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Trigonometric equationsTrigonometryHigh schoolabout 1:43

Find arcsec(-2) and every angle in [0°, 360°) with sec x = -2: animated lesson

A narrated, step-by-step trigonometric equations video lesson with captions. Final answer: arcsec(-2) = 120° (2π/3); solutions in [0°, 360°): 120°, 240°; general: x = ±120° + 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 arcsec(-2) and every angle in [0°, 360°) with sec x = -2.
  2. 2. Steps, part 1
    1. sec⁡x=−2  ⟺  cos⁡x=1−2≈−0.5\sec x = -2 \iff \cos x = \frac{1}{-2} \approx -0.5
      Rewrite sec as a reciprocal
    2. arcsec⁡(−2)=120∘=2π3\operatorname{arcsec}\left(-2\right) = 120^\circ = \frac{2\pi}{3}
      Principal value of arcsec, taken in the range [0, π], excluding π/2
    3. cos⁡(−x)=cos⁡x\cos(-x) = \cos x
      Cosine is even, so x and −x (that is, 360° − x) give the same value
    4. x=120∘, 240∘(2π3, 4π3)x = 120^\circ,\ 240^\circ \quad (\frac{2\pi}{3},\ \frac{4\pi}{3})
      Solutions in one full turn, 0° ≤ x < 360°
  3. 3. Steps, part 2
    1. x=±120∘+360∘nx = \pm 120^\circ + 360^\circ n
      General solution, n any integer
  4. 4. The answer
    arcsec⁡(−2)=120∘=2π3,x=±120∘+360∘n\operatorname{arcsec}\left(-2\right) = 120^\circ = \frac{2\pi}{3},\quad x = \pm 120^\circ + 360^\circ n

    arcsec(-2) = 120°

  5. 5. Check
    • Substituting each solution back: sec(120°) ≈ -2; sec(240°) ≈ -2.

Transcript

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

Steps, part 1. Rewrite sec as a reciprocal. Principal value of arcsec, taken in the range [0, pi], excluding pi/2. 120° is a special angle, so the answer is exact. Cosine is even, so x and negative x (that is, 360° minus x) give the same value. Solutions in one full turn, 0° less than or equal to x less than 360°.

Steps, part 2. General solution, n any integer.

The answer. So the answer is arcsec(negative 2) equals 120° (2pi/3); solutions in [0°, 360°): 120°, 240°; general: x equals plus or minus 120° plus 360°n. An inverse trigonometric function returns one angle, the principal value. Because the trigonometric functions repeat, the equation sec(x) equals negative 2 has infinitely many solutions; the calculator lists those in one turn and the general pattern.

Check. Substituting each solution back: sec(120°) approximately negative 2; sec(240°) approximately negative 2.

About trigonometric equations

Solve trigonometric equations and give every solution.

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