are in the interval?
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Zero and pi are inside. 2 pi is not.