Unit Circle Trig (tan, cot, sec, csc)
Introduction to Trigonometric Functions
Trigonometric Functions Covered: Sine, Cosine, Tangent, Cotangent, Secant, Cosecant.
Key Relationships:
Sine (sin) = Y value on the unit circle.
Cosine (cos) = X value on the unit circle.
Review of Basic Fraction Problems
Dividing fractions:
Example: Simplify 1 divided by 1/2.
Rewrite as: 1 * (2/1).
Result: 2.
Undefined Division:
Example: 1 divided by 0.
Result: Undefined (calculator shows "error").
Zero in Numerator:
Example: 0 divided by -1.
Result: 0 (valid division).
Important Trigonometric Formulas
Pause the video to note down formulas for later use.
Tangent and Cotangent Calculations
Tangent of 90 degrees:
Formula: Tangent = Y/X.
Calculation: Tan(90°) = 1/0 = Undefined.
Concept: Asymptotes in future graphing.
Cotangent of 3π/4:
Formula: Cotangent = X/Y.
Coordinates at 3π/4: (−√2/2, √2/2).
Calculation: Cot(3π/4) = (−√2/2) / (√2/2) = −1.
Secant and Cosecant Calculations
Secant of π/3:
Formula: Secant = 1/X.
X at π/3: 1/2.
Calculation: Sec(π/3) = 1/(1/2) = 2.
Cosecant of −120 degrees:
Formula: Cosecant = 1/Y.
Y-value found by reflecting +120 degrees across the X-axis.
Coordinates: Y = −√3/2.
Calculation:
Csc(−120°) = 1/(−√3/2)
Using Copy, Change, Flip: = 2/(-√3) = -2/√3.
Rationalizing: Result = -2√3/3.
Practice Questions Recap
Question: Coordinate Calculation Result
Answer: 1 (by evaluating negative values).
Question: Secant at 90 degrees.
Answer: 1.
Question: Undefined division at 3π/2.
Answer: Undefined due to division by zero.
Question: Tangent at 300 degrees.
Calculation: Negative√3/2 for Y and 1/2 for X.
Final Answer: -√3.
Question: Undefined Cotangent at π.
Same concept as division by zero.
Question: Cosecant calculation for negative 150 degrees.
Result uses reflection and simplifies to -2.