Trigonometric Functions MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 J. Robert Buchanan Trigonometric Functions
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Trigonometric FunctionsMATH 160, Precalculus
J. Robert Buchanan
Department of Mathematics
Fall 2011
J. Robert Buchanan Trigonometric Functions
Objectives
In this lesson we will learn to:
identify a unit circle and describe its relationship to realnumbers,
evaluate trigonometric functions using the unit circle,
use the domain and period to evaluate sine and cosinefunctions,
use a calculator to evaluate trigonometric functions.
J. Robert Buchanan Trigonometric Functions
Unit Circle
The unit circle is the graph of the equation x2 + y2 = 1.
t
t
H x , y L
Θ
x
y
J. Robert Buchanan Trigonometric Functions
Arc Length
Recall: s = rθ in general. When r = 1 (as in the case of theunit circle) then s = θ.
The arc length will be denoted t and is the arc of the circleintercepted by the central angle whose measure is θ.
The coordinates of the point (x , y) depend on the value of thearc length t .
J. Robert Buchanan Trigonometric Functions
Trigonometric Functions
Definition
Let t be a real number and let (x , y) be the point on the unitcircle corresponding to t .
sin t = y
cos t = x
tan t =yx
, if x 6= 0
cot t =xy
, if y 6= 0
sec t =1x
, if x 6= 0
csc t =1y
, if y 6= 0
J. Robert Buchanan Trigonometric Functions
Unit Circle Values
J. Robert Buchanan Trigonometric Functions
Table of Values
t x y t x y0 1 0 π −1 0π
6
√
32
12
7π
6 −√
32 −1
2π
4
√
22
√
22
5π
4 −√
22 −
√
22
π
312
√
32
4π
3 −12 −
√
32
π
2 0 1 3π
2 0 −12π
3 −12
√
32
5π
312 −
√
32
3π
4 −√
22
√
22
7π
4
√
22 −
√
22
5π
6 −√
32
12
11π
6
√
32 −1
2
J. Robert Buchanan Trigonometric Functions
Examples
Fill in the missing values in the following table.
t cos t sin t tan t cot t sec t csc tπ
35π
65π
411π
6
J. Robert Buchanan Trigonometric Functions
Examples
Fill in the missing values in the following table.
t cos t sin t tan t cot t sec t csc tπ
312
√
32
√3 1
√
32 2
√
35π
65π
411π
6
J. Robert Buchanan Trigonometric Functions
Examples
Fill in the missing values in the following table.
t cos t sin t tan t cot t sec t csc tπ
312
√
32
√3 1
√
32 2
√
35π
6 −√
32
12 − 1
√
3−√
3 − 2√
32
5π
411π
6
J. Robert Buchanan Trigonometric Functions
Examples
Fill in the missing values in the following table.
t cos t sin t tan t cot t sec t csc tπ
312
√
32
√3 1
√
32 2
√
35π
6 −√
32
12 − 1
√
3−√
3 − 2√
32
5π
4 − 1√
2− 1
√
21 1 −
√2 −
√2
11π
6
J. Robert Buchanan Trigonometric Functions
Examples
Fill in the missing values in the following table.
t cos t sin t tan t cot t sec t csc tπ
312
√
32
√3 1
√
32 2
√
35π
6 −√
32
12 − 1
√
3−√
3 − 2√
32
5π
4 − 1√
2− 1
√
21 1 −
√2 −
√2
11π
6
√
32 −1
2 − 1√
3−√
3 2√
3−2
J. Robert Buchanan Trigonometric Functions
Sine and Cosine
The domain of the sine and cosine functions is the set ofall real numbers.The range of the sine and cosine function is the interval[−1, 1].Since cos(−t) = cos t , cosine is an even function.Since sin(−t) = − sin t , sine is an odd function.
J. Robert Buchanan Trigonometric Functions
Sine and Cosine
The domain of the sine and cosine functions is the set ofall real numbers.The range of the sine and cosine function is the interval[−1, 1].Since cos(−t) = cos t , cosine is an even function.Since sin(−t) = − sin t , sine is an odd function.
Question: are the other four trigonometric functions even, odd,or neither?
J. Robert Buchanan Trigonometric Functions
Sine and Cosine
The domain of the sine and cosine functions is the set ofall real numbers.The range of the sine and cosine function is the interval[−1, 1].Since cos(−t) = cos t , cosine is an even function.Since sin(−t) = − sin t , sine is an odd function.
Question: are the other four trigonometric functions even, odd,or neither?
tan(−t) = − tan t (odd)
cot(−t) = − cot t (odd)
sec(−t) = sec t (even)
csc(−t) = − csc t (odd)
J. Robert Buchanan Trigonometric Functions
Peiodic Functions
Definition
A function f is periodic if there exists a positive real number csuch that
f (t + c) = f (t)
for all t in the domain of f . The smallest number c for which f isperiodic is called the period of f .
J. Robert Buchanan Trigonometric Functions
Peiodic Functions
Definition
A function f is periodic if there exists a positive real number csuch that
f (t + c) = f (t)
for all t in the domain of f . The smallest number c for which f isperiodic is called the period of f .
cos(t + 2n π) = cos t
sin(t + 2n π) = sin t
for all t when n is an integer.
Remark: sine and cosine are periodic functions with period 2π.
J. Robert Buchanan Trigonometric Functions
Examples
Use the period of the trigonometric functions to evaluate thefunction.
cos 3π =
sin9π
4=
sin19π
6=
cos(
−9π
4
)
=
J. Robert Buchanan Trigonometric Functions
Examples
Use the period of the trigonometric functions to evaluate thefunction.
cos 3π = −1
sin9π
4=
sin19π
6=
cos(
−9π
4
)
=
J. Robert Buchanan Trigonometric Functions
Examples
Use the period of the trigonometric functions to evaluate thefunction.
cos 3π = −1
sin9π
4=
√2
2
sin19π
6= −1
2
cos(
−9π
4
)
=
√2
2
J. Robert Buchanan Trigonometric Functions
Homework
Read Section 4.2.
Exercises: 1, 5, 9, 13, . . . , 53, 57
J. Robert Buchanan Trigonometric Functions
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