Arctan Equations Quiz: Solving Equations Using Arctan

1) Which of the following are true about the inverse tangent function?

Arctan(1) = π/4

Arctan(−√3/2) = −π/3

Arctan(x) has a period of 2π

Arctan(x) maps negative x to negative values

Arctan(x) is a decreasing function

arctan(1)=π/4, arctan(−√3/2)=−π/3, and arctan(x) preserves sign but is increasing, not decreasing.

Explanation

arctan(1)=π/4, arctan(−√3/2)=−π/3, and arctan(x) preserves sign but is increasing, not decreasing.

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2) For θ = arctan(x), the equation tan(θ) = x holds for all real x.

True

False

By definition, tan(arctan(x))=x for all reals.

Explanation

By definition, tan(arctan(x))=x for all reals.

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4) For y = arctan(x), the domain of this function is ____.

Domain of arctan(x) is (−∞, ∞), since tan(θ) covers all reals in (−π/2, π/2).

Explanation

Domain of arctan(x) is (−∞, ∞), since tan(θ) covers all reals in (−π/2, π/2).

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5) The function y = arctan(x) is increasing on its entire domain.

True

False

arctan(x) increases on all reals since its derivative is positive everywhere.

Explanation

arctan(x) increases on all reals since its derivative is positive everywhere.

6) Select all correct definitions of arctan(x).

Arctan(x) is the inverse of tan(x)

Arctan(x) is the angle whose tangent is x, with θ ∈ (−π/2, π/2)

Arctan(x) always gives a positive result

Arctan(x) maps negative x to negative values

Arctan(x) is periodic with period π

arctan(x) inverts tan(x), gives results in (−π/2, π/2), and is odd (maps negatives to negatives).

Explanation

arctan(x) inverts tan(x), gives results in (−π/2, π/2), and is odd (maps negatives to negatives).

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7) Select all true statements for y = arctan(x).

The domain of arctan(x) is all real numbers

The range of arctan(x) is (−π/2, π/2)

The graph of arctan(x) has vertical asymptotes

Arctan(1) = π/4

Arctan(x) has horizontal asymptotes at y = ±π/2

Domain is all reals; range is (−π/2, π/2). Asymptotes occur at y = ±π/2.

Explanation

Domain is all reals; range is (−π/2, π/2). Asymptotes occur at y = ±π/2.

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9) Select all correct statements about arctan.

Arctan(x) returns the angle whose tangent is x

Arctan(x) has range (−π/2, π/2)

Arctan(x) returns the angle in [0, π]

Arctan(x) is defined for all real x

Arctan(x) is undefined for x > 1

arctan(x) gives θ with tan(θ) = x, where θ ∈ (−π/2, π/2), and is defined for all x.

Explanation

arctan(x) gives θ with tan(θ) = x, where θ ∈ (−π/2, π/2), and is defined for all x.

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10) Solve θ = arctan(0). Which is the correct value for θ?

π/4

−π/4

0

π/2

tan(0) = 0, and 0 lies in the range (−π/2, π/2), so θ = 0.

Explanation

tan(0) = 0, and 0 lies in the range (−π/2, π/2), so θ = 0.

11) Evaluate arcsin(tan(arctan(0.5))). ____

tan(arctan(0.5))=0.5. arcsin(0.5)=π/6, which is valid since 0.5 ∈ [−1,1].

Explanation

tan(arctan(0.5))=0.5. arcsin(0.5)=π/6, which is valid since 0.5 ∈ [−1,1].

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13) Arctan(x) is the inverse of tan(x) for all real values of x.

True

False

arctan(x) reverses tan(x), returning an angle in (−π/2, π/2) for any real x.

Explanation

arctan(x) reverses tan(x), returning an angle in (−π/2, π/2) for any real x.

14) What is the range of the function y = arctan(x)?

−π to π

(−π/2, π/2)

0 to π

−π/4 to π/4

The range of arctan(x) is the open interval (−π/2, π/2).

Explanation

The range of arctan(x) is the open interval (−π/2, π/2).

15) What is the principal range of y = arctan(x)?

[−π, π]

(−π/2, π/2)

[0, π]

(−π/4, π/4)

The inverse tangent function outputs values only in the open interval (−π/2, π/2), since tan(θ) is undefined at ±π/2.

Explanation

The inverse tangent function outputs values only in the open interval (−π/2, π/2), since tan(θ) is undefined at ±π/2.

16) For all real x, tan(arctan(x)) = x.

True

False

tan and arctan are inverses, so tan(arctan(x)) = x for all real x.

Explanation

tan and arctan are inverses, so tan(arctan(x)) = x for all real x.

17) Arctan(−1) = −π/4.

True

False

Since tan(−π/4) = −1, arctan(−1) = −π/4.

Explanation

Since tan(−π/4) = −1, arctan(−1) = −π/4.

18) If θ = arctan(−√3), what is the value of θ in radians?

−π/6

−π/4

−π/3

−π/2

tan(−π/3) = −√3, so arctan(−√3) = −π/3.

Explanation

tan(−π/3) = −√3, so arctan(−√3) = −π/3.

19) Select all true statements for y = arctan(x).

Arctan(x) is defined for all real numbers

Arctan(x) is an odd function

Arctan(x) has a range of (−π, π)

Arctan(x) is periodic with a period of π

Arctan(x) is always increasing

arctan(x) is odd, defined for all x, and increases over its domain; its range is (−π/2, π/2).

Explanation

arctan(x) is odd, defined for all x, and increases over its domain; its range is (−π/2, π/2).

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20) What is the value of arctan(−1)?

π/4

−π/4

π/3

−π/3

tan(−π/4)=−1, so arctan(−1)=−π/4.

Explanation

tan(−π/4)=−1, so arctan(−1)=−π/4.

Arctan Equations Quiz: Solving Equations Using Arctan

1) Which of the following are true about the inverse tangent function?

2)

What first name or nickname would you like us to use?

2) For θ = arctan(x), the equation tan(θ) = x holds for all real x.

3) Arctan(−√3) = ____.

4) For y = arctan(x), the domain of this function is ____.

5) The function y = arctan(x) is increasing on its entire domain.

6) Select all correct definitions of arctan(x).

7) Select all true statements for y = arctan(x).

8) Arctan(1) = ____.

9) Select all correct statements about arctan.

10) Solve θ = arctan(0). Which is the correct value for θ?

11) Evaluate arcsin(tan(arctan(0.5))). ____

12) The value of arctan(√2/2) is equal to ____.

13) Arctan(x) is the inverse of tan(x) for all real values of x.

14) What is the range of the function y = arctan(x)?

15) What is the principal range of y = arctan(x)?

16) For all real x, tan(arctan(x)) = x.

17) Arctan(−1) = −π/4.

18) If θ = arctan(−√3), what is the value of θ in radians?

19) Select all true statements for y = arctan(x).

20) What is the value of arctan(−1)?