Questions: Why is csc θ = csc (θ + n • 360°) true for any angle θ and any integer n? When an integer multiple of [] is added to θ, the resulting angle is [] with θ. The cosecant values of [] angles are equal.

Solution Steps

The cosecant function, csc(θ), is periodic with a period of 360°. This means that adding any integer multiple of 360° to an angle θ will result in an angle that is coterminal with θ. Since coterminal angles have the same trigonometric values, csc(θ) will be equal to csc(θ + n * 360°) for any integer n.

Let \( \theta = 45^\circ \) and \( n = 2 \). We calculate the new angle as follows: \[ \text{new\_angle} = \theta + n \cdot 360^\circ = 45^\circ + 2 \cdot 360^\circ = 765^\circ \]

Next, we compute the cosecant values for both angles: \[ \csc(\theta) = \csc(45^\circ) = \frac{1}{\sin(45^\circ)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \] \[ \csc(\text{new\_angle}) = \csc(765^\circ) = \frac{1}{\sin(765^\circ)} \] Calculating \( \sin(765^\circ) \): \[ 765^\circ \text{ is coterminal with } 765^\circ - 2 \cdot 360^\circ = 45^\circ \] Thus, \( \sin(765^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2} \), leading to: \[ \csc(765^\circ) = \frac{1}{\sin(765^\circ)} = \sqrt{2} \]

We find that: \[ \csc(45^\circ) = \sqrt{2} \quad \text{and} \quad \csc(765^\circ) = \sqrt{2} \] However, due to floating-point precision, the computed values were: \[ \csc(45^\circ) \approx 1.4142 \quad \text{and} \quad \csc(765^\circ) \approx 1.4142 \] The slight difference in the last decimal place indicates that while they are theoretically equal, numerical precision can lead to discrepancies.

Final Answer