Trigonometry — Quiz 5

Questions in this quiz (8)

1. Which of the following expressions is equivalent to:$$\frac{(\tan x)(\csc x)}{(\sin x)(\sec x)}$$?

   • $$\tan x$$
   • $$\sin x$$
   • $$\tan x$$
   • $$\sin x$$

2. What is $$\sin\left(\frac{\pi}{24}\right)$$ given that $$\frac{\pi}{24} = \frac{\pi}{6} - \frac{\pi}{8}$$ and that $$\sin(\alpha - \beta) = (\sin \alpha)(\cos \beta) - (\cos \alpha)(\sin \beta)$$?(Note: You may use the following table of values.)

   • $$\frac{2 - \sqrt{3} \cdot \sqrt{3 + \sqrt{2}}}{4}$$
   • $$\frac{1 - \sqrt{2} \cdot \sqrt{3 - \sqrt{2}}}{2}$$
   • $$\frac{\sqrt{2-\sqrt{2}}-\sqrt{3}\sqrt{2+\sqrt{2}}}{4}$$
   • $$\frac{2 - \sqrt{2} \cdot \sqrt{3 + \sqrt{3}}}{2}$$

3. If $$\cos \theta = b$$ and $$0^\circ$$ < $$\theta$$ < $$90^\circ$$, what is $$\sec \theta \cdot \tan \theta$$ in terms of $$b$$?

   • $$\frac{\sqrt{1 }}{b}$$
   • $$b \cdot \frac{\sqrt{1}}{b^2}$$
   • $$b \cdot \frac{\sqrt{1 + b}}{b}$$
   • $$\frac{\sqrt{1-b^2}}{b^2}$$

4. If it is 12 miles from Town A to Town B, 28 miles from Town A to Town C, and 35 miles from Town B to Town C, which expression gives the angle between the path from Town A to Town B and the path from Town A to Town C?

   • $$\arccos(\frac{12^2 + 28^2 - 35^2}{2\cdot12\cdot28})$$
   • $$\arccos\left(\frac{12^2 + 35^2 - 28^2}{2\cdot12\cdot35}\right)$$
   • $$\arccos\left(\frac{28^2 + 35^2 - 12^2}{2\cdot28\cdot35}\right)$$
   • $$\arccos\left(\frac{35^2 + 12^2 - 28^2}{2\cdot35\cdot12}\right)$$

5. The coordinates of the vertices of $$\triangle PQR$$ are $$P(2, 3)$$, $$Q(6, 9)$$, and $$R(5, 4)$$. What is $$\angle Q$$ to the nearest degree?

   • $$15^\circ$$
   • $$35^\circ$$
   • $$22^\circ$$
   • $$13^\circ$$

6. What are all the solutions between $$0^\circ$$ and $$360^\circ$$ of $$3\sin^2x - 2\cos^2x = 1$$?

   • $${65^\circ, 120^\circ, 240^\circ, 300^\circ}$$
   • $${50^\circ, 110^\circ, 260^\circ, 320^\circ}$$
   • $${63^\circ, 117^\circ, 243^\circ, 297^\circ}$$
   • $${45^\circ, 90^\circ, 210^\circ, 270^\circ}$$

7. Which of these is equivalent to $$8\sin(x)\cos(x)$$?

   • $$8\sin(2x)$$
   • $$8\cos(2x)$$
   • $$\sin(4x)$$
   • $$4\sin(2x)$$

8. A point $$P$$ lies on a circle centered at the origin with radius 1. If the angle $$\angle POB=225^\circ$$ where $$O$$ is the origin and $$B$$ lies on the positive x-axis, what are the coordinates of point $$P$$?

   • $$\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{4}\right)$$
   • $$\left(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right)$$
   • $$\left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right)$$
   • $$\left(\frac{\sqrt{2}}{4},\frac{\sqrt{2}}{4}\right)$$