Degrees And Radians Practice Sheet 1 To Convert From Degrees

Degrees And Radians Practice Sheet 1 To Convert From Degrees To Ra

Convert from degrees to radians: Remember, DO NOT put the degree measure directly into your calculator. Instead, multiply the degree measure by π/180 to obtain the radian measure in exact form. For problems #2-7, convert each degree measure into radians and give exact answers in terms of π.

Convert from radians to degrees: Remember, DO NOT put the radian value directly into your calculator. Instead, multiply the radian measure by 180/π to find the degree measure. For problems #9-14, convert each radian measure into degrees, giving exact answers if possible.

Explain how to find coterminal angles: To find coterminal angles, add or subtract 360° for degree angles, or add or subtract 2π for radian angles. This process differs because degrees use 360° as a full rotation, whereas radians use 2π. The principle is the same: adding or subtracting a full rotation yields a coterminal angle.

Find an angle between 0 and 2π radians that is coterminal with the given angles: For #15-18, identify an equivalent angle in the standard [0, 2π) interval, given in radians, that shares the same terminal side.

Find the measure of each angle: For #20-23, determine the degree measure of the given radian angle.

Draw an angle with the given measure in standard position: For #24-27, sketch the angle starting from the positive x-axis, measure uder in degrees, and position in standard position.

State the quadrant in which the terminal side of the angle lies: For #28-30, determine the quadrant for each angle specified in degrees or radians, considering the angle's measure and position.

Paper For Above instruction

Converting between degrees and radians is fundamental in trigonometry, as different problems and contexts prefer one measure over the other. To convert from degrees to radians, multiply the degree measure by π/180. This involves understanding that 180° equals π radians; thus, the conversion factor is π/180. For example, converting 130° to radians involves calculating 130 × π/180, which simplifies to 13π/18 radians. It is crucial not to input degrees directly into the calculator but rather perform this multiplication to obtain the exact form involving π.

Conversely, to convert from radians to degrees, multiply the radian measure by 180/π. This is based on the understanding that π radians equal 180°, so the conversion factor is 180/π. For instance, converting 12π/5 radians involves calculating (12π/5) × 180/π. The π terms cancel, leaving (12/5) × 180 = 432°. This method ensures an exact degree measure corresponding to the radian input.

Understanding how to find coterminal angles involves adding or subtracting full rotations, which differ depending on whether the angles are in degrees or radians. In degrees, coterminal angles are obtained by adding or subtracting 360°, as full rotations are 360°. In radians, the equivalent full rotation is 2π radians. For example, to find an angle coterminal with 45°, subtract 360° to get -315°, or add 360° to get 405°, both coterminal with the original. Similarly, in radians, subtract or add 2π to find coterminal angles. For the radian measure 3π/2, subtracting 2π yields -π/2, while adding 2π results in 7π/2, both sharing the same terminal side.

For problems involving coterminal angles in the interval [0, 2π), you must find an equivalent angle within this range. For example, for an angle of 5π/3, which is already within [0, 2π), it is coterminal with itself. For an angle like -π/4, adding 2π results in 7π/4, which is within the desired range. This ensures the angle lies in the standard position, traditionally between 0 and 2π radians.

To find the measure of given angles in radians, use the conversion factor or known angle measures. For example, given π/3 radians, the degree measure is (π/3) × 180/π, which simplifies to 60°. Similarly, for 7π/6 radians, the degree measure is (7π/6) × 180/π, simplifying to 210°.

Drawing angles in standard position involves starting from the positive x-axis (0°/0 radians) and rotating counterclockwise for positive measures or clockwise for negative measures. For example, an angle of 280° can be sketched by starting from the x-axis and rotating almost four-quarters around, ending in the fourth quadrant. For an angle of 710°, since 710° exceeds 360°, subtract multiples of 360° to find the equivalent within 0-360°. This process helps accurately position the terminal side.

Determining the quadrant in which the terminal side lies depends on the angle's measure. In degrees, the quadrants are defined as follows: I (0°–90°), II (90°–180°), III (180°–270°), and IV (270°–360°). In radians, the same regions apply: QI (0–π/2), QII (π/2–π), QIII (π–3π/2), QIV (3π/2–2π). For an angle like -510°, adding 2π repeatedly or adjusting appropriately locates it within one of these quadrants.

Overall, mastering these conversions and the concept of coterminal angles enhances understanding of angular measures and their applications in trigonometry and related fields.

References

  • Anton, H., Bivens, I., & Davis, S. (2013). Calculus: Early Transcendentals. John Wiley & Sons.
  • Larson, R., & Edwards, B. H. (2016). Precalculus with Limits: A Graphing Approach. Cengage Learning.
  • Newton, R. (2002). Trigonometry. McGraw-Hill Education.
  • Swokowski, E., & Cole, J. (2012). Trigonometry. Cengage Learning.
  • Lay, D. C. (2012). Linear Algebra and Its Applications. Addison-Wesley.
  • Fowler, D. (2014). Essentials of Trigonometry. Pearson Education.
  • Stewart, J. (2015). Precalculus: Mathematics for Calculus. Cengage Learning.
  • Jones, A. (2000). Trigonometry Part 1. Cambridge University Press.
  • Houghton Mifflin Company. (2004). Trigonometry. Houghton Mifflin Harcourt.
  • Adams, R. A. (2018). Calculus: A Complete Course. Pearson.

Related Assignments