Chapter 7 Trigonometric Functions of Angles

1
Chapter 7 Trigonometric Functions of Angles

• Suppose ? is an angle placed in the standard
  position
• Let (x, y) be a point on the terminal side of ?
• Then the distance r from the origin to (x, y) is

y
(x, y)
r
?
x
2
Chapter 7 Trigonometric Functions of Angles

• Suppose ? is an angle placed in the standard
  position
• Let (x, y) be a point on the terminal side of ?
• Then the distance r from the origin to (x, y) is
• r ? x2 y2

y
(x, y)
r
?
x
3
Chapter 7 Trigonometric Functions of Angles

• Then, the Trigonometric Functions of angle ? are
• sin ? y/r
• cos ? x/r
• tan ? y/x
• cot ? x/y
• sec ? r/x
• csc ? r/y
• The values of these ratios DO NOT DEPEND ON THE
  CHOICE OF THE POINT (x, y)

y
(x, y)
r
?
x
4
Chapter 7 Trigonometric Functions of Angles

• This would have the same values if ? is the same
  
• sin ? y/r
• cos ? x/r
• tan ? y/x
• cot ? x/y
• sec ? r/x
• csc ? r/y

y
?
(x, y)
r
x
5
Chapter 7 Trigonometric Functions of Angles

• Example
• Whats r? Whats r?

y
y
(2, 2)
r
?
?
(1, 1)
r
x
x
6
Chapter 7 Trigonometric Functions of Angles

• Same in both cases
• sin ? y/r
• cos ? x/r
• tan ? y/x

y
y
(2, 2)
r
?
?
(1, 1)
r
x
x
7
MTH 110 Elementary Functions

• First Quadrant
• sin ?
• cos ?
• tan ?
• cot ?
• sec ?
• csc ?

y
x
8
MTH 110 Elementary Functions

• Second Quadrant
• sin ?
• cos ? -
• tan ? -
• cot ? -
• sec ? -
• csc ?

y
x
9
MTH 110 Elementary Functions

• Third Quadrant
• sin ? -
• cos ? -
• tan ?
• cot ?
• sec ? -
• csc ? -

y
x
10
MTH 110 Elementary Functions

• Fourth Quadrant
• sin ? -
• cos ?
• tan ? -
• cot ? -
• sec ?
• csc ? -

y
x
11
MTH 110 Related Angle

• Suppose angle ? is an angle in standard position.
  The related angle ? is the positive acute angle
  that the terminal side of ? makes with the
  x-axis.

y
x
12
MTH 110 Related Angle Theorem - p. 405

• The value of any trigonometric function of an
  angle ? is equal to the value of the
  corresponding trigonometric function of the
  related angle, except possibly for the sign. The
  sign depends on the quadrant ? is in and can be
  determined by using the diagram of Figure 7.13.

y
x
13
MTH 110 Elementary Functions

• Example Given that cos ? 8/17, and ? lies in
  the Fourth Quadrant, find all the other
  trigonometric functions.

y
x
14
MTH 110 Elementary Functions

• Example Given that cos ? 8/17, and ? lies in
  the Fourth Quadrant, find all the other
  trigonometric functions.
• cos ? 8/17 x/r
• r2 x2 y2
• ...
• y ? 15
• (Which is it, and how do we know??)

y
x
15
MTH 110 Elementary Functions

• Example Given that cos ? 8/17, and ? lies in
  the Fourth Quadrant, find all the other
  trigonometric functions.
• sin ? -
• cos ?
• tan ? -
• cot ? -
• sec ?
• csc ? -

y
x
16
Homeworkp. 406

• 1, 3, 5, 7, 15, 17, 19, 21, 27, 29, 31, 41, 43,
  45, 47, 59, 61

17
Chapter 7 Identities

• sin ? y/r
• cos ? x/r
• tan ? y/x
• cot ? x/y
• sec ? r/x
• csc ? r/y

y
?
(x, y)
r
x
18
Chapter 7 Reciprocal Identities

• sin ? csc ? 1
• cos ? sec ? 1
• tan ? cot ? 1

y
?
(x, y)
r
x
19
Chapter 7 Reciprocal Identities

• sin ? 1/csc ?
• cos ? 1/sec ?
• tan ? 1/cot ?

y
?
(x, y)
r
x
20
Chapter 7 Reciprocal Identities

• csc ? 1/sin ?
• sec ? 1/cos ?
• cot ? 1/tan ?

y
?
(x, y)
r
x
21
Chapter 7 Quotient Identities

• tan ? sin ? / cos ?
• cot ? cos ? / sin ?

y
?
(x, y)
r
x
22
Chapter 7 Ranges of Values

• -1 ? cos ? ? 1
• -1 ? sin ? ? 1
• sec ? ? -1 or sec ? ? 1
• csc ? ? -1 or csc ? ? 1

y
?
(x, y)
r
x
23
Homeworkp. 413

• 1, 3, 5, 9, 10, 17, 23, 25, 33, 35, 37