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Detailed Summary
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Chapter 5 Trigonometric Functions
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Chapter 5 Trigonometric Functions
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Chapter 5 Trigonometric Functions
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Chapter 5 Trigonometric Functions
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Chapter 5 Trigonometric Functions
8
The graph of y = sec x =
1
has vertical asymptotes
𝑐𝑜𝑠𝑥
where cos x = 0.
y
4
y
sec x
2
0
x
3
2
2
2
2
4
Period: 2π
Amplitude: None (𝑦 = 𝑠𝑒𝑐𝑥 increases and decreases without
bound)
Domain: {𝑥 |𝑥 ≠
(2𝑥+1)𝜋
2
𝑓𝑜𝑟 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑛 }
Range: (−∞, −1] ∪ [1, ∞) 𝑖. 𝑒. (𝑦| 𝑦 ≤ −1 𝑜𝑟 𝑦 ≥ 1)
𝜋
Vertical asymptotes: x =
(odd multiples of )
2
Symmetric to the y-axis (even function)
Example 2: To graph
(shown in gray) for reference. The
𝜋
asymptotes of y = 𝑠𝑒𝑐 (𝑥 − ) correspond
4
𝜋
to the x-intercepts of y=cos(𝑥 − )
4
To graph variations of y = tan x and y = cot x, graph the function without the
vertical shift first.
1. First graph two consecutive asymptotes.
2. Plot an x-intercept halfway between the asymptotes.
3. Sketch the general shape of the “parent” function between the asymptotes.
4. Apply a vertical shift if applicable.
5. Sketch additional cycles to the right or left as desired.
The graph of y = tan x =
𝒔𝒊𝒏𝒙
𝒄𝒐𝒔𝒙
has
vertical asymptotes where cos x = 0.
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Detailed Summary
one period
y
6
y
tan x
3
x
2
2
3
6
Period: π
Amplitude: None (y = tan x is unbounded)
Domain: { 𝑥| 𝑥 ≠
𝑓𝑜𝑟 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑛 }
Range: All Real numbers
Vertical asymptotes: x =
Symmetric to the origin (odd function)
23π
𝜋
(odd Multiples of )
2
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Chapter 5 Trigonometric Functions
The inverse cosine function (or arccosine), denoted by cos-1 or (arccos,) is defined by
y
y = cos-1 x ⟺ cos y = x for −1 ≤ 𝑥 ≤ 1 𝑎𝑛𝑑 0 ≤ 𝑦 ≤ 𝜋
y
cos
1
x
x
1
1
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Detailed Summary
Example 2:211
3
a.
2
cos2
b5
b. arccos(1) 5 0
The inverse tangent function (or arctangent), denoted by tan-1 or arctan, is defined
by
𝜋
y = tan-1 x ⟺ tan y = x for x ∈ ℝ and − 2 < 𝑦 <
𝜋
2
The graph of the inverse tangent function has horizontal asymptotes y =−
y=
𝜋
2
𝜋
2
and
y
y
2
x
y
tan
1
x
y
2
Example 3:
𝜋
a. arctan(-1) = −
4
b. tan-11 =
To compose a trigonometric function and an inverse trigonometric function
in either order requires that the argument to each function be within the
domain of that function.
Example 42:11
1
a. sin cos
2π
2bb
5 sin a
b. sin21cos
3
b 5 sin21 a 2b 5 2
c. tan(tan21 5) 5 5
d. tan21 a tan
b 5 tan2113 5
b5
2
3
2π
11
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