Math127 The Unit Circle

The unit circle is a circle centered at the origin (0,0) with a radius of 1.
Its equation is:[ x^2 + y^2 = 1 ]

Each point on the unit circle can be defined by coordinates (x, y).
For an angle ( t ):
- ( ext{sin}(t) = y )
- ( ext{cos}(t) = x )
- Coordinates are represented as ( ( ext{cos}(t), ext{sin}(t) ) )
Tangent is defined as:[ ext{tan}(t) = \frac{ ext{sin}(t)}{\text{cos}(t)} = \frac{y}{x} ]
Reciprocal Functions:
- Cosecant: ( ext{cosec}(t) = \frac{1}{y} ) (related to sine)
- Secant: ( ext{sec}(t) = \frac{1}{x} ) (related to cosine)
- Cotangent: ( ext{cot}(t) = \frac{x}{y} ) (reciprocal of tangent)

For point ( p ): ((\frac{\sqrt{3}}{2}, \frac{1}{2}))
- ( ext{sin}(p) = \frac{1}{2}, ext{cos}(p) = \frac{\sqrt{3}}{2} )
- ( ext{tan}(p) = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} )
- ( ext{cosec}(p) = 2, ext{sec}(p) = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}, ext{cot}(p) = \sqrt{3} )
For angle ( t = \frac{\pi}{2} ): (90 degrees)
- Point: ((0, 1))
- ( ext{sin}(t) = 1, ext{cos}(t) = 0 )
- ( ext{tan}(t) = \frac{1}{0} ) is undefined
- Reciprocal functions: ( ext{cosec}(t) = 1, ext{sec}(t) = \frac{1}{0} ) (undefined), ( ext{cot}(t) = 0 )

For angle ( c ): ( \pi ) (180 degrees)
- Point: ((-1, 0))
- Trig values:
  - ( ext{sin}(\pi) = 0, ext{cos}(\pi) = -1, ext{tan}(\pi) = 0 )
  - ( ext{cosec}(\pi) = \text{undefined}, ext{sec}(\pi) = -1, ext{cot}(\pi) = \text{undefined})
For angle ( d ): ( \frac{3\pi}{2} ) (270 degrees)
- Point: ((0, -1))
- Trig values:
  - ( ext{sin}(\frac{3\pi}{2}) = -1, ext{cos}(\frac{3\pi}{2}) = 0, ext{tan}(\frac{3\pi}{2}) = \text{undefined} )
  - ( ext{cosec}(\frac{3\pi}{2}) = -1, ext{sec}(\frac{3\pi}{2}) = \text{undefined}, ext{cot}(\frac{3\pi}{2}) = 0 )

45 degrees (or ( \frac{\pi}{4} )): ( \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) )
60 degrees (or ( \frac{\pi}{3} )): ( \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) )
30 degrees (or ( \frac{\pi}{6} )): ( \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right) )
Relationship between 30 and 60 degrees:
- The sine and cosine values swap their positions between these angles.

Finding exact values based on known angles:
- For ( \sin(45) ): ( \frac{\sqrt{2}}{2} )
- For ( \tan(180) ): returns value of 0
- For ( ext{sec}(\frac{\pi}{4}) = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} ) after rationalizing

Finding sine, cosine, and tangent involves identifying the point's coordinates on the unit circle.
Rationalizing denominators is critical for cosecant and secant calculations.
Understanding quadrant signs and angles is essential when determining the values of sine, cosine, and tangent.

Unit Circle Overview The unit circle is centered at the origin (0,0) with a radius of 1, described by the equation: x² + y² = 1.

Key Points and Definitions

Each point on the unit circle has coordinates (x, y) where:
- sin(t) = y
- cos(t) = x
- Coordinates: (cos(t), sin(t))
- tan(t) = sin(t)/cos(t) = y/x

Reciprocal Functions:

Example Values:

For point (√3/2, 1/2):
- sin(p) = 1/2, cos(p) = √3/2, tan(p) = √3/3
- cosec(p) = 2, sec(p) = 2√3/3, cot(p) = √3
For angle t = π/2 (90°):
- Point = (0, 1); sin(t) = 1, cos(t) = 0; tan(t) = undefined
- cosec(t) = 1, sec(t) = undefined, cot(t) = 0
For angle π (180°):
- Point = (-1, 0); sin(π) = 0, cos(π) = -1, tan(π) = 0
- cosec(π) = undefined, sec(π) = -1, cot(π) = undefined
For angle 3π/2 (270°):
- Point = (0, -1); sin(3π/2) = -1, cos(3π/2) = 0, tan(3π/2) = undefined
- cosec(3π/2) = -1, sec(3π/2) = undefined, cot(3π/2) = 0

Important Angles and Coordinates:

0° (0): (1, 0)
90° (π/2): (0, 1)
180° (π): (-1, 0)
270° (3π/2): (0, -1)
360° (2π): (1, 0)

Key Reference Angles:

Trigonometric Function Calculation Tips:

Identify point coordinates on the unit circle for sine, cosine, and tangent values.
Rationalizing denominators is key for cosecant and secant.
Understand quadrant signs and angles for accurate calculations.