Question

without using a calculator, use the general angle definitions of the trigonometric functions to determine the value of \\( \sin (450^{\circ}) \\) or state that the value is undefined
select the correct choice below and, if necessary, fill in the answer box within your choice
\\( \bigcirc \\) a. \\( \sin (450^{\circ})= \\) (type an integer or a simplified fraction )
\\( \bigcirc \\) b. the answer is undefined.

Explanation:

Step1: Subtract \(360^{\circ}\) from \(450^{\circ}\)

To find the reference angle, we use the formula \(\theta - 360^{\circ}n\) (where \(n\) is an integer). For \(450^{\circ}\), \(n = 1\). So \(450^{\circ}-360^{\circ}=90^{\circ}\).

Step2: Use the sine function value for \(90^{\circ}\)

We know that \(\sin(90^{\circ}) = 1\). Since \(450^{\circ}\) and \(90^{\circ}\) are coterminal angles (angles that have the same terminal side), \(\sin(450^{\circ})=\sin(90^{\circ})\).

Answer:

A. \(\sin(450^{\circ}) = 1\)