tan(β) = cot(α)
sec(β) = csc(α)
|
DrDelMath PRECALCULUS by Barnett/Ziegler/Byleen |
Trigonometry has an enormous variety of applications. The ones mentioned explicitly in textbooks and courses on trigonometry are its uses in practical endeavors such as navigation, land surveying, building, and the like. It is also used extensively in a number of academic fields, primarily mathematics, science and engineering.
Among the scientific fields that make use of trigonometry are:
acoustics, crystallography, economics (in particular in analysis of financial markets), electrical engineering, electronics,
land surveying and geodesy, many ph, architecture, astronomy (and hence navigation, on the oceans, in aircraft, and in space;
in this connection, see great circle distance), biology, cartography, chemistry, civil engineering, computer graphics,
geophysicsysical sciences, mechanical engineering, machining, medical imaging (CAT scans and ultrasound), meteorology,
music theory, number theory (and hence cryptography), oceanography, optics, pharmacology, phonetics, probability theory,
psychology, seismology, statistics, and visual perception.
Referrence
Chapter 6: Trigonometric Identities and Conditional Equations
Section 6.1: Basic Identities and Their Use Page 562
Recommended Exercises Page 569.
If you understand the previous material you should be able to work the following exercises.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized.
7 - 32, 33, 37, 39, 45 - 60, 61 - 90, 103 -108
Section 6.2: Sum, Difference, and CoFunction Identities Page 572
Cofunction Identities:
Definition: Two angles are complementary angles if their sum is 90°.
Property: The sum of the interior angles of a triangle is 180° .
Property: The two acute angles in a right triangle are complementary angles.
| Refer to Figure 1 to observe the following cofunction identities. | |
||
| If α and β are complementary angles, then | |||
|
sin(β) = cos(α) tan(β) = cot(α) sec(β) = csc(α) |
|||
| That is to say: | |||
| The cofunction of an angle equals the function of the complementary angle | |||
| If β is an angle, then its complement is |
|||
| This gives rise to an alternate statement of the so-called cofunction identities. | |||
|
|||
Sum and Difference Identities:
cos (x – y) = cos(x)cos(y) + sin(x)sin(y) These two identities and the cofunction identities permit a development of identities for sum and difference of the sine and tangent functions. sin(x – y) = sin(x)cos(y) – cos(x)sin(y) |
 |
Example: To prove (or derive) the identity for the sine of the difference of two angles:
Recommended Exercises Page 582.
If you understand the previous material you should be able to work the following exercises.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized.
7 - 14, 15 - 22, 23 - 26, 27 - 32, 33, 35, 37, 39, 41, 43, 49, 50, 53, 54, 55
Section 6.3: Double-Angle and Half-Angle Identities Page 584
Double-Angle Identities:
sin(2x) = 2 sin(x)cos(x)
cos(2x) = cos2(x) – sin2(x)
= 1 – 2sin2(x)
= 2cos2(x) – 1
From the three identities for cos(2x) we can derive
Half-Angle Identities:
Consider the double-angle identity cos(2x) = 1 - 2sin2(x).
The choice of variable does not alter the identity, and therefore cos(2y) = 1 - 2sin2(y).
Now if y = x/2, then this last identity becomes cos(x) = 1 - 2sin2(x/2).
Solving this identity for sin(x/2) yields the half-angle identity for sin.
In a similar manner the half-angle identity for cos is derived from
cos(2x) = 2cos2(x) - 1.
The half-angle identity for tan is then derived from the fact that tan is the quotient of sin divided by cos.
Recommended Exercises Page 593.
If you understand the previous material you should be able to work the following exercises.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized.
7 - 12, 13, 15, 17, 23, 29 - 46
Section 6.4: Product-Sum and Sum-Product Identities Page 596
Product-Sum Identities
Addition of the two identities
sin(x + y) = sin(x)cos(y) + sin(y)cos(x) and
sin(x - y) = sin(x)cos(y) - sin(y)cos(x) yields
sin( x + y) + sin(x - y) = 2sin(y)cos(x) from which may be obtained the identity
Subtraction of the second identity from the first yields
sin(x + y) - sin(x - y) = 2sin(y)cos(x) from which may be obtained the identity
Addition of the two identities
cos(x + y) = cos(x)cos(y) - sin(x)sin(y) and
cos(x - y) =
cos(x)cos(y) +sin(x)sin(y) yields
cos(x + y) + cos(x - y) = 2 cos(x)cos(y) from which may be obtained the identity
Subtraction of the second identity from the first yields
cos(x + y) - cos(x - y) = -2sin(x)sin(y) from which may be obtained the identity
Sum-Product Identities
Recommended Exercises Page 508.
If you understand the previous material you should be able to work the following exercises.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized.
Section 6.5: Trignometric Equations Page 604
The following website has an excellent list of examples with great discussion of each example. It also provides practice exercises.
Click Here
Here is another pretty good website Click Here
Some Useful Diagrams
Recommended Exercises Page 615.
If you understand the previous material you should be able to work the following exercises.
Each of the individually listed exercises should be done. In each of the lists (like 5 - 12) work as many as needed to insure that you understand the concept being illustrated or utilized.
3 - 12, 21 - 26, 31, 32, 33, 37, 39, 43, 45, 47 - 52
