precalculus module 3...3.1.1 idea of trigonometry 3.1.2 trigonometric functions 3.1.1 idea of...
TRANSCRIPT
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3. Trigonometry
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3.1 Introduction to Trigonometry
3.2 Trigonometric Examples
3.3 Radians
3.4 Periodicity and Plotting Trigonometric Functions
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3.5 Major Trigonometric Identities
3.6 Inverse Trigonometric Functions
3.7 Domain and Range of Trigonometric Functions
3.8 Laws of Sines and Cosines
3.9 Trigonometric Equations
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3.1 Introduction to Trigonometry
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3.1.1 Idea of Trigonometry
3.1.2 Trigonometric Functions
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3.1.1 Idea of Trigonometry
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• For us, trigonometry is the study of right triangles, i.e. a triangle with one right angle.
• These triangles have two legs, and a hypotenuse opposite the right angle. These are related via the famous Pythagorean Theorem:
If a, b are the lengths of the two legs of a right
triangle, and c is the length of the hypotenuse, then a2 + b2 = c2
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• The fundamental trigonometric functions are functions that take as input angles of a right triangle.
• These functions are defined in terms of ratios of side lengths corresponding to the angle.
• The value of these functions depend only on the angles, not on the particular choice of triangle.
• This is because two right triangles with another common angle are necessarily similar, i.e. side lengths are the same after multiplying them all by the same constant.
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3.1.2 Trigonometric Functions
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• Let be an angle of a right triangle. Assume that is not the right angle.
• There is a side opposite , call its length .
• There is also a side adjacent to , call it .
• Call the hypotenuse .
✓
✓
✓Opp
✓ Adj
Hyp
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sin(✓) =Opp
Hyp
cos(✓) =Adj
Hyp
tan(✓) =Opp
Adj
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Compute sin(✓), cos(✓), tan(✓)
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Compute sin(✓), cos(✓), tan(✓)
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sec(✓) =Hyp
Adj
csc(✓) =Hyp
Opp
cot(✓) =Adj
Opp
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Compute sec(✓), csc(✓), cot(✓)
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Compute sec(✓), csc(✓), cot(✓)
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3.2 Trigonometric Examples
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30-60-90 Triangle
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sin(30) =
1
2
cos(30) =
p3
2
tan(30) =
1p3
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sin(60) =
p3
2
cos(60) =
1
2
tan(60) =
p3
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45-45-90 Triangle
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sin(45) =
1p2
cos(45) =
1p2
tan(45) = 1
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3.3 Radians
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• So far, we have discussed the inputs of trigonometric functions in terms of degrees.
• Another system of inputs is more common and convenient for higher mathematics, such as calculus, called radians.
• Radians replace degrees with numbers according to the following exchange rate:
180 degrees = ⇡ radians
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• To convert from degrees to radians, one multiplies by a factor of
• To convert from radians to degrees, one multiples by a factor of
⇡
180
180
⇡
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Convert from degrees to radians:
360
�
90
�
135
�
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Convert from radians to degrees:
⇡
3
3⇡
2
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3.4 Periodicity and Plotting Trigonometric Functions
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• Recall that a rotation of is the same as a rotation of
• In general, an angle of is the same as a rotation of
• In radians, an angle of magnitude is equivalent to an angle of magnitude
360�
0�
✓�
(360 + ✓)�
✓✓ + 2⇡
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• This implies that the trigonometric functions are periodic.
• They repeat themselves after a fixed interval.
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sin(✓ + 2⇡) = sin(✓)
cos(✓ + 2⇡) = cos(✓)
csc(✓ + 2⇡) = csc(✓)
sec(✓ + 2⇡) = sec(✓)
![Page 32: Precalculus Module 3...3.1.1 Idea of Trigonometry 3.1.2 Trigonometric Functions 3.1.1 Idea of Trigonometry • For us, trigonometry is the study of right triangles, i.e. a triangle](https://reader034.vdocuments.mx/reader034/viewer/2022042621/5f645ee760532d1f7161ed6f/html5/thumbnails/32.jpg)
Plot f(x) = sin(x)
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Plot f(x) = cos(x)
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tan(✓ + ⇡) = tan(✓)
cot(✓ + ⇡) = cot(✓)
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Plot f(x) = tan(x)
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3.5 Major Trigonometric Identities
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• A trigonometric identity is relation between trigonometric functions.
