Lecture on Tangent Function

Introduction

• Speaker: Mr. Bean (filling in for Mr. Bruss)
• Topic: Tangent function, part of Unit 3
• Reminder: SOHCAHTOA principle - Tangent is "TOA" (opposite/adjacent)

Tangent and the Unit Circle

• Tangent on a Unit Circle:
  • Draw a unit circle with an angle theta
  • Tangent is the slope of the terminal ray: ( \frac{y}{x} )
  • In terms of sine and cosine: ( \tan \theta = \frac{\sin \theta}{\cos \theta} )
  • Tangent undefined when cosine is zero

Examples and Practice

1. Problem: Angle (\theta) = (\frac{\pi}{3})

   • Coordinate point: (\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right))
   • Slope of terminal ray = ( \sqrt{3} )

2. Problem: Angle (\theta = \frac{\pi}{4})

   • Coordinate point: ( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) )
   • Slope = 1

3. Problem: Angle (\theta = \frac{5\pi}{6})

   • Coordinate point: ( \left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right) )
   • Slope = ( -\frac{1}{\sqrt{3}} ) (rationalized: ( -\frac{\sqrt{3}}{3} ))

4. Problem: Angle (\theta = \frac{3\pi}{2})

   • Slope is undefined (because ( x = 0 ))

Graphing Tangent

• Behavior of Tangent:

  • Starts at 0, increases to infinity as it approaches (\frac{\pi}{2})
  • Undefined at vertical asymptotes (where cosine = 0)
  • Period of tangent is (\pi)

• Finding Asymptotes:

  • Vertical asymptotes occur at: (\frac{\pi}{2} + k\pi) for integer values of (k)

Tangent Function Characteristics

• Graph Features:
  • Increasing: Always increasing between asymptotes
  • Point of Inflection: Where graph changes concavity
  • Period: (\pi)

Transformations

1. Vertical Scaling (a):

   • Stretch or shrink vertically by (|a|)
   • Reflect over x-axis if (a < 0)

2. Horizontal Dilation (b):

   • Period changes by (\frac{1}{b})
   • Reflect over y-axis if (b < 0)

3. Horizontal Translation (c):

   • Shift left/right by (-c)

4. Vertical Translation (d):

   • Move up/down by (d)

Examples of Transformed Tangent Graphs

• Finding Vertical Asymptotes and Inflection Points:
  • Adjust for phase shifts and vertical shifts
  • Consider period adjustments: (\frac{\pi}{b})
  • Mark vertical asymptotes and use (a) to find inflection points

Practice with Graphing

• Example Problems to practice transformations
  • Analyze shifts, period changes, and scaling

Conclusion

• Understanding the tangent function's graph and transformations is crucial
• Practice sketching by identifying asymptotes and inflection points

Mr. Bean concludes the lesson, encouraging students to practice and prepare for future lessons with Mr. Bruss.