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Chapter 2 Section 3
 The domain of the sine function is the set of all real numbers
 The domain of the cosine function is the set of all real numbers
 Do you know why?
 The domain of the sine function is the set of all real numbers
 The domain of the cosine function is the set of all real numbers
 Do you know why?
 Remember
 sin 𝜃 = 𝑦 and cos 𝜃 = 𝑥
 Aren’t x and y values any real number in the coordinate system?
 The domain of the tangent function is the set of all real numbers, except odd
integer multiples of
𝜋
2
(90°).
 The domain of the secant function is the set of all real numbers, except odd integer
𝜋
2
multiples of (90°).
 Do you know why?
 The domain of the tangent function is the set of all real numbers, except odd
integer multiples of
𝜋
2
(90°).
 The domain of the secant function is the set of all real numbers, except odd integer
𝜋
2
multiples of (90°).
 Do you know why?
 Remember
 tan 𝜃 =
𝑦
𝑥
1
and sec 𝜃 = 𝑥 and at what angles would x = 0?
 The domain of the cotangent function is the set of all real numbers, except integer
multiples of 𝜋 (180°).
 The domain of the cosecant function is the set of all real numbers, except integer
multiples of 𝜋 (180°).
 Do you know why?
 The domain of the cotangent function is the set of all real numbers, except integer
multiples of 𝜋 (180°).
 The domain of the cosecant function is the set of all real numbers, except integer
multiples of 𝜋 (180°).
 Do you know why?
 Remember
 cot 𝜃 =
𝑥
𝑦
1
and csc 𝜃 = 𝑦 and at what angles would y = 0?
 Recall P = (x,y) is any point on the unit circle that corresponds to a particular angle 𝜃
−1 ≤ 𝑥 ≤ 1 and −1 ≤ 𝑦 ≤ 1
 Recall P = (x,y) is any point on the unit circle that corresponds to a particular angle 𝜃
−1 ≤ 𝑥 ≤ 1 and −1 ≤ 𝑦 ≤ 1
−1 ≤ cos(𝜃) ≤ 1 and −1 ≤ sin(𝜃) ≤ 1
 If 𝜃 is not an integer multiple of 𝜋 then csc 𝜃 ≤ −1
 If 𝜃 is not an odd integer multiple of
𝜋
2
or csc 𝜃 ≥ 1
then sec 𝜃 ≤ −1 or sec 𝜃 ≥ 1
 If 𝜃 is not an integer multiple of 𝜋 then csc 𝜃 ≤ −1
 If 𝜃 is not an odd integer multiple of
 Remember tan 𝜃 =

𝑦
𝑥
𝜋
2
or csc 𝜃 ≥ 1
then sec 𝜃 ≤ −1 or sec 𝜃 ≥ 1
𝑥
𝑦
and 𝑐𝑜𝑡 𝜃 = and x and y can be any Real Number
Thus,
−∞ ≤ tan 𝜃 ≤ ∞
−∞ ≤ cot 𝜃 ≤ ∞
 A function 𝑓 is called periodic if there is a positive number 𝑝 such that, whenever 𝜃
is in the domain of 𝑓, so is 𝜃 + 𝑝, and 𝑓 𝜃 + 𝑝 = 𝑓(𝜃)
 A function 𝑓 is called periodic if there is a positive number 𝑝 such that, whenever 𝜃
is in the domain of 𝑓, so is 𝜃 + 𝑝, and 𝑓 𝜃 + 𝑝 = 𝑓(𝜃)
 Example:
17𝜋
sin
4
 Example:
cos 5𝜋
 Example:
5𝜋
tan
4
Quadrant of P
I
II
III
IV
𝒔𝒊𝒏 𝜽 , 𝒄𝒔𝒄(𝜽)
𝒄𝒐𝒔 𝜽 , 𝒔𝒆𝒄(𝜽)
ta𝒏 𝜽 , 𝒄𝒐𝒕(𝜽)
 Example:
 Find the quadrant in which an angle lies when 𝑠𝑖𝑛 𝜃 < 0 𝑎𝑛𝑑 cos 𝜃 < 0
csc 𝜃 =
1
𝑦
= ____________
sec 𝜃 =
1
𝑥
= ____________
cot 𝜃 =
𝑥
𝑦
=
1
𝑦
𝑥
= ____________
 tan 𝜃 = ____________
 cot 𝜃 = ____________
 tan 20° −
sin(20°)
cos(20°)
 sin2
𝜋
12
+
1
sec2
𝜋
12
 Given: sin 𝜃 =
1
3
and cos 𝜃 < 0
 Review
 A function is EVEN if 𝑓 −𝜃 = 𝑓(𝜃)
 A function is ODD if 𝑓 −𝜃 = −𝑓(𝜃)
 Review
 A function is EVEN if 𝑓 −𝜃 = 𝑓(𝜃)
 A function is ODD if 𝑓 −𝜃 = −𝑓(𝜃)
 Using identities
 Review
 A function is EVEN if 𝑓 −𝜃 = 𝑓(𝜃)
 A function is ODD if 𝑓 −𝜃 = −𝑓(𝜃)
Odd
Even
Odd
 Find the exact values for the following
 sin −45°
 cos −𝜋
 cot −
3𝜋
2
 tan −
37𝜋
4