Skip to content
MathSolver Pro
  • Open the math solver
  • Ask the AI tutor
  • Find an AP course
  • Create a quiz
  • Open the calculator
  • Graphing Calculator
  • Browse all calculators
  • Read math guides
↑ ↓Enter to open · Esc to close
Solve a problem
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/6 + 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=π6x = \frac{\pi}{6}
      Solutions in one period [0, 2π/3)
  3. 3. Steps, part 2
    1. x=π6+2πk3,k∈Zx = \frac{\pi}{6} + \frac{2\pi k}{3},\quad k \in \mathbb{Z}
      Add any whole number of periods
  4. 4. The answer
    x=π6+2πk3, k∈Zx = \frac{\pi}{6} + \frac{2\pi k}{3},\ k \in \mathbb{Z}

    x = π/6 + 2πk/3 (k any integer)

  5. 5. Check
    • x = 0.52359878: left side = 1, right side = 1 ✓

Transcript

The problem. Here is the problem: sin(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 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/6 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 0.52359878: left side equals 1, right side equals 1, which checks out.

About trigonometric equations

Solve trigonometric equations and give every solution.

More lessons like this

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