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Trigonometric equationsTrigonometryHigh schoolabout 2:12

Find arcsin(0.3) and every angle in [0°, 360°) with sin x = 0.3: animated lesson

A narrated, step-by-step trigonometric equations video lesson with captions. Final answer: arcsin(0.3) = 17.45760312° (0.304692654 rad); solutions in [0°, 360°): 17.45760312°, 162.5423969°; general: x = 17.45760312° + 360°n or x = 162.5423969° + 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 (0.8 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    Find arcsin(0.3) and every angle in [0°, 360°) with sin x = 0.3.
  2. 2. Step by step
    1. arcsin⁡(0.3)=17.45760312∘=0.304692654 rad\operatorname{arcsin}\left(0.3\right) = 17.45760312^\circ = 0.304692654\text{ rad}
      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=17.45760312∘, 162.5423969∘(0.304692654, 2.8369)x = 17.45760312^\circ,\ 162.5423969^\circ \quad (0.304692654,\ 2.8369)
      Solutions in one full turn, 0° ≤ x < 360°
    4. x=17.45760312∘+360∘n or x=162.5423969∘+360∘nx = 17.45760312^\circ + 360^\circ n \ \text{or}\ x = 162.5423969^\circ + 360^\circ n
      General solution, n any integer
  3. 3. The answer
    arcsin⁡(0.3)=17.45760312∘≈0.304692654,x=17.45760312∘+360∘n or x=162.5423969∘+360∘n\operatorname{arcsin}\left(0.3\right) = 17.45760312^\circ \approx 0.304692654,\quad x = 17.45760312^\circ + 360^\circ n \ \text{or}\ x = 162.5423969^\circ + 360^\circ n

    arcsin(0.3) = 17.45760312°

  4. 4. Check
    • Substituting each solution back: sin(17.45760312°) ≈ 0.3; sin(162.5423969°) ≈ 0.3.

Transcript

The problem. Here is the problem: Find arcsin(0.3) and every angle in [0°, 360°) with sin x equals 0.3. Find the angle whose sin is 0.3: the principal value of arcsin, every solution of sin(x) equals 0.3 in one turn, and the general solution.

Step by step. Principal value of arcsin, taken in the range [−pi/2, pi/2]. The value is not one of the standard exact values, so the angle is given as a decimal. 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(0.3) equals 17.45760312° (0.304692654 rad); solutions in [0°, 360°): 17.45760312°, 162.5423969°; general: x equals 17.45760312° plus 360°n or x equals 162.5423969° plus 360°n. An inverse trigonometric function returns one angle, the principal value. Because the trigonometric functions repeat, the equation sin(x) equals 0.3 has infinitely many solutions; the calculator lists those in one turn and the general pattern.

Check. Substituting each solution back: sin(17.45760312°) approximately 0.3; sin(162.5423969°) approximately 0.3.

About trigonometric equations

Solve trigonometric equations and give every solution.

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