Arcsin Values Quiz: Evaluating Common Arcsin Values

  • 11th Grade
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Quizzes Created: 8156 | Total Attempts: 9,588,805
| Attempts: 11 | Questions: 20 | Updated: Dec 17, 2025
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Question 1 / 21
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1) State the principal range of arcsin(x): ____.

Explanation

By definition, arcsin maps [−1,1] to angles between −π/2 and π/2 inclusive.

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About This Quiz
Arcsin Values Quiz: Evaluating Common Arcsin Values - Quiz

How can you evaluate common arcsin values with confidence? In this quiz, you’ll work through problems that focus on interpreting angles, recognizing familiar ratios, and connecting arcsin outputs to the unit circle. You’ll practice identifying principal angles, avoiding common pitfalls, and using symmetry to check your reasoning. Each question helps... see morestrengthen your comfort with inverse trig evaluation, building fluency with values that frequently appear across trigonometry, geometry, and applied problem-solving contexts.
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2) Select all true statements about arcsin compositions and values.

Explanation

The first four are standard identities and exact values. arcsin(2) has no real value since 2 is outside the domain.

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3) Arcsin is an odd function: arcsin(−x) = −arcsin(x) for all x in [−1, 1].

Explanation

Sine is odd and arcsin is its inverse on a symmetric interval about 0, so arcsin(−x)=−arcsin(x).

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4) Compute sin(arcsin(√3/2)) = ____.

Explanation

Let θ = arcsin(√3/2) ∈ [−π/2, π/2]. Then by definition sinθ = √3/2, so sin(arcsin(√3/2)) = √3/2.

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5) Arcsin(x) is strictly increasing on [−1, 1].

Explanation

Its derivative is d/dx[arcsin x] = 1/√(1−x^2) > 0 for x∈(−1,1), so arcsin is strictly increasing on the closed interval.

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6) Which values of x have negative principal angles under arcsin? Select all that apply.

Explanation

For negative inputs in [−1,0), arcsin(x) is negative: arcsin(−1/2)=−π/6, arcsin(−√2/2)=−π/4, arcsin(−√3/2)=−π/3. arcsin(0)=0 and arcsin(1/2)>0.

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7) Evaluate arcsin(0) = ____.

Explanation

Since sin(0)=0 and 0 is in the principal range, arcsin(0)=0.

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8) Evaluate arcsin(√2/2).

Explanation

sin(π/4)=√2/2 and π/4 is in [−π/2, π/2], so arcsin(√2/2)=π/4.

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9) Evaluate arcsin(−√2/2).

Explanation

sin(−π/4)=−√2/2 and −π/4 belongs to [−π/2, π/2], hence arcsin(−√2/2)=−π/4.

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10) The equation arcsin(1.2) is undefined over the reals.

Explanation

Because the domain of arcsin is [−1,1], inputs with |x|>1 have no real output.

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11) Arcsin(√2/2) = 3π/4.

Explanation

3π/4 has sine √2/2, but it is not in the principal range. arcsin returns π/4 instead, the value in [−π/2, π/2].

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12) Arcsin(0) = 0.

Explanation

0 is in the range of sine and in the principal interval. Because sin(0)=0 and 0∈[−π/2, π/2], arcsin(0)=0.

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13) Evaluate arcsin(sin(5π/6)).

Explanation

sin(5π/6)=1/2. arcsin returns the principal angle with that sine, which is π/6, not 5π/6.

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14) Evaluate arcsin(1). Choose the exact principal value.

Explanation

arcsin returns the angle θ in [−π/2, π/2] with sinθ = 1. Since sin(π/2)=1 and π/2 lies in the principal range, arcsin(1)=π/2.

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15) Select all correct exact evaluations.

Explanation

All listed except E hold within the principal range. Since sin(π)=0 but π is not in [−π/2, π/2], arcsin(0)=0, not π.

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16) Arcsin(−1) = ____.

Explanation

sin(−π/2)=−1 and −π/2 is within [−π/2, π/2], so arcsin(−1)=−π/2.

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17) Select all x for which arcsin(x) is real-valued.

Explanation

The domain of arcsin is [−1,1]. Values −1, −√2/2, and 0 are within this interval. 5/4 and −3/2 lie outside, so arcsin is not real there.

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18) Evaluate arcsin(−1/2).

Explanation

sin(−π/6)=−1/2 and −π/6 is in [−π/2, π/2]; therefore arcsin(−1/2)=−π/6.

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19) Evaluate arcsin(−√3/2) = ____.

Explanation

sin(−π/3)=−√3/2 and −π/3 is within [−π/2, π/2], so arcsin(−√3/2)=−π/3.

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20) Compute cos(arcsin(1/2)).

Explanation

Let θ=arcsin(1/2) so sinθ=1/2 with θ∈[−π/2,π/2]. Then cosθ=√(1−sin^2θ)=√(1−1/4)=√(3/4)=√3/2 (nonnegative in the principal range).

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State the principal range of arcsin(x): ____.
Select all true statements about arcsin compositions and values.
Arcsin is an odd function: arcsin(−x) = −arcsin(x) for all x in...
Compute sin(arcsin(√3/2)) = ____.
Arcsin(x) is strictly increasing on [−1, 1].
Which values of x have negative principal angles under arcsin? Select...
Evaluate arcsin(0) = ____.
Evaluate arcsin(√2/2).
Evaluate arcsin(−√2/2).
The equation arcsin(1.2) is undefined over the reals.
Arcsin(√2/2) = 3π/4.
Arcsin(0) = 0.
Evaluate arcsin(sin(5π/6)).
Evaluate arcsin(1). Choose the exact principal value.
Select all correct exact evaluations.
Arcsin(−1) = ____.
Select all x for which arcsin(x) is real-valued.
Evaluate arcsin(−1/2).
Evaluate arcsin(−√3/2) = ____.
Compute cos(arcsin(1/2)).
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