Comprehensive Study Notes on Unit Circle Trigonometry

Fundamental Definitions of Trigonometric Functions

• Operational Definitions on a General Circle:

  • The Cosine function takes an angle $\theta$ and outputs the ratio of the horizontal position to the radius:     $\cos(\theta) = \frac{x}{r}$
  • The Sine function takes an angle $\theta$ and outputs the ratio of the vertical position to the radius:     $\sin(\theta) = \frac{y}{r}$

• Components and Terminology:

  • $\theta$: The angle of rotation measured in standard position (starting from the positive $x$-axis).
  • $r$: The radius of the circle centered at the origin $(0,0)$.
  • $x$: The horizontal coordinate of the point where the terminal side intersects the circle.
  • $y$: The vertical coordinate of the point where the terminal side intersects the circle.
  • Initial Side: Fixed ray along the positive $x$-axis where an angle begins.
  • Terminal Side: The position of the radius ray after rotating by angle $\theta$. The terminal side moves based on the magnitude of the rotation and can land in any quadrant or along any axis.
  • Abbreviations: Cosine is abbreviated as cos, and Sine is abbreviated as sin.

Evaluating Trigonometric Functions at 90 Degrees

• Example 1: Evaluation on a Circle of Radius $r = 7$

  • Circle Parameters: Radius $r = 7$, centered at $(0,0)$.
  • Key Points on Axes: Positive $x$-intercept at $(7,0)$, negative $x$-intercept at $(-7,0)$.
  • Rotation Process: Angle $\theta = 90^\circ$ rotates counterclockwise by a right angle from the positive $x$-axis.
  • Terminal Position: The terminal side lands directly on the positive $y$-axis.
  • Intersecting Point Coordinates: $(x, y) = (0, 7)$.
  • Cosine Evaluation:     $\cos(90^\circ) = \frac{x}{r} = \frac{0}{7} = 0$
  • Sine Evaluation:     $\sin(90^\circ) = \frac{y}{r} = \frac{7}{7} = 1$

• Example 2: Evaluation on the Unit Circle ($r = 1$

  • Circle Parameters: Radius $r = 1$, centered at $(0,0)$.
  • Intersecting Point Coordinates at $90^\circ$: $(x, y) = (0, 1)$.
  • Cosine Evaluation:     $\cos(90^\circ) = \frac{x}{r} = \frac{0}{1} = 0$
  • Sine Evaluation:     $\sin(90^\circ) = \frac{y}{r} = \frac{1}{1} = 1$

• Invariance Under Radius Changes:

  • Trigonometric function values depend exclusively on the angle $\theta$ and are completely independent of the circle's radius $r$.
  • Changing the radius rescales both coordinates and radius proportionally, preserving the ratio.

• Simplified Unit Circle Definitions:

  • The word "unit" means one ($1$). A unit circle is specifically a circle of radius $r = 1$
  • On the unit circle, dividing by $r = 1$ leaves the numerator unchanged:     $\cos(\theta) = x$$\sin(\theta) = y$
  • Thus, on the unit circle, $\cos(\theta)$ is simply the $x$-coordinate and $\sin(\theta)$ is simply the $y$-coordinate of the point on the circle.

Derivations of Special Angles on the Unit Circle

• Derivation for $\theta = 45^\circ$ ($\frac{\pi}{4}$ radians):

  • Geometry: Draw a right triangle inside the unit circle ($r = 1$) with horizontal leg $x$, vertical leg $y$, and hypotenuse $r = 1$.
  • Isosceles Property: Since one acute angle is $45^\circ$, the other acute angle is also $45^\circ$. This forms an isosceles right triangle, meaning $x = y$
  • Equation of the Unit Circle:     $x^2 + y^2 = 1$
  • Substitution and Solution:     $x^2 + x^2 = 1$$2x^2 = 1$$x^2 = \frac{1}{2}$$x = \pm \sqrt{\frac{1}{2}} = \pm \frac{1}{\sqrt{2}}$
  • Quadrant Selection: Because $\theta = 45^\circ$ lies in Quadrant I, $x$ and $y$ must be positive:     $x = \frac{1}{\sqrt{2}}$
  • Rationalizing the Denominator:     $x = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
  • Coordinates and Trigonometric Values:     $x = \frac{\sqrt{2}}{2}, \quad y = \frac{\sqrt{2}}{2}$$\cos(45^\circ) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$$\sin(45^\circ) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$
  • Point on Unit Circle: $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$

• Derivation for $\theta = 30^\circ$ ($\frac{\pi}{6}$ radians):

  • Geometry: Form a $30^\circ-60^\circ-90^\circ$ triangle inside the unit circle ($r = 1$).
  • Geometric Fact: The side opposite the $30^\circ$ angle is half the hypotenuse, giving $y = \frac{1}{2}$.
  • Substitution into Unit Circle Equation:     $x^2 + y^2 = 1$$x^2 + \left(\frac{1}{2}\right)^2 = 1$$x^2 + \frac{1}{4} = 1$$x^2 = 1 - \frac{1}{4} = \frac{3}{4}$$x = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}$
  • Trigonometric Values:     $\cos(30^\circ) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$$\sin(30^\circ) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$
  • Point on Unit Circle: $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$

