Introduction to Radians

Basic Concepts and Comparison with Degrees

  • Understand that radian is another way of measuring angles, which is based on the radius of a circle.
  • Get comfortable with the relationship: 180 degrees is equivalent to π radians and hence, 1 radian is approximately 57.3 degrees.
  • Develop an understanding of why radians are often more practical in mathematics than degrees. This is largely due to their natural division of a circle into equal parts by its radius.

Conversion between Radians and Degrees

  • Be proficient in converting angles given in degrees into radians and vice versa. A useful rule: multiple the number of degrees by π/180 or divide the number of radians by 180/π to perform the conversions.
  • Be able to simplify expressions involving pi (π) in radians.

Properties of Radians

Circles and Arcs

  • Understand the application of radian measure in finding lengths of arcs in a circle. Learn that an arc length of a circle is , where r is the radius and θ is the angle in radians.
  • Understand that the area of a sector of a circle is 1/2 *r²θ, where r is the radius and θ is the angle in radians.

Trigonometric Functions and Radians

  • Appreciate that trigonometric functions, i.e., sine, cosine and tangent, are often defined in terms of radians rather than degrees, especially for calculus.
  • Recognise the unit circle as a tool for understanding trigonometric functions using radians.

Real World Applications

  • Understand that radian measure is frequently used in any science involving wave motion, linear and angular speeds and in calculus.
  • Be familiar with situations in physics and engineering where radian measure is more practical, specifically in rotational dynamics and simple harmonic motion.

Graphical Representation

  • Be capable of sketching trigonometric functions with arguments in radians.
  • Understand how changing the measure from degrees to radians affects the graphs and the periodicity of sine, cosine and tangent functions.