Inverse Tangent Quiz: Inverse Tangent Fundamentals

pi/4 lies within the principal range of arctan. When the input angle is already in the principal range, arctan(tan(theta)) returns theta directly. tan(pi/4) = 1 and arctan(1) = pi/4. Option A gives pi/2, never returned by arctan. Option B gives -pi/4, the output for a negative input. Option C gives pi, outside the principal range.

tan(-pi/4) = -1 and -pi/4 lies within the principal range, so arctan(-1) = -pi/4. Option A gives pi/4, corresponding to arctan(1). Option C gives pi/3, whose tangent is positive sqrt(3). Option D gives -pi/3, whose tangent is -sqrt(3), not -1. This also follows from the odd function property since arctan(1) = pi/4 implies arctan(-1) = -pi/4.

arctan is defined for every real number with no domain restrictions, confirming A. Since arctan(-x) = -arctan(x) for all x, the function is odd, confirming B. Option C is false — the range is the open interval from -pi/2 to pi/2, not from -pi to pi. Option D is false — arctan is not periodic and has no repeating cycle.

tan(-pi/3) = -sqrt(3) and -pi/3 lies within the principal range, so arctan(-sqrt(3)) = -pi/3. Option A gives -pi/6, whose tangent is -1/sqrt(3). Option B gives -pi/4, whose tangent is -1. Option D gives -pi/2, which is a horizontal asymptote and never returned by arctan.

The answer is True. tan(-pi/4) = -1 and -pi/4 lies within the principal range. Both conditions of the arctan definition are satisfied. This also follows from the odd function property: since arctan(1) = pi/4, it follows that arctan(-1) = -pi/4 by symmetry.

The answer is True. arctan(x) returns the unique angle theta in the open interval from -pi/2 to pi/2 such that tan(theta) = x. Applying tan immediately recovers x. Because arctan is defined for every real number and tan undoes arctan exactly, this identity holds without exception across the entire real number line.

tan(x) is undefined at -pi/2 and pi/2 because cos(x) = 0 there, causing division by zero. Since arctan reverses tangent, it can never return an angle where tangent is undefined. The values -pi/2 and pi/2 are therefore excluded, making the principal range an open interval.

The range of arctan is the open interval from -pi/2 to pi/2. The endpoints are excluded because they are horizontal asymptotes that the graph approaches but never reaches. Option A is too wide. Option C describes the range of arccos. Option D is too narrow, covering only a quarter of the actual range.

The answer is True. Tangent is one-to-one on the open interval from -pi/2 to pi/2, allowing a proper inverse to be defined. The resulting arctan accepts every real number as input because tangent produces every real number as output on that interval. arctan returns the unique corresponding angle in the open interval from -pi/2 to pi/2.

tan(pi/6) = 1/sqrt(3) = sqrt(3)/3 and pi/6 lies within the principal range, so arctan(sqrt(3)/3) = pi/6. Option A gives pi/4, whose tangent is 1. Option B gives pi/3, whose tangent is sqrt(3). Option C gives pi/2, which is never returned by arctan since the range is open.

tan(pi/4) = 1 and pi/4 lies in the principal range, confirming A. tan(-pi/3) = -sqrt(3) and -pi/3 lies in the principal range, confirming B. Option C is false — arctan has no period at all, it is not a periodic function. Since arctan is an odd function, every negative input produces a negative output, confirming D.

tan(0) = 0 and 0 lies within the principal range, so arctan(0) = 0. Option A gives pi/4, the output of arctan(1). Option B gives -pi/4, the output of arctan(-1). Option D gives pi/2, which lies on the boundary of the open range and is never returned by arctan.

arctan returns the unique angle in the open principal range whose tangent equals x, confirming A and B. arctan accepts all real numbers as input, confirming D. Option C is false — the interval from 0 to pi describes the range of arccos, not arctan.

tan(pi/4) = 1 and pi/4 lies within the principal range, so arctan(1) = pi/4. Option A gives -pi/4, the output of arctan(-1). Option B gives pi/3, whose tangent is sqrt(3), not 1. Option C gives pi/2, which lies on the boundary of the open range and is never returned by arctan.

arctan accepts every real number as input with no domain restrictions, confirming A. The range is the open interval from -pi/2 to pi/2 where endpoints are approached but never reached, confirming B. Option C is false — arctan has no vertical asymptotes because it is defined for every real input. tan(pi/4) = 1 and pi/4 is in the principal range, confirming D.

arctan is the inverse of tangent by definition, confirming A. It returns the unique angle in the open principal range satisfying tan(theta) = x, confirming B. Since arctan is an odd function, negative inputs always produce negative outputs, confirming D. Option C is false because arctan(0) = 0 and all negative inputs give negative outputs.

The answer is True. The derivative of arctan(x) is 1 divided by (1 plus x squared), which is always positive for every real value of x. A consistently positive derivative confirms the function never decreases anywhere on its domain. Unlike sine and cosine, arctan never decreases, oscillates, or levels off anywhere on the real number line.

The tangent function produces every real number as output, so every real number is a valid input for arctan. Unlike arcsin and arccos, arctan has no domain restriction. Option A describes the domain of arcsin. Option B is the range of arctan, not the domain. Option D incorrectly excludes all negative inputs.

tan(-pi/3) = -sqrt(3) and -pi/3 lies within the principal range, so arctan(-sqrt(3)) = -pi/3. Option A gives pi/3, producing a positive tangent. Option B gives -pi/6, whose tangent is -1/sqrt(3). Option C gives pi/6, also producing a positive tangent. Only -pi/3 satisfies the definition.

The answer is True. By definition, arctan(x) returns the unique angle theta in the open interval from -pi/2 to pi/2 such that tan(theta) = x. Applying tan to that angle immediately recovers x. Since the domain of arctan is all real numbers, this composition holds without exception for every real value of x.