Question

question 25 (1 point) which pair of angles between 0° and 360° satisfies \\( \sin \theta=-\frac{1}{\sqrt{2}} \\)? a) \\( 45^{circ} \\) and \\( 0^{circ} \\) b) \\( -45^{circ} \\) and \\( 225^{circ} \\) c) \\( 45^{circ} \\) and \\( 135^{circ} \\) d) \\( 225^{circ} \\) and \\( 315^{circ} \\)

Explanation:

Step1: Recall the unit - circle values

We know that \(\sin45^{\circ}=\frac{1}{\sqrt{2}}\). Since \(\sin\theta =-\frac{1}{\sqrt{2}}\), we need to find angles in the third and fourth quadrants (where sine is negative).

Step2: Calculate the angles in the third and fourth quadrants

For the third - quadrant angle: \(\theta = 180^{\circ}+45^{\circ}=225^{\circ}\)
For the fourth - quadrant angle: \(\theta = 360^{\circ}-45^{\circ}=315^{\circ}\)

Answer:

D. \(225^{\circ}\) and \(315^{\circ}\)