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Trigonometric equationsTrigonometryHigh schoolabout 1:09

Solve tan(3x) = 1: animated lesson

A narrated, step-by-step trigonometric equations video lesson with captions. Final answer: x = π/12 + πk/3 (k any integer).

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Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.4 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Solving an equation
$\tan\left(3~ x\right) = 1$
2. Steps, part 1
1. $\tan\left(3~ x\right) = 1$
  Start with the equation
2. $f(x) = \tan\left(3~ x\right)-1 = 0$
  Rewrite as f(x) = 0
3. The left side repeats every π/3, so find the solutions in one period
4. $x = \frac{\pi}{12}$
  Solutions in one period [0, π/3)
3. Steps, part 2
1. $x = \frac{\pi}{12} + \frac{\pi k}{3},\quad k \in \mathbb{Z}$
  Add any whole number of periods
4. The answer
$x = \frac{\pi}{12} + \frac{\pi k}{3},\ k \in \mathbb{Z}$
x = π/12 + πk/3 (k any integer)
5. Check
- x = 0.26179939: left side = 1, right side = 1 ✓

Transcript

The problem. Here is the problem: tan(3x) equals 1. A periodic equation in x: it has infinitely many solutions, one family for each solution in a single period.

Steps, part 1. Start with the equation. Rewrite as f(x) equals 0. The left side repeats every pi/3, so find the solutions in one period. f(x plus pi/3) equals f(x) for every x, so every solution is one of the solutions in [0, pi/3) plus a whole number of periods. Solutions in one period [0, pi/3).

Steps, part 2. Add any whole number of periods.

The answer. So the answer is x equals pi/12 plus pik/3 (k any integer). Each solution in one period was found numerically, matched to an exact value and confirmed by substitution; adding whole periods gives all solutions.

Check. x equals 0.26179939: left side equals 1, right side equals 1, which checks out.

About trigonometric equations

Solve trigonometric equations and give every solution.

Solve tan(3x) = 1: Animated Step-by-Step Lesson 10