The Unit Circle and Reference Angles

Circular Definitions

  • The unit circle is a geometric tool used to determine the values of trigonometric functions for any given angle.

    • Center: The origin (0, 0).
    • Radius: 1 unit.

    - Notation: Point P(x, y) represents a point on the circle.

    • The unit circle is graphically represented in Cartesian coordinates, where the positive x-axis serves as the initial ray for angles in standard position.

  • Standard Position of an Angle:

    • An angle in standard position has its initial ray on the positive x-axis.
    • The angle intersects the circumference of the unit circle, creating a right triangle with:
    • Hypotenuse = 1 (the radius).
    • Legs equal to the x (adjacent side) and y (opposite side) coordinates of the intersection point.

  • Trigonometric Functions in Terms of the Unit Circle:

    • Sine (sin) and Cosine (cos): These can be defined based on the coordinates of the intersection point:
    • sin(θ) = y (opposite), cos(θ) = x (adjacent).
    • Tangent (tan): Defined as the ratio of the sine and cosine:
    • tan(θ) = opposite/adjacent = y/x.

  • Reciprocal Functions:

    • Secant (sec): Reciprocal of cosine (cos).
    • Cosecant (csc): Reciprocal of sine (sin).
    • Cotangent (cot): Reciprocal of tangent (tan).
    • The circular definitions for these functions are derived from their triangular counterparts when the triangle is inscribed in the unit circle.

Reference Angles

  • A reference angle is defined as the acute angle formed by the terminal side of an angle and the x-axis.
    • For angles in the first quadrant, the reference angle is equal to the angle itself.
  • Using Reference Angles:
    • Reference angles help to determine trigonometric ratios across all quadrants.
  • Calculation for Reference Angles:
    • The method to determine the measure of a reference angle varies depending on the quadrant in which the terminal ray of the angle lies:
    • Quadrant I: a = θ.
    • Quadrant II: a = 180° - θ.
    • Quadrant III: a = θ - 180°.
    • Quadrant IV: a = 360° - θ.
    • In radians, replace:
    • 180° with extπ1\frac{ ext{π}}{1} and 360° with 2extπ1\frac{2 ext{π}}{1}.

Examples of Reference Angles

  1. Reference Angle for 225°:
    • Terminal ray lies in Quadrant III, 45° past the negative x-axis.
      • Reference angle = 45°.
  2. Reference Angle for -π:
    • Terminal ray lies in Quadrant II.
      • A clockwise rotation gives an angle formed by the terminal ray and the x-axis.
      • Reference angle determined similarly based on its quadrant location.
  • Practice Problems:
    1. Reference angle for 195°.
    2. Reference angle for -191°.
    3. Reference angle for 2extπ3\frac{2 ext{π}}{3}.
    4. Reference angle for 7extπ12\frac{7 ext{π}}{12}.

The Unit Circle - All Four Quadrants

  • Understanding reference angles allows us to calculate sine and cosine for many special angles based on their reference angles:
    • Special angles include 30°, 45°, and 60° in the first quadrant.
    • These angles are often reflected across the x-axis and y-axis, producing additional angles in other quadrants.
    • Each reflection alters the sign of the sine and cosine values according to the quadrant rules shown in the unit circle.

Trigonometric Function Values by Quadrant

  • To determine the sign of trigonometric function values in different quadrants, use the mnemonic: "All Students Take Calculus."
    • Quadrant I: All trig values are positive.
    • Quadrant II: Sine is positive; cosine and tangent are negative.
    • Quadrant III: Tangent is positive; sine and cosine are negative.
    • Quadrant IV: Cosine is positive; sine and tangent are negative.

Determining Trigonometric Functions for Points Not on the Unit Circle

  • When a point of intersection lies outside the unit circle:
    • Draw a right triangle with one vertex at the point and a leg perpendicular to the x-axis.
    • Label the sides using the coordinates (x, y) of the point.
    • Use the Pythagorean theorem to determine the hypotenuse of the triangle:
    • r=exthypotenuse=x2+y2r = ext{hypotenuse} = \sqrt{x^2 + y^2}.
    • Use SOH CAH TOA to deduce trigonometric function values:
    • SOH: sin is opposite over hypotenuse.
    • CAH: cos is adjacent over hypotenuse.
    • TOA: tan is opposite over adjacent.

Example When Point is Off-Unit Circle

  1. Point (-1, 4): Find cos(θ).
    • Triangle has sides defined by the coordinates of the point.
    • Determine side lengths and apply the appropriate trigonometric function.
  2. Point (-3, -6): Find csc(θ).
    • Analogous method to find sine and its reciprocal.

Practice Problems for Points Off the Unit Circle

  1. For point (2, -3), find sec(θ).
  2. For point (-2, -5), find tan(θ).
  3. For point (-5, 4), find sin(θ).