Math 1250 - Exam 4 - Additional Practice Problems Study Guide

Question 1: Angles, Quadrants, and Reference Angles

• For each of the following angles, you need to determine the quadrant and the reference angle:

  • (a) Angle: \frac{17\pi}{6}

  • Quadrant:

  • To find the quadrant, first convert the angle to degrees:

  • \frac{17\pi}{6} \times \frac{180}{\pi} = 510^\circ

  • 510^\circ is greater than 360^\circ, so subtract 360^\circ:

  • 510^\circ - 360^\circ = 150^\circ

  • Since 150^\circ is in the second quadrant, the angle is in the Second Quadrant.

  • Reference Angle:

  • The reference angle is found by subtracting 150^\circ from 180^\circ:

  • Reference Angle = 180^\circ - 150^\circ = 30^\circ

  • (b) Angle: 510^\circ

  • Quadrant:

  • As previously calculated, 510^\circ is equivalent to 150^\circ, hence the Second Quadrant.

  • Reference Angle:

  • Reference Angle = 180^\circ - 150^\circ = 30^\circ

  • (c) Angle: -\frac{20\pi}{3}

  • Quadrant:

  • First, convert to a positive angle by adding 2\pi:

  • -\frac{20\pi}{3} + 2\pi \times \frac{3}{3} = -\frac{20\pi}{3} + \frac{6\pi}{3} = -\frac{14\pi}{3}

  • Adding 2\pi again:

  • -\frac{14\pi}{3} + \frac{6\pi}{3} = -\frac{8\pi}{3}

  • A final addition of 2\pi:

  • -\frac{8\pi}{3} + \frac{6\pi}{3} = -\frac{2\pi}{3} + 2\pi = \frac{4\pi}{3}

  • \frac{4\pi}{3} is in the Third Quadrant.

  • Reference Angle:

  • The reference angle is \frac{4\pi}{3} - \pi = \frac{4\pi}{3} - \frac{3\pi}{3} = \frac{\pi}{3}

  • (d) Angle: -150^\circ

  • Quadrant:

  • Convert to a positive angle by adding 360^\circ:

  • -150^\circ + 360^\circ = 210^\circ

  • 210^\circ is located in the Third Quadrant.

  • Reference Angle:

  • Reference Angle = 210^\circ - 180^\circ = 30^\circ

Question 2: Trigonometric Values

• Calculate the following trigonometric expressions:

  • (a) \sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}

  • (b) \cos\left(\frac{9\pi}{4}\right) = \frac{\sqrt{2}}{2}

  • (c) \tan\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{3}

  • (d) \sec\left(\frac{4\pi}{3}\right) = -2

Question 3: Inverse Trigonometric Functions

• Solve the equations involving inverse trigonometric functions:

  • (a) \tan\left(\sin^{-1}\left(\frac{4}{5}\right)\right) = \frac{4}{3}

  • (b) \arcsin\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3}

  • (c) \sin^{-1}\left(\cos\left(\frac{8\pi}{3}\right)\right) = -\frac{\pi}{6}

  • (d) \tan^{-1}(1) = \frac{\pi}{4}

  • (e) \sin\left(\sin^{-1}\left(-\frac{\pi}{2}\right)\right) = \text{Undefined}

  • (f) \cos\left(\sec^{-1}\left(\frac{13}{5}\right)\right) = \frac{5}{13}

Question 4: Using Trigonometric Functions

• Given that \cos(\theta) = \frac{5}{11} and \sin(\theta) < 0, find the following:

  • (a) \sin(\theta):

  • Using the relationship \sin^2(\theta) + \cos^2(\theta) = 1:

    • \sin^2(\theta) + \left(\frac{5}{11}\right)^2 = 1

    • \sin^2(\theta) + \frac{25}{121} = 1

    • \sin^2(\theta) = 1 - \frac{25}{121} = \frac{96}{121}

    • \sin(\theta) = -\sqrt{\frac{96}{121}} = -\frac{4\sqrt{6}}{11}

  • (b) \tan(\theta):

  • \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{-\frac{4\sqrt{6}}{11}}{\frac{5}{11}} = -\frac{4\sqrt{6}}{5}

  • (c) \sec(\theta):

  • \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{11}{5}

  • (d) \csc(\theta):

  • \csc(\theta) = \frac{1}{\sin(\theta)} = -\frac{11}{4\sqrt{6}}

  • (e) \cot(\theta):

  • \cot(\theta) = \frac{1}{\tan(\theta)} = -\frac{5}{4\sqrt{6}}

Question 5: Function Properties

• For the function f(x) = -3 \sin\left(\frac{\pi x}{2} + \pi\right), identify the following:

  1. Amplitude:

  • Amplitude = |-3| = 3

  1. Period:

  • Period = \frac{2\pi}{\frac{\pi}{2}} = 4

  1. Phase Shift:

  • Phase Shift = -\frac{\pi}{\frac{\pi}{2}} = -2

  1. Vertical Shift:

  • Vertical Shift = 0

  1. Key Points:

  • Calculate f(x) at specific intervals:

    • List of 5 key points:

      • (-2, f(-2)), (-1, f(-1)), (0, f(0)), (1, f(1)), (2, f(2))

  1. Graph:

  • Generate the graph for y = f(x) over specified intervals.

Question 6: Trigonometric Function Analysis

• For the function f(x) = 2 \cot\left(\frac{x}{3}\right) - 1, determine:

  1. Vertical Stretch:

  • Vertical Stretch = 2

  1. Period:

  • Period = \frac{\pi}{\frac{1}{3}} = 3\pi

  1. Phase Shift:

  • Phase Shift = 0

  1. Vertical Shift:

  • Vertical Shift = -1

  1. Key Points:

  • Determine 5 key points for the function:

    • List of key points at specific intervals.

  1. Graph:

  • Generate the graph for y = f(x) over specified intervals.