Unit Circle

1. The Unit Circle

The unit circle is a circle with a radius of 11 unit, centered at the origin (0,0)(0,0) of a coordinate plane. It is a fundamental tool for understanding trigonometric functions.

  • Equation: The equation of the unit circle is x2+y2=1x^2 + y^2 = 1.
  • Points on the Circle: Any point (x,y)(x,y) on the unit circle can be represented by an angle hetaheta measured counterclockwise from the positive x-axis.
2. Angles in the Unit Circle

Angles in the unit circle are typically measured in radians, though degrees can also be used.

  • Standard Position: An angle is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis.
  • Terminal Side: The ray that rotates from the initial side is called the terminal side. The point where the terminal side intersects the unit circle determines the trigonometric values for that angle.
  • Radians vs. Degrees:
    • 2π radians=360∘2\pi \text{ radians} = 360^{\circ}
    • π radians=180∘\pi \text{ radians} = 180^{\circ}
    • To convert degrees to radians: radians=degrees×π180∘\text{radians} = \text{degrees} \times \frac{\pi}{180^{\circ}}
    • To convert radians to degrees: degrees=radians×180∘π\text{degrees} = \text{radians} \times \frac{180^{\circ}}{\pi}
3. Trigonometric Functions and the Unit Circle

For any angle hetaheta in standard position, let (x,y)(x,y) be the point where its terminal side intersects the unit circle. The six trigonometric functions are defined as follows:

3.1 Primary Functions
  • Sine (sin): The y-coordinate of the point on the unit circle.
    • sin⁡(θ)=y\sin(\theta) = y
  • Cosine (cos): The x-coordinate of the point on the unit circle.
    • cos⁡(θ)=x\cos(\theta) = x
  • Tangent (tan): The ratio of the y-coordinate to the x-coordinate (provided x≠0x \neq 0).
    • tan⁡(θ)=yx=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{y}{x} = \frac{\sin(\theta)}{\cos(\theta)}
3.2 Reciprocal Functions
  • Cosecant (csc): The reciprocal of sine (provided y≠0y \neq 0).
    • csc⁡(θ)=1y=1sin⁡(θ)\csc(\theta) = \frac{1}{y} = \frac{1}{\sin(\theta)}
  • Secant (sec): The reciprocal of cosine (provided x≠0x \neq 0).
    • sec⁡(θ)=1x=1cos⁡(θ)\sec(\theta) = \frac{1}{x} = \frac{1}{\cos(\theta)}
  • Cotangent (cot): The reciprocal of tangent (provided y≠0y \neq 0).
    • cot⁡(θ)=xy=cos⁡(θ)sin⁡(θ)=1tan⁡(θ)\cot(\theta) = \frac{x}{y} = \frac{\cos(\theta)}{\sin(\theta)} = \frac{1}{\tan(\theta)}
4. Special Angles and Their Values

It's useful to memorize the coordinates (x,y)(x,y) for common angles on the unit circle, as these correspond to (cos⁡(θ),sin⁡(θ))(\cos(\theta), \sin(\theta)).

  • 0 or 2π (0∘ or 360∘):0 \text{ or } 2\pi \text{ (}0^{\circ} \text{ or } 360^{\circ}\text{):} (1,0)(1, 0)
  • π6 (30∘):\frac{\pi}{6} \text{ (}30^{\circ}\text{):} (32,12)(\frac{\sqrt{3}}{2}, \frac{1}{2})
  • π4 (45∘):\frac{\pi}{4} \text{ (}45^{\circ}\text{):} (22,22)(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})
  • π3 (60∘):\frac{\pi}{3} \text{ (}60^{\circ}\text{):} (12,32)(\frac{1}{2}, \frac{\sqrt{3}}{2})
  • π2 (90∘):\frac{\pi}{2} \text{ (}90^{\circ}\text{):} (0,1)(0, 1)
  • π (180∘):\pi \text{ (}180^{\circ}\text{):} (−1,0)(-1, 0)
  • 3π2 (270∘):\frac{3\pi}{2} \text{ (}270^{\circ}\text{):} (0,−1)(0, -1)
5. Quadrants and Signs of Trigonometric Functions

The signs of the trigonometric functions depend on the quadrant in which the terminal side of the angle hetaheta lies.

  • Quadrant I (0<θ<π20 < \theta < \frac{\pi}{2}): All functions (sin, cos, tan, csc, sec, cot) are positive.
  • Quadrant II (π2<θ<π\frac{\pi}{2} < \theta < \pi): Only sine and its reciprocal (cosecant) are positive. Cosine and tangent are negative.
  • Quadrant III (π<θ<3π2\pi < \theta < \frac{3\pi}{2}): Only tangent and its reciprocal (cotangent) are positive. Sine and cosine are negative.
  • Quadrant IV (3π2<θ<2π\frac{3\pi}{2} < \theta < 2\pi): Only cosine and its reciprocal (secant) are positive. Sine and tangent are negative.

A mnemonic device to remember positive functions in each quadrant, starting from Q1: "All Students Take Calculus" (All, Sine, Tangent, Cosine).