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3.11 · Definitions
Write the definitions of sec x, csc x, and cot x as reciprocals.
Which original function does each reciprocal pair with?
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3.11 · cot Alt Form
Write cotangent in terms of sin and cos (not using tan).
cot = ? / ?
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3.11 · Asymptotes
Where does y=sec x have vertical asymptotes? Where does y=csc x? Where does y=cot x?
Set the denominator = 0
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3.11 · Ranges
What is the range of sec x? Of csc x? Of cot x?
Can sec/csc ever equal 0.5?
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3.11 · Ex1 — sec VA
f(x)=3sec(2x). Find a vertical asymptote.
Set cos(2x)=0 and solve for x.
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3.11 · Ex2 — csc VA
g(x)=4−2csc(πx). Find a vertical asymptote.
Set sin(πx)=0 and solve for x.
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3.11 · Ex3 — Range
h(θ)=3csc(θ/2). What is the range of h?
Range of csc = (−∞,−1]∪[1,∞). Multiply by 3.
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3.11 · Ex4 — cot VA
k(x)=−5cot(2πx). Find a vertical asymptote.
Set sin(2πx)=0 and solve for x.
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✓ cot in terms of sin/cos
cot x = cos x / sin x (undefined when sin x = 0)
sec x = 1/cos x · csc x = 1/sin x · cot x = 1/tan x = cos x/sin x. Each is the reciprocal of its corresponding trig function.
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✓ Definitions
sec=1/cos · csc=1/sin · cot=1/tan=cos/sin
Memory trick: co-secant pairs with sine (non-co), secant pairs with cosine. 'The co functions share the co.'
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✓ Ranges
sec: (−∞,−1]∪[1,∞) · csc: (−∞,−1]∪[1,∞) · cot: (−∞,∞)
sec and csc can never be between −1 and 1 (dividing 1 by a value ≤1 gives ≥1 in magnitude). cot has all reals as range, like tan.
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✓ Asymptotes
sec: x=π/2+πk · csc: x=πk · cot: x=πk
sec VA where cos=0 (x=π/2+πk). csc and cot VAs where sin=0 (x=πk). csc and cot share the SAME asymptote locations.
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✓ Ex2 — csc VA
x = 1 (answer D)
sin(πx)=0 → πx=πk → x=k. All integers are asymptotes: x=0,±1,±2,… So x=1 is a valid asymptote. Note: x=π/2 is NOT an asymptote — sin(π·π/2)≠0.
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✓ Ex1 — sec VA
x = π/4 (answer C)
cos(2x)=0 → 2x=π/2+πk → x=π/4+πk/2. Simplest positive: x=π/4. Check: cos(2·π/4)=cos(π/2)=0 ✓
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✓ Ex4 — cot VA
x = 1/2 (answer B)
sin(2πx)=0 → 2πx=πk → x=k/2. Asymptotes at x=0,±1/2,±1,… So x=1/2 is valid. Note: x=1/4 gives sin(π/2)=1≠0.
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✓ Ex3 — Range
(−∞,−3]∪[3,∞) (answer C)
csc range (−∞,−1]∪[1,∞). Multiply by 3: values ≤−1 become ≤−3, values ≥1 become ≥3. Answer [−3,3] is completely wrong — sec/csc never produce values between −1 and 1.
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3.11 · Ex5 Solve
f(x)=4csc(x)+3=11. Find x in [0,2π).
Isolate csc → flip to sin → unit circle
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3.11 · Ex6 Solve
h(x)=3−(1/2)sec(x)=4. Find x in [0,2π).
Isolate sec → flip to cos → unit circle