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Trigonometric equationsTrigonometryHigh schoolabout 1:11

Solve tan(2x) = 2: animated lesson

A narrated, step-by-step trigonometric equations video lesson with captions. Final answer: x ≈ 0.5535743589 + πk/2 (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 (0.5 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Solving an equation
$\tan\left(2~ x\right) = 2$
2. Steps, part 1
1. $\tan\left(2~ x\right) = 2$
  Start with the equation
2. $f(x) = \tan\left(2~ x\right)-2 = 0$
  Rewrite as f(x) = 0
3. The left side repeats every π/2, so find the solutions in one period
4. $x \approx 0.5535743589$
  Solutions in one period [0, π/2)
3. Steps, part 2
1. $x \approx 0.5535743589 + \frac{\pi k}{2},\quad k \in \mathbb{Z}$
  Add any whole number of periods
4. The answer
$x \approx 0.5535743589 + \frac{\pi k}{2},\ k \in \mathbb{Z}$
x ≈ 0.5535743589 + πk/2 (k any integer)
5. Check
- x = 0.55357436: left side = 2, right side = 2 ✓

Transcript

The problem. Here is the problem: tan(2x) equals 2. 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/2, so find the solutions in one period. f(x plus pi/2) equals f(x) for every x, so every solution is one of the solutions in [0, pi/2) plus a whole number of periods. Solutions in one period [0, pi/2).

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

The answer. So the answer is x approximately 0.5535743589 plus pik/2 (k any integer). Each solution in one period is a numerical approximation (about 10 significant digits) confirmed by substitution; adding whole periods gives all solutions.

Check. x equals 0.55357436: left side equals 2, right side equals 2, which checks out.

About trigonometric equations

Solve trigonometric equations and give every solution.

Solve tan(2x) = 2: Animated Step-by-Step Lesson 10