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3.11 ยท 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 ยท cot Alt Form
Write cotangent in terms of sin and cos (not using tan).
cot = ? / ?
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3.11 ยท 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 ยท 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 ยท Ex1 โ sec VA
f(x)=3sec(2x). Find a vertical asymptote.
Set cos(2x)=0 and solve for x.
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3.11 ยท Ex2 โ csc VA
g(x)=4โ2csc(ฯx). Find a vertical asymptote.
Set sin(ฯx)=0 and solve for x.
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3.11 ยท Ex3 โ Range
h(ฮธ)=3csc(ฮธ/2). What is the range of h?
Range of csc = (โโ,โ1]โช[1,โ). Multiply by 3.
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3.11 ยท Ex4 โ cot VA
k(x)=โ5cot(2ฯx). Find a vertical asymptote.
Set sin(2ฯx)=0 and solve for x.
2/10
✓ cot in terms of sin/cos
cot x = cos x / sin x (undefined when sin x = 0)
sec x = 1/cos x ยท csc x = 1/sin x ยท cot x = 1/tan x = cos x/sin x. Each is the reciprocal of its corresponding trig function.
1/10
✓ Definitions
sec=1/cos ยท csc=1/sin ยท cot=1/tan=cos/sin
Memory trick: co-secant pairs with sine (non-co), secant pairs with cosine. 'The co functions share the co.'
4/10
✓ Ranges
sec: (โโ,โ1]โช[1,โ) ยท csc: (โโ,โ1]โช[1,โ) ยท cot: (โโ,โ)
sec and csc can never be between โ1 and 1 (dividing 1 by a value โค1 gives โฅ1 in magnitude). cot has all reals as range, like tan.
3/10
✓ Asymptotes
sec: x=ฯ/2+ฯk ยท csc: x=ฯk ยท cot: x=ฯk
sec VA where cos=0 (x=ฯ/2+ฯk). csc and cot VAs where sin=0 (x=ฯk). csc and cot share the SAME asymptote locations.
6/10
✓ Ex2 โ csc VA
x = 1 (answer D)
sin(ฯx)=0 โ ฯx=ฯk โ x=k. All integers are asymptotes: x=0,ยฑ1,ยฑ2,โฆ So x=1 is a valid asymptote. Note: x=ฯ/2 is NOT an asymptote โ sin(ฯยทฯ/2)โ 0.
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✓ Ex1 โ sec VA
x = ฯ/4 (answer C)
cos(2x)=0 โ 2x=ฯ/2+ฯk โ x=ฯ/4+ฯk/2. Simplest positive: x=ฯ/4. Check: cos(2ยทฯ/4)=cos(ฯ/2)=0 โ
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✓ Ex4 โ cot VA
x = 1/2 (answer B)
sin(2ฯx)=0 โ 2ฯx=ฯk โ x=k/2. Asymptotes at x=0,ยฑ1/2,ยฑ1,โฆ So x=1/2 is valid. Note: x=1/4 gives sin(ฯ/2)=1โ 0.
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✓ Ex3 โ Range
(โโ,โ3]โช[3,โ) (answer C)
csc range (โโ,โ1]โช[1,โ). Multiply by 3: values โคโ1 become โคโ3, values โฅ1 become โฅ3. Answer [โ3,3] is completely wrong โ sec/csc never produce values between โ1 and 1.
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3.11 ยท Ex5 Solve
f(x)=4csc(x)+3=11. Find x in [0,2ฯ).
Isolate csc โ flip to sin โ unit circle
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3.11 ยท Ex6 Solve
h(x)=3โ(1/2)sec(x)=4. Find x in [0,2ฯ).
Isolate sec โ flip to cos โ unit circle