• Derivation for $\theta = 60^\circ$ ($\frac{\pi}{3}$ radians):

  • Geometry: This triangle is a reflection/orientation swap of the $30^\circ-60^\circ-90^\circ$ triangle.
  • Role Swap: The horizontal and vertical sides swap roles compared to the $30^\circ$ angle:     $x = \frac{1}{2}, \quad y = \frac{\sqrt{3}}{2}$
  • Trigonometric Values:     $\cos(60^\circ) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$$\sin(60^\circ) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$
  • Point on Unit Circle: $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$

Fundamental Trigonometric Identities

• Derivation of the Pythagorean Identity:

  • Equation of the unit circle: $x^2 + y^2 = 1$
  • Substituting $x = \cos(\theta)$ and $y = \sin(\theta)$:     $(\cos(\theta))^2 + (\sin(\theta))^2 = 1$

• Standard Shorthand Notation:

  • To eliminate unnecessary parentheses, trigonometric powers are written with the exponent on the function name:     $\cos^2(\theta) + \sin^2(\theta) = 1$
  • The expression $\cos^2(\theta)$ explicitly means $(\cos(\theta))^2$, where the entire output value of the cosine function is squared.

Reference Angles

• Definition:

  • A reference angle (denoted $\theta_{\text{ref}}$) is the measure of the smallest acute angle formed between the terminal side of an angle $\theta$ and the horizontal axis ($x$-axis).

• Quadrant-by-Quadrant Rules and Examples:

  • Quadrant I:
  • The angle itself is acute and touches the horizontal axis directly.
  • Formula: $\theta_{\text{ref}} = \theta$
  • Quadrant II Example ($\theta = 167^\circ$):
  • Terminal side lies between $90^\circ$ and $180^\circ$.
  • Closest horizontal axis: Negative $x$-axis ($180^\circ$).
  • Calculation:       $\theta_{\text{ref}} = 180^\circ - 167^\circ = 13^\circ$
  • Quadrant III Example ($\theta = 210^\circ$):
  • Terminal side rotates past $180^\circ$ into Quadrant III.
  • Closest horizontal axis: Negative $x$-axis ($180^\circ$).
  • Calculation:       $\theta_{\text{ref}} = 210^\circ - 180^\circ = 30^\circ$
  • Quadrant IV Example ($\theta = 310^\circ$):
  • Terminal side rotates past $270^\circ$ into Quadrant IV.
  • Closest horizontal axis: Positive $x$-axis ($360^\circ$).
  • Calculation:       $\theta_{\text{ref}} = 360^\circ - 310^\circ = 50^\circ$

• General Rule for Terminal Side Boundaries:

  • Reference angle calculation always measures distance to the nearest horizontal axis ($180^\circ$ or $360^\circ$ / $0^\circ$), never the vertical axis ($90^\circ$ or $270^\circ$).

Questions and Discussion

• Etymology of Sine and Cosine:

  • Question: What does "co" or "sine" mean?
  • Answer: The prefix "co" typically denotes "with" or "together" (as in complementary angles), though full historical roots involve mistranslations from Sanskrit and Arabic terms for bowstrings.

• Clarification on Terminal Side:

  • Question: Is the terminal side basically the radius?
  • Answer: Yes, it is the line segment or ray representing where the radius lands after sweeping out angle $\theta$

• Reason for Simplicity in Initial Example:

  • Question: What's the catch? Why did this seem so simple?
  • Answer: The initial example used $\theta = 90^\circ$, where coordinates $(0,7)$ or $(0,1)$ are obvious. For non-quadrantal angles, determining $(x,y)$ coordinates requires trigonometric derivations.

• Arithmetic of Squaring Fractions:

  • Question: Why is $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$?
  • Answer: Multiplying a fraction by itself means taking a fraction of a fraction: half of a half is one quarter ($\frac{1}{4}$).

• Position of Reference Angles:

  • Question: Is the reference angle always outside the given angle?
  • Answer: No. In Quadrant III, the reference angle lies inside the total swept angle as the leftover past $180^\circ$. In Quadrants II and IV, it represents the remaining angle needed to reach the horizontal axis.

Administrative and Assessment Details

• Course Submissions:

  • Worksheets covering sections 5.1 and 5.2 are due and collected.

• In-Class Quiz Specifications:

  • Format: Single-sided sheet (one page total).
  • Content: Three main problems (Problem 1, Problem 2 parts a and b) plus one bonus problem.
  • Time Limit: Strict deadline at 12:00 PM due to facility/class scheduling.
  • Rules: Closed-book, closed-notes, no calculators allowed.