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3.8 · tan Definition
Write all three equivalent definitions of tan θ.
Ratio · slope · coordinate
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3.8 · Why Asymptotes?
Why does tan θ have vertical asymptotes at θ = π/2 + kπ?
What is tan θ = sin θ / cos θ when cos θ = 0?
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3.8 · Period of tan
What is the period of y = tan θ? How does it compare to sin and cos?
NOT the same as sin/cos — half as long
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3.8 · Period Formula
f(θ) = a·tan(bθ) + d. Write the formula for the period. What is different from sin/cos?
P = ?
tan base period is π, not 2π
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3.8 · Asymptote Formula
g(x) = 2tan(x/3) − 1. Write the formula for the vertical asymptotes.
Set argument = π/2 + kπ
b = 1/3 here
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3.8 · Example 1
Graph of f has asymptotes at x = ±π/2 and f(π/8) ≈ 1. f(x) = a·tan(bx)+d. Find a, b, d.
Period from asymptotes first
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3.8 · Example 2
h(x) = 4tan(2x) + 5. What is the period?
P = π/|b|
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3.8 · tan Values Q1
State the exact values: tan(π/6), tan(π/4), tan(π/3).
Use sin/cos values from unit circle
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✓ Why Asymptotes
tan θ = sin θ/cos θ. When cos θ = 0, division by zero → undefined. cos θ = 0 at θ = π/2 + kπ, where the terminal ray is vertical and slope is infinite.
3 definitions: tan θ = sin θ/cos θ · tan θ = slope of terminal ray · tan θ = y/x for point (x,y) on circle.
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✓ tan Definition
tan is undefined at θ = π/2 + kπ (where cos=0). Each half-turn around the unit circle creates another asymptote.
tan θ = sin θ/cos θ = y/x = slope of terminal ray. All three are equivalent.
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✓ Period Formula
P = π/|b| ← uses π, not 2π! Example: tan(2x) → P=π/2. Example: tan(x/3) → P=3π.
tan base period = π. sin/cos base period = 2π. Same b formula structure, different base: P=π/|b| vs P=2π/|b|.
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✓ Period of tan
Period of tan θ = π. Half as long as sin and cos (period=2π). One full pattern of tan completes in π radians.
tan goes from −∞ to +∞ within each interval of width π between asymptotes.
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✓ Example 1 — a,b,d
a = 1 · b = 2 · d = 0 → f(x) = tan(2x)
Period = π/2 (asymptotes π apart). π/|b|=π/2 → b=2. No shift → d=0. f(π/8)=tan(π/4)=1 → a=1.
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✓ Asymptote Formula
x = 3π/2 + 3πk (k any integer)
Set x/3 = π/2+kπ → x = 3π/2+3kπ. Period=3π confirmed. d=−1 shifts graph down but asymptotes stay at same x-values.
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✓ tan Values Q1
tan(π/6) = 1/√3 · tan(π/4) = 1 · tan(π/3) = √3
π/6: (1/2)÷(√3/2)=1/√3. π/4: (√2/2)÷(√2/2)=1. π/3: (√3/2)÷(1/2)=√3. Each from sin÷cos.
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✓ Example 2 — Period
Period = π/2
b=2 → P=π/|b|=π/2. The a=4 and d=5 don't affect the period.
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3.8 · tan Undefined
For which angles is tan θ undefined? For which is tan θ = 0?
Think about when cos θ = 0 and when sin θ = 0
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3.8 · ASTC for tan
In which quadrants is tan θ positive? In which is it negative?
tan = sin/cos — same sign means positive
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✓ ASTC for tan
Q1: tan>0 · Q2: tan<0 · Q3: tan>0 · Q4: tan<0
tan=sin/cos. Same signs → positive (Q1: both+, Q3: both−). Opposite signs → negative (Q2, Q4).
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✓ tan Undefined/Zero
Undefined at θ=π/2+kπ (cos=0) · Zero at θ=kπ (sin=0, cos≠0)
tan=sin/cos. Undefined when denominator=0: cos θ=0 → θ=π/2+kπ. Zero when numerator=0: sin θ=0 → θ=kπ.