Precalculus Module 3
3. Trigonometry
3.1 Introduction to Trigonometry 3.2 Trigonometric Examples 3.3 Radians 3.4 Periodicity and Plotting Trigonometric Functions
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
3.1 Introduction to Trigonometry
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 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 2 2 2 triangle, and c is the length of the hypotenuse, then a + b = c
•
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.
3.1.2 Trigonometric Functions
•
Let ✓ be an angle of a right triangle. Assume that ✓ is not the right angle.
•
There is a side opposite ✓, call its length Opp .
•
There is also a side adjacent to ✓ , call it Adj.
•
Call the hypotenuse Hyp .
Opp sin(✓) = Hyp
Adj cos(✓) = Hyp
Opp tan(✓) = Adj
Compute sin(✓), cos(✓), tan(✓)
Compute sin(✓), cos(✓), tan(✓)
Hyp sec(✓) = Adj
Hyp csc(✓) = Opp
Adj cot(✓) = Opp
Compute sec(✓), csc(✓), cot(✓)
Compute sec(✓), csc(✓), cot(✓)
3.2 Trigonometric Examples
30-60-90 Triangle
1 sin(30) = 2p
3 cos(30) = 2 1 tan(30) = p 3
p
3 sin(60) = 2 1 cos(60) = 2 p tan(60) = 3
45-45-90 Triangle
1 sin(45) = p 2 1 cos(45) = p 2 tan(45) = 1
3.3 Radians
•
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
•
To convert from degrees to radians, one multiplies by a factor of ⇡
180 •
To convert from radians to degrees, one multiples by a factor of 180
⇡
Convert from degrees to radians: 360 90 135
Convert from radians to degrees: ⇡ 3 3⇡ 2
3.4 Periodicity and Plotting Trigonometric Functions
•
Recall that a rotation of 360 is the same as a rotation of 0
•
In general, an angle of (360 + ✓) is the same as a rotation of ✓
•
In radians, an angle of magnitude ✓ is equivalent to an angle of magnitude ✓ + 2⇡
•
This implies that the trigonometric functions are periodic.
•
They repeat themselves after a fixed interval.
sin(✓ + 2⇡) = sin(✓) cos(✓ + 2⇡) = cos(✓) csc(✓ + 2⇡) = csc(✓) sec(✓ + 2⇡) = sec(✓)
Plot f (x) = sin(x)
Plot f (x) = cos(x)
tan(✓ + ⇡) = tan(✓) cot(✓ + ⇡) = cot(✓)
Plot f (x) = tan(x)
3.5 Major Trigonometric Identities
•
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
Reciprocal Identities 1 sin(✓) = csc(✓) 1 cos(✓) = sec(✓) 1 tan(✓) = cot(✓)
Simplify sec(✓) cos(✓)
Simplify csc(✓) tan(✓)
Pythagorean Identities 2
2
sin (✓) + cos (✓) =1 2
2
2
2
tan (✓) + 1 = sec (✓) cot (✓) + 1 = csc (✓)
Simplify
s
1
2
sin (✓) 2 sin
2
2
Simplify csc (✓) tan (✓)
Double Angle Identities sin(2✓) =2 sin(✓) cos(✓) 2
cos(2✓) = cos (✓)
2
sin (✓)
Compute sin(120 )
3.6 Inverse Trigonometric Functions
•
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 1 the function f (x) = sin(x), call it f (x) = arcsin(x)
•
This function has the property that on its domain, sin(arcsin(x)) = x
✓ ◆ 1 arcsin =30 2 p ! 3 arcsin =60 2 arcsin (1) =90 arcsin (0) =0
•
Similarly, one may define inverse trigonometric functions to: cos(x), tan(x), sec(x), csc(x), cot(x)
Compute arccos(1)
p
Compute arctan( 3)
3.7 Domain and Range of Trigonometric Functions
3.7.1 Domain and Range of Trigonometric Functions 3.7.2 Domain and Range of Inverse Trigonometric Functions
3.7.1 Domain and Range of Trigonometric Functions
•
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.
dom(sin) =( 1, 1)
range(sin) =[ 1, 1]
Compute the domain and range of f (x) = 2 sin(2x) + 1
dom(cos) =( 1, 1)
range(cos) =[ 1, 1]
Compute the domain and range of f (x) = sec(x)
⇡k dom(tan) ={x | x 6= , k an integer} 2 range(tan) =( 1, 1)
Compute the domain and range of f (x) = tan
✓
2x ⇡
◆
3.7.2 Domain and Range of Inverse Trigonometric Functions
•
Normally to find the domain and range of f , one 1 simply switches the domain and range of f .
•
However, the inverse trigonometric functions fail the horizontal line test on their full domains.
•
This means that their inverse functions are not welldefined unless the domain of the original function is restricted.
dom(arcsin) =[ 1, 1] h ⇡ ⇡i range(arcsin) = , 2 2
Compute the domain and range of f (x) =
arcsin(x + 1)
dom(arccos) =[ 1, 1] range(arccos) =[0, ⇡]
Compute the domain and range of f (x) = 2 + arccos(3x)
dom(arctan) =( 1, 1) ⇣ ⇡ ⇡⌘ range(arctan) = , 2 2
Compute the domain and range of f (x) = arctan(x
1)
3.8 Law of Sines and Cosines
•
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.
Law of Sines a b c = = sin(A) sin(B) sin(C)
Law of Cosines
3.9 Trigonometric Equations
•
There are problems involving trigonometric functions.
•
In some cases, inverse trigonometric functions will be useful.
