Two consecutive asymptotes are θ = - π / 20 and θ = 3π / 20 and the period is π / 5.
In this question we must look for two consecutive vertical asymptotes related to a trigonometric function. A function becomes undefined when its denominator is zero. Now we proceed to simplify the expression by trigonometric formulas:
y = 4 · csc (5θ - 3π / 4)
y = 4 / sin (5θ - 3π / 4)
y = 4 / (sin 5θ · cos 3π / 4 - cos 5θ · sin 3π / 4)
y = 4 / [(- √2 / 2) · sin 5θ - (√2 / 2) · cos 5θ]
The function is undefined for (- √2 / 2) · sin 5θ - (√2 / 2) · cos 5θ = 0:
(- √2 / 2) · sin 5θ - (√2 / 2) · cos 5θ = 0
(- √2 / 2) · sin 5θ = (√2 / 2) · cos 5θ
- sin 5θ = cos 5θ
tan 5θ = - 1
5θ = - π / 4 + i · π
θ = - π / 20 + i · π / 5
θ = (π / 5) · (- 1 / 4 + i)
There are two consecutive asymptotes:
i = 0
θ = - π / 20
i = 1
θ = 3π / 20
And the period is ΔT = 3π / 20 - (- π / 20) = π / 5.
To learn more on trigonometric equations: https://brainly.com/question/22624805
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