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Trigonometric equationsTrigonometryHigh schoolabout 1:10

Solve sin(3x) = -1: animated lesson

A narrated, step-by-step trigonometric equations video lesson with captions. Final answer: x = π/2 + 2πk/3 (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
    sin⁡(3 x)=−1\sin\left(3~ x\right) = -1
  2. 2. Steps, part 1
    1. sin⁡(3 x)=−1\sin\left(3~ x\right) = -1
      Start with the equation
    2. f(x)=sin⁡(3 x)+1=0f(x) = \sin\left(3~ x\right)+1 = 0
      Rewrite as f(x) = 0
    3. The left side repeats every 2π/3, so find the solutions in one period
    4. x=π2x = \frac{\pi}{2}
      Solutions in one period [0, 2π/3)
  3. 3. Steps, part 2
    1. x=π2+2πk3,k∈Zx = \frac{\pi}{2} + \frac{2\pi k}{3},\quad k \in \mathbb{Z}
      Add any whole number of periods
  4. 4. The answer
    x=π2+2πk3, k∈Zx = \frac{\pi}{2} + \frac{2\pi k}{3},\ k \in \mathbb{Z}

    x = π/2 + 2πk/3 (k any integer)

  5. 5. Check
    • x = 1.5707963: left side = -1, right side = -1 ✓

Transcript

The problem. Here is the problem: sin(3x) equals negative 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 2pi/3, so find the solutions in one period. f(x plus 2pi/3) equals f(x) for every x, so every solution is one of the solutions in [0, 2pi/3) plus a whole number of periods. Solutions in one period [0, 2pi/3).

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

The answer. So the answer is x equals pi/2 plus 2pik/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 1.5707963: left side equals negative 1, right side equals negative 1, which checks out.

About trigonometric equations

Solve trigonometric equations and give every solution.

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Solve sin(3x) = -1: Trigonometric equations Lesson