Cotangent Sign Monotonicity Quiz: Cotangent Sign & Monotonicity

  • 11th Grade
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| Questions: 20 | Updated: Dec 16, 2025
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1) As θ → 0^+, what is the behavior of cotθ?

Explanation

Near 0 with θ>0, sinθ≈θ>0 and cosθ≈1, so cotθ≈1/θ→+∞.

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About This Quiz
Cotangent Sign Monotonicity Quiz: Cotangent Sign & Monotonicity - Quiz

What causes cotangent’s sign and slope to shift across the coordinate plane? In this quiz, you’ll explore how cotangent behaves within different quadrants, analyze intervals where it increases or decreases, and interpret its graph through geometric reasoning. You’ll connect angle behavior to the signs of sine and cosine, visualize transitions... see morenear asymptotes, and understand why cotangent’s pattern repeats predictably. Step by step, you’ll build strong intuition for cotangent’s variation and monotonicity.
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2) What is the fundamental period of y = cotθ?

Explanation

cot(θ+π)=cos(θ+π)/sin(θ+π)=(-cosθ)/(-sinθ)=cosθ/sinθ=cotθ, so the least positive period is π.

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3) Select all expressions equivalent to cotθ (where defined).

Explanation

By definition cotθ=cosθ/sinθ=1/tanθ. On the unit circle x=cosθ, y=sinθ so x/y=cotθ. In right triangles, cotθ=adjacent/opposite. secθ/cscθ=(1/cos)/(1/sin)=sin/cos=tanθ.

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4) The derivative d/dθ(cotθ) = −csc^2θ is negative wherever defined, so cotθ decreases on each open interval between asymptotes.

Explanation

Since csc^2θ>0 where sinθ≠0, we have −csc^2θ

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5) Give the domain of y = cotθ.

Explanation

Asymptotes of cot occur where sinθ=0, namely θ=kπ. Exclude these points from the domain.

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6) Compute the derivative: d/dθ[cotθ] =

Explanation

Differentiate cotθ = cosθ/sinθ or use the identity: derivative of cotθ is −csc^2θ.

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7) Select all intervals in [0, 2π) where cotθ < 0.

Explanation

cotθ is negative when cos and sin have opposite signs: Quadrants II and IV, i.e., (π/2,π) and (3π/2,2π).

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8) State the general locations of vertical asymptotes of y = cotθ.

Explanation

cotθ=cosθ/sinθ is undefined when sinθ=0, which occurs at θ=kπ for any integer k.

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9) On (3π/2, 2π), cotθ is negative.

Explanation

Quadrant IV has cosθ>0 and sinθ

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10) As θ → π^−, what is the behavior of cotθ?

Explanation

Approaching π from the left, sinθ→0^+ while cosθ→−1. Then cotθ=cosθ/sinθ≈(−1)/(small +)→−∞.

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11) Cotθ is strictly decreasing on every interval (kπ, (k+1)π).

Explanation

Between asymptotes at kπ and (k+1)π, sinθ≠0 and d/dθ(cotθ)=−csc^2θ

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12) On (0, π), which statements about y = cotθ are true? Select all that apply.

Explanation

From d/dθ(cotθ)=−csc^2θ

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13) On any interval between consecutive asymptotes, cotθ is ______.

Explanation

Since d/dθ(cotθ)=−csc^2θ

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14) Where are the zeros of y = cotθ?

Explanation

cotθ=0 ⇔ cosθ=0 with sinθ≠0. cosθ=0 at θ=π/2 + kπ.

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15) Solve cotθ = 1 on [0, 2π).

Explanation

cotθ=1 when cosθ=sinθ≠0. This occurs at θ=π/4 (QI) and θ=5π/4 (QIII).

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16) What is the range of y = cotθ?

Explanation

Between asymptotes, cotθ decreases continuously from +∞ to −∞, attaining every real value exactly once per period.

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17) Select all intervals in [0, 2π) where cotθ > 0.

Explanation

cotθ=cosθ/sinθ is positive when cos and sin have the same sign: Quadrants I and III, i.e., (0,π/2) and (π,3π/2).

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18) State the sign of cotθ in Quadrant II.

Explanation

In Quadrant II, cosθ0, so cotθ=cosθ/sinθ is negative.

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19) Which summary correctly describes sign intervals for cotθ over one period?

Explanation

Within one period (0,π): QI gives cos>0, sin>0 so cot>0; QII gives cos0 so cot

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20) Cot(θ+π) = cotθ for all θ where both sides are defined.

Explanation

cot(θ+π)=cos(θ+π)/sin(θ+π)=(-cosθ)/(-sinθ)=cosθ/sinθ=cotθ. Hence period π.

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As θ → 0^+, what is the behavior of cotθ?
What is the fundamental period of y = cotθ?
Select all expressions equivalent to cotθ (where defined).
The derivative d/dθ(cotθ) = −csc^2θ is negative wherever defined,...
Give the domain of y = cotθ.
Compute the derivative: d/dθ[cotθ] =
Select all intervals in [0, 2π) where cotθ < 0.
State the general locations of vertical asymptotes of y = cotθ.
On (3π/2, 2π), cotθ is negative.
As θ → π^−, what is the behavior of cotθ?
Cotθ is strictly decreasing on every interval (kπ, (k+1)π).
On (0, π), which statements about y = cotθ are true? Select all that...
On any interval between consecutive asymptotes, cotθ is ______.
Where are the zeros of y = cotθ?
Solve cotθ = 1 on [0, 2π).
What is the range of y = cotθ?
Select all intervals in [0, 2π) where cotθ > 0.
State the sign of cotθ in Quadrant II.
Which summary correctly describes sign intervals for cotθ over one...
Cot(θ+π) = cotθ for all θ where both sides are defined.
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