3.1 Introduction to Trigonometry 3.2 Trigonometric Examples 3.3 Radians 3.4 Periodicity and Plotting Trigonometric Functions
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
3.1 Introduction to Trigonometry
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 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 2 2 2 triangle, and c is the length of the hypotenuse, then a + b = c
•
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.
3.1.2 Trigonometric Functions
•
Let ✓ be an angle of a right triangle. Assume that ✓ is not the right angle.
•
There is a side opposite ✓, call its length Opp .
•
There is also a side adjacent to ✓ , call it Adj.
•
Call the hypotenuse Hyp .
Opp sin(✓) = Hyp
Adj cos(✓) = Hyp
Opp tan(✓) = Adj
Compute sin(✓), cos(✓), tan(✓)
Compute sin(✓), cos(✓), tan(✓)
Hyp sec(✓) = Adj
Hyp csc(✓) = Opp
Adj cot(✓) = Opp
Compute sec(✓), csc(✓), cot(✓)
Compute sec(✓), csc(✓), cot(✓)
3.2 Trigonometric Examples
30-60-90 Triangle
1 sin(30) = 2p
3 cos(30) = 2 1 tan(30) = p 3
p
3 sin(60) = 2 1 cos(60) = 2 p tan(60) = 3
45-45-90 Triangle
1 sin(45) = p 2 1 cos(45) = p 2 tan(45) = 1
3.3 Radians
•
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
•
To convert from degrees to radians, one multiplies by a factor of ⇡
180 •
To convert from radians to degrees, one multiples by a factor of 180
⇡
Convert from degrees to radians: 360 90 135
Convert from radians to degrees: ⇡ 3 3⇡ 2
3.4 Periodicity and Plotting Trigonometric Functions
•
Recall that a rotation of 360 is the same as a rotation of 0
•
In general, an angle of (360 + ✓) is the same as a rotation of ✓
•
In radians, an angle of magnitude ✓ is equivalent to an angle of magnitude ✓ + 2⇡
•
This implies that the trigonometric functions are periodic.
•
They repeat themselves after a fixed interval.
sin(✓ + 2⇡) = sin(✓) cos(✓ + 2⇡) = cos(✓) csc(✓ + 2⇡) = csc(✓) sec(✓ + 2⇡) = sec(✓)
Plot f (x) = sin(x)
Plot f (x) = cos(x)
tan(✓ + ⇡) = tan(✓) cot(✓ + ⇡) = cot(✓)
Plot f (x) = tan(x)
3.5 Major Trigonometric Identities
•
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
Reciprocal Identities 1 sin(✓) = csc(✓) 1 cos(✓) = sec(✓) 1 tan(✓) = cot(✓)
Simplify sec(✓) cos(✓)
Simplify csc(✓) tan(✓)
Pythagorean Identities 2
2
sin (✓) + cos (✓) =1 2
2
2
2
tan (✓) + 1 = sec (✓) cot (✓) + 1 = csc (✓)
Simplify
s
1
2
sin (✓) 2 sin
2
2
Simplify csc (✓) tan (✓)
Double Angle Identities sin(2✓) =2 sin(✓) cos(✓) 2
cos(2✓) = cos (✓)
2
sin (✓)
Compute sin(120 )
3.6 Inverse Trigonometric Functions
•
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 1 the function f (x) = sin(x), call it f (x) = arcsin(x)
•
This function has the property that on its domain, sin(arcsin(x)) = x
✓ ◆ 1 arcsin =30 2 p ! 3 arcsin =60 2 arcsin (1) =90 arcsin (0) =0
•
Similarly, one may define inverse trigonometric functions to: cos(x), tan(x), sec(x), csc(x), cot(x)
Compute arccos(1)
p
Compute arctan( 3)
3.7 Domain and Range of Trigonometric Functions
3.7.1 Domain and Range of Trigonometric Functions 3.7.2 Domain and Range of Inverse Trigonometric Functions
3.7.1 Domain and Range of Trigonometric Functions
•
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.
dom(sin) =( 1, 1)
range(sin) =[ 1, 1]
Compute the domain and range of f (x) = 2 sin(2x) + 1
dom(cos) =( 1, 1)
range(cos) =[ 1, 1]
Compute the domain and range of f (x) = sec(x)
⇡k dom(tan) ={x | x 6= , k an integer} 2 range(tan) =( 1, 1)
Compute the domain and range of f (x) = tan
✓
2x ⇡
◆
3.7.2 Domain and Range of Inverse Trigonometric Functions
•
Normally to find the domain and range of f , one 1 simply switches the domain and range of f .
•
However, the inverse trigonometric functions fail the horizontal line test on their full domains.
•
This means that their inverse functions are not welldefined unless the domain of the original function is restricted.
dom(arcsin) =[ 1, 1] h ⇡ ⇡i range(arcsin) = , 2 2
Compute the domain and range of f (x) =
arcsin(x + 1)
dom(arccos) =[ 1, 1] range(arccos) =[0, ⇡]
Compute the domain and range of f (x) = 2 + arccos(3x)
dom(arctan) =( 1, 1) ⇣ ⇡ ⇡⌘ range(arctan) = , 2 2
Compute the domain and range of f (x) = arctan(x
1)
3.8 Law of Sines and Cosines
•
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.
Law of Sines a b c = = sin(A) sin(B) sin(C)
Law of Cosines
3.9 Trigonometric Equations
•
There are problems involving trigonometric functions.
•
In some cases, inverse trigonometric functions will be useful.
