Skip to content
MathSolver Pro
  • Open the math solver
  • Ask the AI tutor
  • Find an AP course
  • Create a quiz
  • Open the calculator
  • Graphing Calculator
  • Browse all calculators
  • Read math guides
↑ ↓Enter to open · Esc to close
Solve a problem
Trigonometric equationsTrigonometryHigh schoolabout 1:38

Find arctan(-1) and every angle in [0°, 360°) with tan x = -1: animated lesson

A narrated, step-by-step trigonometric equations video lesson with captions. Final answer: arctan(-1) = -45° (-π/4); solutions in [0°, 360°): 135°, 315°; general: x = -45° + 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 (0.6 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    Find arctan(-1) and every angle in [0°, 360°) with tan x = -1.
  2. 2. Step by step
    1. arctan⁡(−1)=−45∘=−π4\operatorname{arctan}\left(-1\right) = -45^\circ = -\frac{\pi}{4}
      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=135∘, 315∘(3π4, 7π4)x = 135^\circ,\ 315^\circ \quad (\frac{3\pi}{4},\ \frac{7\pi}{4})
      Solutions in one full turn, 0° ≤ x < 360°
    4. x=−45∘+180∘nx = -45^\circ + 180^\circ n
      General solution, n any integer
  3. 3. The answer
    arctan⁡(−1)=−45∘=−π4,x=−45∘+180∘n\operatorname{arctan}\left(-1\right) = -45^\circ = -\frac{\pi}{4},\quad x = -45^\circ + 180^\circ n

    arctan(-1) = -45°

  4. 4. Check
    • Substituting each solution back: tan(135°) ≈ -1; tan(315°) ≈ -1.

Transcript

The problem. Here is the problem: Find arctan(negative 1) and every angle in [0°, 360°) with tan x equals negative 1. Find the angle whose tan is negative 1: the principal value of arctan, every solution of tan(x) equals negative 1 in one turn, and the general solution.

Step by step. Principal value of arctan, taken in the range (−pi/2, pi/2). negative 45° 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(negative 1) equals negative 45° (-pi/4); solutions in [0°, 360°): 135°, 315°; general: x equals negative 45° plus 180°n. An inverse trigonometric function returns one angle, the principal value. Because the trigonometric functions repeat, the equation tan(x) equals negative 1 has infinitely many solutions; the calculator lists those in one turn and the general pattern.

Check. Substituting each solution back: tan(135°) approximately negative 1; tan(315°) approximately negative 1.

About trigonometric equations

Solve trigonometric equations and give every solution.

More lessons like this

Find arctan(-1) and every angle in [0°, 360°) with tan x =…