Trigonometry Rotations and Radians

Rotations

  • Always start rotations on the positive x-axis.
  • Rotating 30 degrees, 45 degrees, 60 degrees, and so forth, all start from the positive x-axis and go counterclockwise.

Negative Rotations

  • A negative rotation goes in the clockwise direction.
  • Example: Rotating -45 degrees ends in the middle of the fourth quadrant.
  • Negative 90 degrees, negative 135 degrees, and so on.
  • Positive rotations: counterclockwise.
  • Negative rotations: clockwise.

Radians

Why Radians?

  • Degrees are not real numbers or distances, which makes them difficult to work with in periodic functions.
  • Degrees are divided into 60 minutes, and minutes into 60 seconds, making calculations cumbersome.
  • Radians are a more natural way to measure angles because they relate directly to distances on a circle.

Definition of a Radian

  • Consider the unit circle (radius = 1).
  • Circumference of a circle: C=2πrC = 2 \pi r
  • For the unit circle, C=2π(1)=2πC = 2 \pi (1) = 2 \pi
  • Traveling all the way around the unit circle is 2π2 \pi units.
  • Traveling half way around the unit circle is π\pi units.
  • Traveling a quarter of the way is π2\frac{\pi}{2} units.

Radian Measure

  • The radian measure of an angle is the distance along the curve of the unit circle from the positive x-axis to the point on the circle corresponding to that angle.
  • Radian measure is a distance, so it's a number that can be treated like any other number.

Converting Degrees to Radians

  • 180 degrees=π radians180 \text{ degrees} = \pi \text{ radians}
  • 1 degree=π180 radians1 \text{ degree} = \frac{\pi}{180} \text{ radians}
  • 1 radian=180π degrees1 \text{ radian} = \frac{180}{\pi} \text{ degrees}

Conversion Examples

  • To convert from degrees to radians, multiply by π180\frac{\pi}{180} (a small number).
  • To convert from radians to degrees, multiply by 180π\frac{180}{\pi} (a big number).
Example 1: 30 degrees to radians
  • 30 degrees=30⋅π180=π6 radians30 \text{ degrees} = 30 \cdot \frac{\pi}{180} = \frac{\pi}{6} \text{ radians}
  • At this angle, the point has coordinates (32,12)(\frac{\sqrt{3}}{2}, \frac{1}{2}).
Example 2: 60 degrees to radians
  • 60 degrees=60⋅π180=π3 radians60 \text{ degrees} = 60 \cdot \frac{\pi}{180} = \frac{\pi}{3} \text{ radians}
Example 3: 45 degrees to radians
  • 45 degrees=45⋅π180=π4 radians45 \text{ degrees} = 45 \cdot \frac{\pi}{180} = \frac{\pi}{4} \text{ radians}

Key Angles

  • 30 degrees = π6\frac{\pi}{6} radians
  • 60 degrees = π3\frac{\pi}{3} radians
  • 45 degrees = π4\frac{\pi}{4} radians

Thinking in Fractions

  • Analogous to a number line, divide the distance π\pi into equal parts to represent angles in radians.
  • Example: Divide π\pi into six equal parts: π6,2π6,3π6,4π6,5π6,6π6\frac{\pi}{6}, \frac{2\pi}{6}, \frac{3\pi}{6}, \frac{4\pi}{6}, \frac{5\pi}{6}, \frac{6\pi}{6}
  • Reduce fractions where possible: π6,π3,π2,2π3,5π6,π\frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}, \frac{2\pi}{3}, \frac{5\pi}{6}, \pi

Common Angles in Radians

  • π6\frac{\pi}{6} radian angle
  • π3\frac{\pi}{3} radian angle
  • π2\frac{\pi}{2} radian angle
  • 7π6\frac{7\pi}{6} radian angle
  • 8π6\frac{8\pi}{6} radian angle
  • 9π6\frac{9\pi}{6} radian angle
  • 10π6\frac{10\pi}{6} radian angle
  • 11π6\frac{11\pi}{6} radian angle
  • 12π6=2π\frac{12\pi}{6} = 2\pi (all the way around the circle)

Quarters

  • One fourth: π4\frac{\pi}{4}
  • Two fourths: 2π4\frac{2\pi}{4}
  • Three fourths: 3π4\frac{3\pi}{4}
  • Four fourths: 4π4\frac{4\pi}{4}

Converting Radians to Degrees

  • Multiply by 180π\frac{180}{\pi}.
Example: Convert 4π5\frac{4\pi}{5} radians to degrees
  • 4π5⋅180π=4⋅1805=4⋅36=144 degrees\frac{4\pi}{5} \cdot \frac{180}{\pi} = \frac{4 \cdot 180}{5} = 4 \cdot 36 = 144 \text{ degrees}
  • The π\pi's cancel out.
  • 5 goes into 180, 36 times.
  • 4 times 36 is 144 degrees. Therefore an angle of 4π5\frac{4\pi}{5} radians is a 144 degree angle.