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Trigonometric equationsTrigonometryHigh schoolabout 1:41

Find arcsin(1/2) and every angle in [0°, 360°) with sin x = 1/2: animated lesson

A narrated, step-by-step trigonometric equations video lesson with captions. Final answer: arcsin(1/2) = 30° (π/6); solutions in [0°, 360°): 30°, 150°; general: x = 30° + 360°n or x = 150° + 360°n.

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 (0.7 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    Find arcsin(1/2) and every angle in [0°, 360°) with sin x = 1/2.
  2. 2. Step by step
    1. arcsin⁡(1/2)=30∘=π6\operatorname{arcsin}\left(1/2\right) = 30^\circ = \frac{\pi}{6}
      Principal value of arcsin, taken in the range [−π/2, π/2]
    2. sin⁡(180∘−x)=sin⁡x\sin(180^\circ - x) = \sin x
      Sine is also positive or negative in a second quadrant, so use x and 180° − x
    3. x=30∘, 150∘(π6, 5π6)x = 30^\circ,\ 150^\circ \quad (\frac{\pi}{6},\ \frac{5\pi}{6})
      Solutions in one full turn, 0° ≤ x < 360°
    4. x=30∘+360∘n or x=150∘+360∘nx = 30^\circ + 360^\circ n \ \text{or}\ x = 150^\circ + 360^\circ n
      General solution, n any integer
  3. 3. The answer
    arcsin⁡(1/2)=30∘=π6,x=30∘+360∘n or x=150∘+360∘n\operatorname{arcsin}\left(1/2\right) = 30^\circ = \frac{\pi}{6},\quad x = 30^\circ + 360^\circ n \ \text{or}\ x = 150^\circ + 360^\circ n

    arcsin(1/2) = 30°

  4. 4. Check
    • Substituting each solution back: sin(30°) ≈ 0.5; sin(150°) ≈ 0.5.

Transcript

The problem. Here is the problem: Find arcsin(1 over 2) and every angle in [0°, 360°) with sin x equals 1 over 2. Find the angle whose sin is 1 over 2: the principal value of arcsin, every solution of sin(x) equals 1 over 2 in one turn, and the general solution.

Step by step. Principal value of arcsin, taken in the range [−pi/2, pi/2]. 30° is a special angle, so the answer is exact. Sine is also positive or negative in a second quadrant, so use x and 180° minus x. Solutions in one full turn, 0° less than or equal to x less than 360°. General solution, n any integer.

The answer. So the answer is arcsin(1 over 2) equals 30° (pi/6); solutions in [0°, 360°): 30°, 150°; general: x equals 30° plus 360°n or x equals 150° plus 360°n. An inverse trigonometric function returns one angle, the principal value. Because the trigonometric functions repeat, the equation sin(x) equals 1 over 2 has infinitely many solutions; the calculator lists those in one turn and the general pattern.

Check. Substituting each solution back: sin(30°) approximately 0.5; sin(150°) approximately 0.5.

About trigonometric equations

Solve trigonometric equations and give every solution.

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