Circular Measure: Radians, Degrees, Unit Circle

• Circular measure expresses angles as the ratio of arc length to the radius of a circle.
• Radians are the standard unit of angular measure; one radian is the angle subtended at the center of a circle by an arc equal to the radius in length.
• Degrees, another unit of angular measure, divide a full rotation into 360 equal parts.

Radian vs Degree

• To convert degrees to radians, multiply by π/180.
• To convert radians to degrees, multiply by 180/π.
• One full rotation is 360 degrees or 2π radians.
• 1 radian is approximately 57.3 degrees.

Unit Circle Properties

• The unit circle is a circle with a radius of 1, centered at (0,0) in the Cartesian coordinate system.
• Points on the unit circle are defined by (cos θ, sin θ), where θ is the angle measured counterclockwise from the positive x-axis.
• The equation of the unit circle is x² + y² = 1.
• The unit circle is useful for understanding the periodic nature of trigonometric functions.
• Coordinates on the unit circle directly give the cosine and sine values of corresponding angles.
• For any angle θ:
  • cos θ = x-coordinate of the point on the unit circle
  • sin θ = y-coordinate of the point on the unit circle
  • tan θ = sin θ / cos θ = y/x
• The signs of trigonometric functions vary in different quadrants:
  • Quadrant I (0 < θ < π/2): All trigonometric functions are positive.
  • Quadrant II (π/2 < θ < π): Sine is positive, cosine and tangent are negative.
  • Quadrant III (π < θ < 3π/2): Tangent is positive, sine and cosine are negative.
  • Quadrant IV (3π/2 < θ < 2π): Cosine is positive, sine and tangent are negative.
• The CAST rule helps recall which trigonometric functions are positive in each quadrant (Cosine, All, Sine, Tangent).
• Angles measured clockwise from the positive x-axis are considered negative angles.
• Trigonometric functions of negative angles have these properties:
  • sin(-θ) = -sin(θ) (sine is an odd function)
  • cos(-θ) = cos(θ) (cosine is an even function)
  • tan(-θ) = -tan(θ) (tangent is an odd function)

Trigonometric Identities

• Trigonometric identities are equations involving trigonometric functions that are true for all values for which the functions are defined.
• Pythagorean Identities:
  • sin²θ + cos²θ = 1
  • 1 + tan²θ = sec²θ
  • 1 + cot²θ = csc²θ
• Quotient Identities:
  • tan θ = sin θ / cos θ
  • cot θ = cos θ / sin θ
• Reciprocal Identities:
  • csc θ = 1 / sin θ
  • sec θ = 1 / cos θ
  • cot θ = 1 / tan θ
• Sum and Difference Formulas:
  • sin(A + B) = sin A cos B + cos A sin B
  • sin(A - B) = sin A cos B - cos A sin B
  • cos(A + B) = cos A cos B - sin A sin B
  • cos(A - B) = cos A cos B + sin A sin B
  • tan(A + B) = (tan A + tan B) / (1 - tan A tan B)
  • tan(A - B) = (tan A - tan B) / (1 + tan A tan B)
• Double Angle Formulas:
  • sin 2θ = 2 sin θ cos θ
  • cos 2θ = cos²θ - sin²θ = 2 cos²θ - 1 = 1 - 2 sin²θ
  • tan 2θ = (2 tan θ) / (1 - tan²θ)
• Half Angle Formulas:
  • sin (θ/2) = ±√((1 - cos θ) / 2)
  • cos (θ/2) = ±√((1 + cos θ) / 2)
  • tan (θ/2) = ±√((1 - cos θ) / (1 + cos θ)) = (1 - cos θ) / sin θ = sin θ / (1 + cos θ)
• Product-to-Sum Formulas:
  • sin A cos B = 1/2 [sin(A + B) + sin(A - B)]
  • cos A sin B = 1/2 [sin(A + B) - sin(A - B)]
  • cos A cos B = 1/2 [cos(A + B) + cos(A - B)]
  • sin A sin B = -1/2 [cos(A + B) - cos(A - B)]
• Sum-to-Product Formulas:
  • sin A + sin B = 2 sin((A + B) / 2) cos((A - B) / 2)
  • sin A - sin B = 2 cos((A + B) / 2) sin((A - B) / 2)
  • cos A + cos B = 2 cos((A + B) / 2) cos((A - B) / 2)
  • cos A - cos B = -2 sin((A + B) / 2) sin((A - B) / 2)
• These identities are used to simplify trigonometric expressions, solve trigonometric equations, and prove other identities.
• Understanding the unit circle is essential for deriving and remembering these identities.