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Trigonometric equationsTrigonometryHigh schoolabout 2:09

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

A narrated, step-by-step trigonometric equations video lesson with captions. Final answer: arcsin(-√2/2) = -45° (-π/4); solutions in [0°, 360°): 225°, 315°; general: x = -45° + 360°n or x = 225° + 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(-√2/2) and every angle in [0°, 360°) with sin x = -√2/2.
  2. 2. Step by step
    1. arcsin⁡(−2/2)=−45∘=−π4\operatorname{arcsin}\left(-\sqrt{2}/2\right) = -45^\circ = -\frac{\pi}{4}
      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=225∘, 315∘(5π4, 7π4)x = 225^\circ,\ 315^\circ \quad (\frac{5\pi}{4},\ \frac{7\pi}{4})
      Solutions in one full turn, 0° ≤ x < 360°
    4. x=−45∘+360∘n or x=225∘+360∘nx = -45^\circ + 360^\circ n \ \text{or}\ x = 225^\circ + 360^\circ n
      General solution, n any integer
  3. 3. The answer
    arcsin⁡(−2/2)=−45∘=−π4,x=−45∘+360∘n or x=225∘+360∘n\operatorname{arcsin}\left(-\sqrt{2}/2\right) = -45^\circ = -\frac{\pi}{4},\quad x = -45^\circ + 360^\circ n \ \text{or}\ x = 225^\circ + 360^\circ n

    arcsin(-√2/2) = -45°

  4. 4. Check
    • Substituting each solution back: sin(225°) ≈ -0.7071067812; sin(315°) ≈ -0.7071067812.

Transcript

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

Step by step. Principal value of arcsin, taken in the range [−pi/2, pi/2]. negative 45° 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(negative the square root of 2 over 2 equals negative 45° (-pi/4); solutions in [0°, 360°): 225°, 315°; general: x equals negative 45° plus 360°n or x equals 225° plus 360°n. An inverse trigonometric function returns one angle, the principal value. Because the trigonometric functions repeat, the equation sin(x) equals negative the square root of 2 over 2 has infinitely many solutions; the calculator lists those in one turn and the general pattern.

Check. Substituting each solution back: sin(225°) approximately negative 0.7071067812; sin(315°) approximately negative 0.7071067812.

About trigonometric equations

Solve trigonometric equations and give every solution.

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