Question

which of the following is true of the location of the terminal side of an angle, θ, whose sine value is \\(\frac{1}{2}\\)?

• θ has a reference angle of 30° and is in quadrant i or ii
• θ has a reference angle of 30° and is in quadrant i or iv
• θ has a reference angle of 60° and is in quadrant i or ii
• θ has a reference angle of 60° and is in quadrant i or iv

Explanation:

Step1: Recall sine reference angle

We know that \(\sin\theta=\frac{1}{2}\). The reference angle \(\alpha\) for which \(\sin\alpha = \frac{1}{2}\) is \(30^{\circ}\) (since \(\sin30^{\circ}=\frac{1}{2}\)).

Step2: Determine quadrants for positive sine

The sine function is positive in Quadrant I and Quadrant II (because in Quadrant I, all trigonometric functions are positive, and in Quadrant II, sine is positive as \(y\)-coordinate is positive in the unit circle definition of sine: \(\sin\theta = y/r\) where \(r>0\)). So the angle \(\theta\) with \(\sin\theta=\frac{1}{2}\) has a reference angle of \(30^{\circ}\) and is in Quadrant I or II.

Answer:

\(\boldsymbol{\theta}\) has a reference angle of \(30^{\circ}\) and is in Quadrant I or II