Chapter 5: Trigonometric  Functions

Section 5.0: Preliminary Functions (supplemental)

An Introduction to Circular Functions Click here

Section 5.1: Angles and Their Measure Page 386

Recommended Exercises Page 394.
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.
1-6, 11-16, 17, 19, 21, 23, 31, 35, 39, 41, 45, 49, 51-70.

Section 5.2: Trigonometric Functions Page 397

Definitions of the Six Circular Functions

Comments About Notation
Recall that the product of two functions f and g is defined to be a function fg whose rule is fg(x) = f(x)g(x).
Of course this includes the special case where both factors of the product are the same function. In which case we have the product function ff whose rule is ff(x) = f(x)f(x).
Conventional notation permits us to write this as f²(x)=(f(x))².
The use of this notation is quite prevalent in trigonometry.

For any real number x, the definitions of the sine and cosine functions identify sin(x) and cos(x) as coordinates of a point on the unit circle, from which it follows that (sin(x))² + (cos(x))² = 1. Note this is an identity involving range values for the sine and cosine functions.
However, from the above discussion it follows that (sin(x))² =sin²(x) and (cos(x))² = cos²(x), consequently the identity (sin(x))² + (cos(x))² = 1 is usually written as sin²(x) + cos²(x) = 1 to emphasize the fact that it is the sine function and the cosine funtion which are squared. In fact one could write sin² + cos² = one where one is the name of the function whose rule is one(x) = 1. This last form is rarely written, but is frequently heard in verbal conversation.

Some Immediate Observations
The squared, or Pythagorean Identities are consequences of the equation x² + y² = 1 of the unit circle and the definition of sin and cos.

Identities for negatives follow from the fact that W(x) and W(–x) are symmetrical with respect to the x –axis.

The following diagram may help to visualize the effect of the wrapping function W on a real number and its opposite. Note the domain of the wrapping function is that "third" real number line (the one in blue) in the picture. I believe this shows more clearly the effect of wrapping the positive portion of the real line in the counterclockwise direction and wrapping the negative portion of the real number line in the clockwise direction.

Recommended Exercises Page 406.
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.
1-16, 17-32, 33-38, 39, 41, 45, 49, 51, 53, 57, 59-62, 63-66, 67-76, 83-88.

Section 5.3: Solving Right Triangles Page 408

In terms of a right triangle, as shown in Figure 1, the six trigonometric functions are defined by:

1)  If any two of the lengths a, b, or c are known, then the third is easily determined using the Pythagorean Theorem.

2)  If all three of the lengths are known, then the angles may be determined using the sin^-1, cos^-1, or tan^-1 keys on your calculator, as detailed here:

Other combinations of the inverse trigonometric functions which can also be used. However, your calculator probably only has inverse function keys for sin, cos, and tan so those are the three to use.

3)  If an angle and the length of the hypotenuse are known, then the sine function may be used to find the opposite side. The cosine function may be used to find the adjacent side. Simply plug the known values into the defining equations shown above.

4)  If an angle and its opposite side are known, the sine function may be used to determine the hypotenuse. Simply plug the known values into the defining equations above.

5)  If an angle and its adjacent side are known, the cosine function may be used to determine the hypotenuse. Simply plug the known values into the defining equations above.

6)  If only the three angles are known, then there are infinitely many triangles satisfying that condition. Consequently we usually claim that the triangle cannot be solved.

Click HERE for a summary in PDF format. Click HERE for a few worked out examples.

Recommended Exercises Page 414.
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.
1-6, 7-12, 19-30(all), 31-36(all), 45, 47.

