Unit Circle: Ultimate Guide

  • Understanding the Unit Circle

  • The y values for the coordinates are the sine theta values.

  • The x values for the coordinates are for the cosine theta values.

  • Example: On the unit circle, the cosine of 0 degrees has to be 1, because at 0 degrees on the coordinate the x-value is 1.

Periodic Properties: The Secret Sauce of Trig!

Trig functions are like your favorite Netflix series—they keep repeating no matter how far you go! Here's how it works:

  1. Sine & Cosine: Always Coming Back Around

    • Think of the unit circle as a 360° runway (or 2π2\pi2π radians for our math besties). After a full turn, sine and cosine repeat their values.

    • Formula: sin⁡(θ+2πk)=sin⁡(θ)cos⁡(θ+2πk)=cos⁡(θ)\sin(\theta + 2\pi k) = \sin(\theta) \cos(\theta + 2\pi k) = \cos(\theta)sin(θ+2πk)=sin(θ)cos(θ+2πk)=cos(θ)

      • kk is any whole number (positive or negative), like rewatching your fave episode.

    • So, if you have a crazy angle like sin⁡(450∘) just subtract 360 degrees.

  2. Tangent: Your Repeat Queen

    • Tangent is the party girl—she only needs half the circle (or π\piπ radians) to start repeating!

    • Formula: tan(θ+πk)=tan(θ)

    • Example: Want to find tan⁡(210∘)\)? Add or subtract 180° until you land on a familiar angle:

Reflection Time: Flipping Trig Over

Angles can live in all four quadrants, but their trig values get sign-flipped depending on where they are:

  • ASTC Rule: "All Students Take Calculus" tells us which functions are positive:

    • A: All trig functions are positive in Quadrant I.

    • S: Only sine (and cosecant) are positive in Quadrant II.

    • T: Only tangent (and cotangent) are positive in Quadrant III.

    • C: Only cosine (and secant) are positive in Quadrant IV.

So, if you’ve got cos⁡(240∘)\cos(240^\circ)cos(240∘):

  1. Subtract 180°: 240∘−180∘=60∘240^\circ - 180^\circ = 60^\circ240∘−180∘=60∘.

  2. Quadrant III = cosine is negative, so cos⁡(240∘)=−cos⁡(60∘)=−12\cos(240^\circ) = -\cos(60^\circ) = -\frac{1}{2}cos(240∘)=−cos(60∘)=−21​.

Pro Tips for Girlypop Math Mastery 💅

  • Big angles? No problem! Just subtract or add 360 until you’re in the 0°–360° range.

  • Negative angles? Add 360 ∘ to make them positive—it’s like taking a second look in the mirror to feel fabulous.

  • Radians-only vibes? Subtract or add 2π instead of 360°.

This way, you’ll always know how to evaluate trig values—even when they aren’t directly on the unit circle. Trig is just a little repetitive queen who loves patterns. 🌈