WEBVTT

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Here is the problem: Find arcsin(0.3) and every angle in [0°, 360°) with sin x equals 0.3.

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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.

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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.

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Sine is also positive or negative in a second quadrant, so use x and 180° minus x.

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Solutions in one full turn, 0° less than or equal to x less than 360°.

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General solution, n any integer.

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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.

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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.

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