Section 5.4: Properties of Trigonometric Functions Page 416

SINE FUNCTION	The domain of sin is all real numbers. The range of sin is [-1, 1] The zeros of sin are the multiples of The sin function is periodic with period 2 The sin function is positive in quadrants I and II and negative in quadrants III and IV The sin function is not one-to-one (does not pass the horizontal line test) and therefore has no inverse. The sin function with its domain restricted to is one_to_one and has an inverse sin-1 Another symbol used for the inverse of sin is arcsin
COSINE FUNCTION	The domain of cos is all real numbers. The range of cos is [-1, 1] The zeros of cos are the odd multiples of The cos function is periodic with period 2 The cos function is positive in quadrants I and IV and negative in quadrants II and III The cos function is not one-to-one (does not pass the horizontal line test) and therefore has no inverse. The cos function with its domain restricted to is one_to_one and has an inverse cos-1 Another symbol used for the inverse of cos is arccos
TANGENT FUNCTION	Recall that the domain of a rational function is all real numbers except the zeros of its denominator. Since the denominator of the tangent function is the cosine function, the domain of tan is all real numbers except the zeros of the cos function. The domain of tan is all real numbers except odd multiples of . Furthermore tan has vertical asymptotes at the odd multiples of . The range of tan is all real numbers. The tan function is periodic with period The tan function is positive in quadrants I and III and negative in quadrants II and IV The tan function is increasing everywhere it is defined. The tan function is not one-to-one (does not pass the horizontal line test) and therefore has no inverse. The tan function with its domain restricted to is one_to_one and has an inverse cos-1 Another symbol used for the inverse of tan is arctan
COTANGENT FUNCTION	Recall that the domain of a rational function is all real numbers except the zeros of its denominator. Since the denominator of the cotangent function is the sine function, the domain of cot is all real numbers except the zeros of the sin function. The domain of cot is all real numbers except multiples of . Furthermore cot has vertical asymptotes at the odd multiples of . The range of cot is all real numbers. The cot function is periodic with period The cot function is positive in quadrants I and III and negative in quadrants II and IV The cot function is decreasing everywhere it is defined. The cot function is not one-to-one (does not pass the horizontal line test) and therefore has no inverse. The cot function with its domain restricted to [0, ]is one_to_one and has an inverse cot -1 Another symbol used for the inverse of cot is arccot
SECANT FUNCTION	Since the denominator of the secant function is the cosine function, the domain of secant is all real numbers except the zeros of the cos function. The domain of secant is all real numbers except odd multiples of . Furthermore secant has vertical asymptotes at the odd multiples of . The range of secant is all real numbers not in the interval [-1,1]. The secant function is periodic with period 2 The sec function is positive in quadrants I and IV and negative in quadrants II and III The secant function is not one-to-one (does not pass the horizontal line test) and therefore has no inverse. The secant function with its domain restricted to [0, ]is one_to_one and has an inverse cos-1 Another symbol used for the inverse of secant is arcsec
COSECANT FUNCTION	Since the denominator of the cosecant function is the sine function, the domain of cosecant is all real numbers except the zeros of the sin function. The domain of cosecant is all real numbers except odd multiples of . Furthermore cosecant has vertical asymptotes at the odd multiples of . The range of cosecant is all real numbers not in the interval [-1,1]. The cosecant function is periodic with period 2 The csc function is positive in quadrants I and II and negative in quadrants III and IV The cosecant function is not one-to-one (does not pass the horizontal line test) and therefore has no inverse. The cosecant function with its domain restricted to is one_to_one and has an inverse csc-1 Another symbol used for the inverse of cosecant is arccsc

The following diagram as well as Table 1 on Page 419 of the text are helpful in remembering where the various trig functions are positive and where they are negative.

Recommended Exercises Page 428.
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.
1-10, 25, 27, 29, 31, 33, 35, 37, 39

Section 5.5: General Trigonometric Functions Page 432

To begin the analysis in this section it is helpful to review a few simple and fundamental principles. To begin: Consider a single point (a, b) in the coordinate plane.

Vertical Translations

Where is the new point if a positive number k is added to the second coodinate ?
The new point (a, b + k) is k units above the original point. The original point is shifted up by k units.

Where is the new point if a negative number k is added to the second coordinate ?
The new point (a, b + k) is k units below the original point. The original point is shifted down by k units.

When we use function notation to write f(x) we mean the unique range element associated with the domain element x by the function f.
The graph of a function f consists of all those points (x, f(x)) and only those points whose second coordinate is the unique range element associated with the first coordinate.

Compare the graph of a function f and a function g whose rule is g(x) = k + f(x).
The graph of g is the result of shifting the graph of f up or down (depending on whether k is positive or negative).
This happens because k is added to the second coordinate of each and every point on the graph of the function f.

The action of creating a new function by adding the same value to every range element of the orginal function is called a translation.

If the function happens to be a trig function such as sin, the very same translation takes place. For example, the graph of the function f whose rule is f(x) 3 + sin(x) is the sin function shifted up three units.

Vertical Stretching

Where is the new point if the second coordinate of the original point (a, b) is multiplied by a positive number k > 1 ?
The distance between the x-axis and the point (a, kb) has been stretched by a factor of k.

Where is the new point if the second coordinate of the original point (a, b) is multiplied by a positive number k < 1 ?
The distance between the x-axis and the point (a, kb) has been shrunk by a factor of k.

Where is the new point if the second coordinate of the original point (a, b) is multiplied by -1 ?
The new point (a, -b) is on the opposite side of the x-axis and on the opposite side of the original point(a, b).

When we use function notation to write f(x) we mean the unique range element associated with the domain element x by the function f.
The graph of a function f consists of all those points (x, f(x)) and only those points whose second coordinate is the unique range element associated with the first coordinate.

Compare the graph of a function f and a function g whose rule is g(x) = k f(x).
The graph of g is the result of stretching (or shrinking), and possibly rotating about the x-axis, the graph of f .
This happens because the second coordinate of each and every point on the graph of the function f is multiplied by the same number k.

The action of creating a new function by multiplying every range element of the orginal function by a fixed number is called stretching. The words expansion and contraction are also used

If the function happens to be a trig function such as sin, the very same stretching takes place. For example, three graphs illustrating stretches of the sine function are shown here.

When a trig function is multiplied by a constant so the resulting function has the form f(x) = Asin(x), the multiplier is called the amplitude.

Horizontal Translations

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Horizontal Stretching

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Recommended Exercises Page 441.
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.
1-12, 13, 14, 17, 18, 21, 23, 29-44.

Section 5.6: Inverse Trigonometric Functions Page 446

Recommended Exercises Page 458.
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.
1-12, 13, 15, 17, 19, 20, 23, 24, 25, 26, 31, 32, 33, 34, 35-52(all)

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