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Trigonometric equationsTrigonometryHigh schoolabout 1:51

Find arctan(√3) and every angle in [0°, 360°) with tan x = √3: animated lesson

A narrated, step-by-step trigonometric equations video lesson with captions. Final answer: arctan(√3) = π/3 (60°); solutions in [0°, 360°): π/3, 4π/3; general: x = 60° + 180°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 arctan(√3) and every angle in [0°, 360°) with tan x = √3.
  2. 2. Step by step
    1. arctan⁡(3)=60∘=π3\operatorname{arctan}\left(\sqrt{3}\right) = 60^\circ = \frac{\pi}{3}
      Principal value of arctan, taken in the range (−π/2, π/2)
    2. tan⁡(x+180∘)=tan⁡x\tan(x + 180^\circ) = \tan x
      tan repeats every 180°, so add multiples of 180°
    3. x=60∘, 240∘(π3, 4π3)x = 60^\circ,\ 240^\circ \quad (\frac{\pi}{3},\ \frac{4\pi}{3})
      Solutions in one full turn, 0° ≤ x < 360°
    4. x=60∘+180∘nx = 60^\circ + 180^\circ n
      General solution, n any integer
  3. 3. The answer
    arctan⁡(3)=60∘=π3,x=60∘+180∘n\operatorname{arctan}\left(\sqrt{3}\right) = 60^\circ = \frac{\pi}{3},\quad x = 60^\circ + 180^\circ n

    arctan(√3) = 60°

  4. 4. Check
    • Substituting each solution back: tan(60°) ≈ 1.732050808; tan(240°) ≈ 1.732050808.

Transcript

The problem. Here is the problem: Find arctan(the square root of 3 and every angle in [0°, 360°) with tan x equals the square root of 3. Find the angle whose tan is the square root of 3: the principal value of arctan, every solution of tan(x) equals the square root of 3 in one turn, and the general solution.

Step by step. Principal value of arctan, taken in the range (−pi/2, pi/2). 60° is a special angle, so the answer is exact. tan repeats every 180°, so add multiples of 180°. 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 arctan(the square root of 3 equals pi/3 (60°); solutions in [0°, 360°): pi/3, 4pi/3; general: x equals 60° plus 180°n. An inverse trigonometric function returns one angle, the principal value. Because the trigonometric functions repeat, the equation tan(x) equals the square root of 3 has infinitely many solutions; the calculator lists those in one turn and the general pattern.

Check. Substituting each solution back: tan(60°) approximately 1.732050808; tan(240°) approximately 1.732050808.

About trigonometric equations

Solve trigonometric equations and give every solution.

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