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Trigonometric equationsTrigonometryHigh schoolabout 1:07

Solve tan(2x) = 1: animated lesson

A narrated, step-by-step trigonometric equations video lesson with captions. Final answer: x = π/8 + πk/2 (k any integer).

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. 1. The problem
    Solving an equation
    tan⁡(2 x)=1\tan\left(2~ x\right) = 1
  2. 2. Steps, part 1
    1. tan⁡(2 x)=1\tan\left(2~ x\right) = 1
      Start with the equation
    2. f(x)=tan⁡(2 x)−1=0f(x) = \tan\left(2~ x\right)-1 = 0
      Rewrite as f(x) = 0
    3. The left side repeats every π/2, so find the solutions in one period
    4. x=π8x = \frac{\pi}{8}
      Solutions in one period [0, π/2)
  3. 3. Steps, part 2
    1. x=π8+πk2,k∈Zx = \frac{\pi}{8} + \frac{\pi k}{2},\quad k \in \mathbb{Z}
      Add any whole number of periods
  4. 4. The answer
    x=π8+πk2, k∈Zx = \frac{\pi}{8} + \frac{\pi k}{2},\ k \in \mathbb{Z}

    x = π/8 + πk/2 (k any integer)

  5. 5. Check
    • x = 0.39269908: left side = 1, right side = 1 ✓

Transcript

The problem. Here is the problem: tan(2x) 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/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 equals pi/8 plus pik/2 (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 0.39269908: left side equals 1, right side equals 1, which checks out.

About trigonometric equations

Solve trigonometric equations and give every solution.

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Solve tan(2x) = 1: Animated Step-by-Step Lesson 10