Principal Values: Defining Inverse Trig & Restricted Domains

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| Questions: 20 | Updated: Nov 10, 2025
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1) Which function has an inverse only if its domain is restricted to [−π/2, π/2]?

Explanation

sin(x) is not one-to-one over ℝ because it repeats every 2π.

Restricting to [−π/2, π/2] makes sin(x) strictly increasing and one-to-one, so it has an inverse, arcsin(x).

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Principal Values: Defining Inverse Trig & Restricted Domains - Quiz

Learn how inverse trigonometric functions are defined by carefully restricting domains to make them valid functions. This quiz focuses on identifying correct domain and range pairs for sine, cosine, and tangent, and recognizing which intervals make each one-to-one. Students explore key relationships such as sin(arcsin x) = x and arcsin(sin... see morex) = x (within the restricted range). It reinforces conceptual understanding of principal values and helps build fluency with inverse notation, valid intervals, and their geometric interpretation. see less

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2) Which restriction makes cosine one-to-one so it can have an inverse?

Explanation

cos(x) is strictly decreasing on [0, π].

Restricting the domain to this interval allows a unique inverse, arccos(x).

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3) The inverse of sin(x) is called:

Explanation

The standard name is arcsin(x) (also written sin⁻¹(x)).

It returns the angle θ ∈ [−π/2, π/2] with sin(θ) = x.

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4) What is the range of arcsin(x)?

Explanation

arcsin outputs principal angles from −π/2 to π/2, inclusive.

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5) What is the domain of arccos(x)?

Explanation

Only numbers between −1 and 1 can be cosines of real angles,

so arccos(x) accepts inputs in [−1, 1].

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6) The graph of y = arctan(x) has horizontal asymptotes at:

Explanation

As x → ±∞, arctan(x) → ±π/2.

These horizontal lines bound the outputs but are never reached.

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7) Which equation defines the relationship between sine and arcsine?

Explanation

For any x ∈ [−1, 1], sin(arcsin(x)) = x.

The reverse, arcsin(sin(x)) = x, only holds when x ∈ [−π/2, π/2].

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8) Which function has range (0, π)?

Explanation

arccos(x) returns angles from 0 to π (endpoints included).

Among the listed, it’s the inverse trig function with outputs spanning 0 to π.

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9) Which of the following is NOT defined for x = 2?

Explanation

arcsin(x) requires x ∈ [−1, 1], so arcsin(2) is undefined.

arctan(2), arccot(2), and arctan(−2) are defined.

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10) What is the principal range of arctan(x)?

Explanation

arctan outputs angles strictly between −π/2 and π/2.

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11) The equation arcsin(x) + arccos(x) = π/2 holds for:

Explanation

For any valid sine/cosine value x, the complementary angles satisfy

arcsin(x) + arccos(x) = π/2.

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12) If f(x) = sin(x) on [−π/2, π/2], then f⁻¹(x) =

Explanation

On that restricted domain, sin(x) is one-to-one, and its inverse is arcsin(x).

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13) What is the range of arccos(x)?

Explanation

arccos(x) returns angles from 0 to π, inclusive.

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14) The expression arctan(0) equals:

Explanation

tan(0) = 0, so arctan(0) = 0.

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15) Which of these pairs correctly matches function and inverse?

Explanation

With the proper domain restriction, sin(x) inverts to arcsin(x).

The other pairings mismatch functions.

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16) What is the inverse of tan(x) when its domain is restricted to (−π/2, π/2)?

Explanation

On (−π/2, π/2), tan(x) is one-to-one and its inverse is arctan(x).

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17) Arcsin(−1) equals:

Explanation

sin(−π/2) = −1, and −π/2 is in the arcsin range.

Thus arcsin(−1) = −π/2.

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18) What is the largest output of arctan(x)?

Explanation

arctan(x) approaches π/2 but never reaches it.

π/2 is the maximum limiting value (horizontal asymptote).

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19) Arccos(1/2) equals:

Explanation

cos(π/3) = 1/2, and π/3 ∈ [0, π], the arccos range.

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20) The inverse trig functions are all defined by restricting:

Explanation

Inverse functions must be one-to-one.

So we restrict each trig function’s domain to an interval where it’s injective.

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Which function has an inverse only if its domain is restricted to...
Which restriction makes cosine one-to-one so it can have an inverse?
The inverse of sin(x) is called:
What is the range of arcsin(x)?
What is the domain of arccos(x)?
The graph of y = arctan(x) has horizontal asymptotes at:
Which equation defines the relationship between sine and arcsine?
Which function has range (0, π)?
Which of the following is NOT defined for x = 2?
What is the principal range of arctan(x)?
The equation arcsin(x) + arccos(x) = π/2 holds for:
If f(x) = sin(x) on [−π/2, π/2], then f⁻¹(x) =
What is the range of arccos(x)?
The expression arctan(0) equals:
Which of these pairs correctly matches function and inverse?
What is the inverse of tan(x) when its domain is restricted to...
Arcsin(−1) equals:
What is the largest output of arctan(x)?
Arccos(1/2) equals:
The inverse trig functions are all defined by restricting:
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