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Trigonometric equationsTrigonometryHigh schoolabout 2:10

Find arccos(0.25) and every angle in [0°, 360°) with cos x = 0.25: animated lesson

A narrated, step-by-step trigonometric equations video lesson with captions. Final answer: arccos(0.25) = 75.52248781° (1.318116072 rad); solutions in [0°, 360°): 75.52248781°, 284.4775122°; general: x = ±75.52248781° + 360°n.

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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)

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1. The problem
Worked example
Find arccos(0.25) and every angle in [0°, 360°) with cos x = 0.25.
2. Step by step
1. $\operatorname{arccos}\left(0.25\right) = 75.52248781^\circ = 1.318116072\text{ rad}$
  Principal value of arccos, taken in the range [0, π]
2. $\cos(-x) = \cos x$
  Cosine is even, so x and −x (that is, 360° − x) give the same value
3. $x = 75.52248781^\circ,\ 284.4775122^\circ \quad (1.318116072,\ 4.965069236)$
  Solutions in one full turn, 0° ≤ x < 360°
4. $x = \pm 75.52248781^\circ + 360^\circ n$
  General solution, n any integer
3. The answer
$\operatorname{arccos}\left(0.25\right) = 75.52248781^\circ \approx 1.318116072,\quad x = \pm 75.52248781^\circ + 360^\circ n$
arccos(0.25) = 75.52248781°
4. Check
- Substituting each solution back: cos(75.52248781°) ≈ 0.25; cos(284.4775122°) ≈ 0.25.

Transcript

The problem. Here is the problem: Find arccos(0.25) and every angle in [0°, 360°) with cos x equals 0.25. Find the angle whose cos is 0.25: the principal value of arccos, every solution of cos(x) equals 0.25 in one turn, and the general solution.

Step by step. Principal value of arccos, taken in the range [0, pi]. The value is not one of the standard exact values, so the angle is given as a decimal. 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°. General solution, n any integer.

The answer. So the answer is arccos(0.25) equals 75.52248781° (1.318116072 rad); solutions in [0°, 360°): 75.52248781°, 284.4775122°; general: x equals plus or minus 75.52248781° plus 360°n. An inverse trigonometric function returns one angle, the principal value. Because the trigonometric functions repeat, the equation cos(x) equals 0.25 has infinitely many solutions; the calculator lists those in one turn and the general pattern.

Check. Substituting each solution back: cos(75.52248781°) approximately 0.25; cos(284.4775122°) approximately 0.25.

About trigonometric equations

Solve trigonometric equations and give every solution.

Find arccos(0.25) and every angle in [0°, 360°) with cos x…