Cotangent Sign Monotonicity Quiz: Cotangent Sign & Monotonicity

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

True

False

Between asymptotes at kπ and (k+1)π, sinθ≠0 and d/dθ(cotθ)=−csc^2θ<0, so the function is strictly decreasing.

Explanation

Between asymptotes at kπ and (k+1)π, sinθ≠0 and d/dθ(cotθ)=−csc^2θ<0, so the function is strictly decreasing.

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

True

False

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

Explanation

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

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

Positive on (0, π/2), negative on (π/2, π)

Negative on (0, π/2), positive on (π/2, π)

Positive on (0, π), negative nowhere

Negative on (0, π/2), (π, 3π/2)

Within one period (0,π): QI gives cos>0, sin>0 so cot>0; QII gives cos<0, sin>0 so cot<0. Pattern repeats each period.

Explanation

Within one period (0,π): QI gives cos>0, sin>0 so cot>0; QII gives cos<0, sin>0 so cot<0. Pattern repeats each period.

Submit

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

(0, π/2)

(π/2, π)

(π, 3π/2)

(3π/2, 2π)

{π/2}

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).

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).

Submit

6) What is the range of y = cotθ?

All real numbers

(−1, 1)

[0, ∞)

All real numbers except 0

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

Explanation

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

7) Solve cotθ = 1 on [0, 2π).

θ = π/4, 5π/4

θ = 3π/4, 7π/4

θ = π/2, 3π/2

θ = 0, π

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

Explanation

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

8) Where are the zeros of y = cotθ?

θ = π/2 + kπ

θ = kπ

θ = π + 2kπ

No zeros

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

Explanation

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

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

Since d/dθ(cotθ)=−csc^2θ<0 where defined, cotθ is strictly decreasing on each such interval.

Explanation

Since d/dθ(cotθ)=−csc^2θ<0 where defined, cotθ is strictly decreasing on each such interval.

Submit

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

Lim_{θ→0^+} cotθ = +∞

Cotθ is decreasing on (0, π)

Cotθ = 0 at θ = π/2

Lim_{θ→π^-} cotθ = −∞

Cotθ > 0 on (π/2, π)

From d/dθ(cotθ)=−csc^2θ<0, cot is decreasing on (0,π). It diverges to +∞ at 0^+, crosses 0 at π/2, and tends to −∞ at π^−. On (π/2,π) it is negative.

Explanation

From d/dθ(cotθ)=−csc^2θ<0, cot is decreasing on (0,π). It diverges to +∞ at 0^+, crosses 0 at π/2, and tends to −∞ at π^−. On (π/2,π) it is negative.

Submit

11) As θ → 0^+, what is the behavior of cotθ?

+∞

−∞

0

1

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

Explanation

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

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

−∞

+∞

0

Undefined oscillation

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

Explanation

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

13) On (3π/2, 2π), cotθ is negative.

True

False

Quadrant IV has cosθ>0 and sinθ<0, so cotθ=cosθ/sinθ<0.

Explanation

Quadrant IV has cosθ>0 and sinθ<0, so cotθ=cosθ/sinθ<0.

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

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

Explanation

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

Submit

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

(0, π/2)

(π/2, π)

(π, 3π/2)

(3π/2, 2π)

{0}

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

Explanation

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

Submit

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

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

Explanation

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

Submit

17) Give the domain of y = cotθ.

θ ∈ ℝ \ {kπ}

θ ∈ ℝ \ {π/2 + kπ}

θ ∈ ℝ \ {0}

θ ∈ ℝ

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

Explanation

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

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

True

False

Since csc^2θ>0 where sinθ≠0, we have −csc^2θ<0. Therefore cotθ is strictly decreasing on every interval between consecutive asymptotes.

Explanation

Since csc^2θ>0 where sinθ≠0, we have −csc^2θ<0. Therefore cotθ is strictly decreasing on every interval between consecutive asymptotes.

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

Cosθ/sinθ

1/tanθ

X/y (on unit circle)

Secθ/cscθ

Adjacent/opposite

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.

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.

Submit