Question

i) determine the angle(s), \\( \theta \\), in the domain \\( - 360 ^ { \circ } \leq \theta \leq 360 ^ { \circ } \\), which is coterminal with the given angle
ii) write an expression involving the principal angle that represents all angles in the domain \\( \theta \in r \\) that are coterminal with the given angle
b) \\( - 81 ^ { \circ } \\)
c) \\( 415 ^ { \circ } \\)

Explanation:

Step1: Find coterminal angle for -81°

Add 360° to -81°: $-81^\circ + 360^\circ = 279^\circ$

Step2: Find coterminal angle for 415°

Subtract 360° from 415°: $415^\circ - 360^\circ = 55^\circ$

Step3: General expression for coterminal angles

Coterminal angles = given angle + $360^\circ k$ (k is integer)

Answer:

i) For -81°: 279°; For 415°: 55°
ii) For -81°: $\theta = -81^\circ + 360^\circ k$; For 415°: $\theta = 55^\circ + 360^\circ k$ (k ∈ ℤ)