πŸ“

Tangent Function
Quiz Adventure!

Math 41 · Trigonometry · Tangent Functions 🌊

πŸ“– Key Concepts
Definition
\(\tan(x) = \dfrac{\sin(x)}{\cos(x)}\)
Period
\(T = \pi\)
Domain
\(x \neq \dfrac{\pi}{2} + n\pi\)
Range
\((-\infty, +\infty)\)
Asymptotes
\(x = \dfrac{\pi}{2} + n\pi\)
Midpoint
\(\tan(0) = 0\)
General Form
\(f(x) = A\tan(Bx - C) + D\)
Period
\(T = \dfrac{\pi}{|B|}\)
Phase Shift
\(\dfrac{C}{B}\)
Vertical Shift
\(D\) units up/down
Amplitude
None (range = ℝ)
Stretch |A|
Vertical stretch

✏️ Example: f(x) = tan(3x) βˆ’ 1

  • 1B = 3, so Period = Ο€/3
  • 2C = 0, so Phase Shift = 0
  • 3D = βˆ’1, so Vertical Shift = down 1
  • 4Domain: x β‰  Ο€/6 + nΟ€/3
  • 5Range: (βˆ’βˆž, +∞) (all reals)
tan(0)
0
tan(Ο€/6)
\(\dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3}\)
tan(Ο€/4)
1
tan(Ο€/3)
\(\sqrt{3}\)
tan(Ο€/2)
Undefined
tan(3Ο€/4)
βˆ’1
tan(5Ο€/6)
\(-\dfrac{\sqrt{3}}{3}\)
tan(Ο€)
0

✏️ How to evaluate tan(2Ο€/3)

  • 12Ο€/3 is in Quadrant II (Ο€/2 < 2Ο€/3 < Ο€)
  • 2Reference angle = Ο€ βˆ’ 2Ο€/3 = Ο€/3
  • 3tan(Ο€/3) = √3
  • 4In Q II, tan is NEGATIVE β†’ tan(2Ο€/3) = βˆ’βˆš3
🎯 Choose Your Problems!

Click individual problems to solve, or tackle all 10 at once!

Problem 1
πŸŽ‰
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