DOMAIN AND RANGE FOR TRIG GRAPHS AND INVERSE TRIG GRAPHS AND EXPLANATION

CHEAT SHEET FOR GRAPHING TRIGS

THE CHART

Function	Domain	Range
(\sin x)	((-\infty,\infty))	([-1,1])
(\cos x)	((-\infty,\infty))	([-1,1])
(\tan x)	(x\neq \frac{\pi}{2}+k\pi)	((-\infty,\infty))
(\csc x)	(x\neq k\pi)	((-\infty,-1]\cup[1,\infty))
(\sec x)	(x\neq \frac{\pi}{2}+k\pi)	((-\infty,-1]\cup[1,\infty))
(\cot x)	(x\neq k\pi)	((-\infty,\infty))
(\arcsin x)	([-1,1])	([-\frac{\pi}{2},\frac{\pi}{2}])
(\arccos x)	([-1,1])	([0,\pi])
(\arctan x)	((-\infty,\infty))	((-\frac{\pi}{2},\frac{\pi}{2}))
(\arccsc x)	((-\infty,-1]\cup[1,\infty))	([-\frac{\pi}{2},0)\cup(0,\frac{\pi}{2}])
(\arcsec x)	((-\infty,-1]\cup[1,\infty))	([0,\frac{\pi}{2})\cup(\frac{\pi}{2},\pi])
(\arccot x)	((-\infty,\infty))	((0,\pi))

Sinx explanation

1. sin⁡x

• Domain: All real numbers, because sine is defined for every angle.

Domain: (−∞,∞)

• Range: Sine outputs values on the y-axis from −1 to 1.

Range: [−1,1]How to see it: On the unit circle, sine = y-coordinate. y always between −1 and 1.

Cosx explanation

• Domain: All real numbers (cosine exists for every angle).

• Range: Cosine = x-coordinate on the unit circle. Always between −1 and 1.

Domain: (−∞,∞),Range: [−1,1]

Tanx explanation

3. tan⁡x

• Definition: tan⁡x=sin⁡x/cos⁡x

• Domain: Cosine cannot be zero (division by zero). Cos = 0 at x=π/2

Domain: x≠π/2+nπ

• Range: Tangent can take any real number, so (−∞,∞)

Visual tip: Vertical asymptotes where cos = 0.

CSCx explanation

4. csc⁡x

• Definition: csc⁡x=1/sin⁡x

• Domain: Cannot divide by 0 → sin x ≠ 0 → x≠nπ

• Range: Reciprocal of sine → values outside [−1,1] Range: (−∞,−1]∪[1,∞)\

SECX

5. sec⁡x

• Definition: sec⁡x=1/cos⁡x​

• Domain: Cosine ≠ 0 → x≠π/2+nπ

• Range: Reciprocal of cosine → (−∞,−1]∪[1,∞)

Tip: Secant “bounces” above and below y=1 and y=-1 with vertical asymptotes where cos=0.

COTX

6. cot⁡x

• Definition: cot⁡x=cos⁡x/sin⁡x

• Domain: Sin ≠ 0 → x≠nπ

• Range: Any real number → (−∞,∞)(-\infty, \infty)(−∞,∞)

ARCSINX explanation

7. arcsin⁡x\

• Definition: Inverse of sine. arcsin⁡y=angle whose sine = y

• Domain: Input must be valid sine value → [−1,1]

• Range: Output angles restricted to principal values → [−π/2,π/2]

Tip: Arcsin is the “undo” of sine. Range chosen so it’s a function (one output per input).

ARCCOSX explanation

• Inverse of cosine

• Domain: x-values must be in [-1,1]

• Range: Angles in [0, π] (Quadrant I & II)

Tip: Ensures one output per input (function).

ARCTANX explanation

9. arctan⁡x

• Inverse of tangent

• Domain: Any real number (tan can output anything)

• Range: Angles in (−π/2,π/2)

Tip: Tan never actually reaches ±π/2 → open interval.

CSCX explanation

10. arccscx

• Inverse of cosecant

• Domain: Must be outside [-1,1] → x≤−1 or x≥1x

• Range: Angles in Quadrants I & IV, avoid 0 → [−π/2,0)∪(0,π/2]

SECX explanation

• Inverse of secant

• Domain: x ≤ -1 or x ≥ 1

• Range: Angles in [0, π], excluding π/2 → [0,π/2)∪(π/2,π)

ARCCOTX explanation

• Inverse of cotangent

• Domain: Any real number (cot can output any number)

• Range: Angles in (0, π) → one output per input