• They are essential for understanding the behavior of trigonometric functions.
• They can make calculations easier.
• There are many such identities, but we focus on just a few of the most important
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Reciprocal Identities
sin(✓) =1
csc(✓)
cos(✓) =1
sec(✓)
tan(✓) =1
cot(✓)
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Simplify sec(✓) cos(✓)
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Simplify csc(✓) tan(✓)
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Pythagorean Identities
sin
2(✓) + cos
2(✓) =1
tan
2(✓) + 1 = sec
2(✓)
cot
2(✓) + 1 = csc
2(✓)
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Simplify
s1� sin2(✓)
sin2
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Simplify csc2(✓) tan2(✓)
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Double Angle Identities
sin(2✓) =2 sin(✓) cos(✓)
cos(2✓) = cos
2(✓)� sin
2(✓)
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Compute sin(120
�)
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3.6 Inverse Trigonometric Functions
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• Recall that our trigonometric functions input an angle and output a number, based on ratios of sides of triangles.
• We can define inverse trigonometric functions that input a number and out put an angle.
• For example, we can define an inverse function to the function , call it
• This function has the property that on its domain,
f(x) = sin(x)f
�1(x) = arcsin(x)
sin(arcsin(x)) = x
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arcsin
✓1
2
◆=30�
arcsin
p3
2
!=60�
arcsin (1) =90�
arcsin (0) =0�
![Page 49: Precalculus Module 3...3.1.1 Idea of Trigonometry 3.1.2 Trigonometric Functions 3.1.1 Idea of Trigonometry • For us, trigonometry is the study of right triangles, i.e. a triangle](https://reader034.vdocuments.mx/reader034/viewer/2022042621/5f645ee760532d1f7161ed6f/html5/thumbnails/49.jpg)
• Similarly, one may define inverse trigonometric functions to:cos(x), tan(x), sec(x), csc(x), cot(x)
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Compute arccos(1)
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Compute arctan(
p3)
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3.7 Domain and Range of Trigonometric Functions
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3.7.1 Domain and Range of Trigonometric Functions
3.7.2 Domain and Range of Inverse Trigonometric Functions
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3.7.1 Domain and Range of Trigonometric Functions
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• What kinds of number make sense as inputs to the trigonometric functions?
• What kinds of numbers can be outputs? In other words, what are their domains and ranges?
• We will plot the basic functions, then determine domain and range.
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dom(sin) =(�1,1)
range(sin) =[�1, 1]
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Compute the domain and range of f(x) = 2 sin(2x) + 1
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dom(cos) =(�1,1)
range(cos) =[�1, 1]
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Compute the domain and range of f(x) = sec(x)
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dom(tan) ={x | x 6= ⇡k
2
, k an integer}
range(tan) =(�1,1)
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Compute the domain and range of f(x) = tan
✓2x
⇡
◆
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3.7.2 Domain and Range of Inverse Trigonometric Functions
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• Normally to find the domain and range of , one simply switches the domain and range of .
• However, the inverse trigonometric functions fail the horizontal line test on their full domains.
• This means that their inverse functions are not well-defined unless the domain of the original function is restricted.
f�1f
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dom(arcsin) =[�1, 1]
range(arcsin) =
h�⇡
2
,⇡
2
i
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Compute the domain and range of f(x) = � arcsin(x+ 1)
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dom(arccos) =[�1, 1]
range(arccos) =[0,⇡]
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Compute the domain and range of f(x) = 2 + arccos(3x)
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dom(arctan) =(�1,1)
range(arctan) =
⇣�⇡
2
,⇡
2
⌘
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Compute the domain and range of f(x) = arctan(x� 1)
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3.8 Law of Sines and Cosines
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• Trigonometric Functions can also be used to study triangles that are not right triangles.
• The laws of sines and cosines provide relationships between the sides and angles of a generic triangle.
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Law of Sines
a
sin(A)=
b
sin(B)=
c
sin(C)
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Law of Cosines
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3.9 Trigonometric Equations
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• There are problems involving trigonometric functions.
• In some cases, inverse trigonometric functions will be useful.
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