1.
Item 1. Which of the following is a reasonable measure for angle \(\theta\) shown in standard position?
Solution:The angle starts on the positive \(x\)-axis and rotates clockwise, so it is a negative angle.The terminal arm is between \(-270^\circ\) and \(-360^\circ\).Therefore, the reasonable measure is \(-315^\circ\).Answer: B
2.
Item 2. Determine three angles greater than \(45^\circ\) and less than \(360^\circ\) that have a reference angle of \(45^\circ\).
Solution:Reference angle is \(45^\circ\).Quadrant II: \(180^\circ-45^\circ=135^\circ\).Quadrant III: \(180^\circ+45^\circ=225^\circ\).Quadrant IV: \(360^\circ-45^\circ=315^\circ\).Answer: C
3.
Item 6. Determine the exact value of \(\tan315^\circ\).
Solution:The reference angle is \(360^\circ-315^\circ=45^\circ\).\(\tan45^\circ=1\).Since \(315^\circ\) is in Quadrant IV, tangent is negative.Therefore, \(\tan315^\circ=-1\).Answer: C
4.
Item 3. Point \(P(0,-4)\) is on the terminal arm of angle \(\theta\) in standard position. Calculate \(\tan\theta\).
Solution:Use \(\tan\theta=\frac{y}{x}\).For \(P(0,-4)\), \(x=0\) and \(y=-4\).\(\tan\theta=\frac{-4}{0}\).Division by zero is undefined.Answer: C
5.
Item 4. Angle \(\theta\) lies in Quadrant III and \(\tan\theta=1\). Find the exact values of the other two trigonometric ratios.
Solution:Since \(\tan\theta=1\), the opposite and adjacent sides have equal lengths.Use a \(45^\circ-45^\circ-90^\circ\) triangle with side lengths \(1,1,\sqrt{2}\).In Quadrant III, both \(x\) and \(y\) are negative.So choose point \((-1,-1)\), and \(r=\sqrt{(-1)^2+(-1)^2}=\sqrt{2}\).\(\sin\theta=\frac{y}{r}=\frac{-1}{\sqrt{2}}=-\frac{\sqrt{2}}{2}\).\(\cos\theta=\frac{x}{r}=\frac{-1}{\sqrt{2}}=-\frac{\sqrt{2}}{2}\).Answer: C
6.
Item 5. Determine the exact value of \(\tan30^\circ\).
Solution:Use the \(30^\circ-60^\circ-90^\circ\) special triangle.For \(30^\circ\), opposite side is \(1\), adjacent side is \(\sqrt{3}\).\(\tan30^\circ=\frac{\text{opposite}}{\text{adjacent}}=\frac{1}{\sqrt{3}}\).Answer: D
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