Chapter 6 Part 1: Class Notes (October 2, 2023)

Welcome to the first part of Chapter 6! In today's lecture, we covered a wide range of topics related to trigonometry and geometry. Let's recap the key concepts:

  • Similar Polygons: Understanding the properties of similar polygons and how to find missing side lengths using proportions. For example, if pentagons $JKLMN$ and $PQRST$ are similar, we can set up ratios like $\frac{4}{5} = \frac{x}{4}$ to solve for unknown lengths.
  • Indirect Measurement: Using similar triangles to find distances that are difficult to measure directly, such as the width of a river. We can set up proportions based on corresponding sides of the triangles: $\frac{64}{28} = \frac{x}{20.7}$. Solving for $x$ gives us the approximate width.
  • Converting Degrees-Minutes-Seconds (DMS) to Decimal Degrees: Converting angles from DMS format to decimal degrees. For instance, $7^{\circ}51'23''$ can be converted to decimal degrees by $7 + \frac{51}{60} + \frac{23}{3600} \approx 7.856^{\circ}$.
  • Converting Decimal Degrees to Degrees-Minutes-Seconds (DMS): Converting angles from decimal degrees to DMS format. For example, $12.725^{\circ}$ becomes $12^{\circ} + 0.725 \times 60' = 12^{\circ}43.5'$. Then $0.5' \times 60'' = 30''$, so $12.725^{\circ} = 12^{\circ}43'30''$.
  • Converting Degrees to Radians and Radians to Degrees:
    • Converting degrees to radians using the conversion factor $\frac{\pi}{180^{\circ}}$. For example, to convert $510^{\circ}$ to radians, we use: $510^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{17\pi}{6}$ radians.
    • Converting radians to degrees using the conversion factor $\frac{180^{\circ}}{\pi}$. For example, to convert $\frac{5\pi}{2}$ radians to degrees, we use: $\frac{5\pi}{2} \times \frac{180^{\circ}}{\pi} = 450^{\circ}$.
  • Sketching Angles in Standard Position: Visualizing angles in standard position, both in degrees and radians. Remember that standard position means the initial side is on the positive x-axis.
  • Finding Coterminal Angles: Determining angles that share the same terminal side. To find coterminal angles, add or subtract multiples of $360^{\circ}$ (or $2\pi$ radians). For instance, to find an angle between $0$ and $2\pi$ that is coterminal with $\frac{29\pi}{12}$, subtract $2\pi$ : $\frac{29\pi}{12} - 2\pi = \frac{5\pi}{12}$.
  • Arc Length and Central Angle Measure: Understanding the relationship between arc length ($s$), radius ($r$), and central angle ($\theta$ in radians): $s = r\theta$.
  • Area of a Sector of a Circle: Calculating the area of a sector using the formula $A = \frac{1}{2}r^2\theta$, where $\theta$ is in radians. For example, if a circle has radius $r=8$mm and the sector has angle $\theta = \frac{4\pi}{3}$ radians, the sector area is $A = \frac{1}{2}(\frac{4\pi}{3})8^2 \approx 134.0$ mm$^2$.
  • Angular and Linear Speed: Distinguishing between angular speed ($\omega = \frac{\theta}{t}$) and linear speed ($v = r\omega$). Remember to use consistent units!

Keep practicing these concepts, and you'll master them in no time! Don't hesitate to review the notes and re-watch the lecture if needed. Good luck with your studies!