Trigonometric Functions

Finney Chapter 1.6

Radian Measure

• The arc length of the unit circle (where ) is defined as

where is the arc angle

• If we define , then the formula becomes

where is in radians

• Thus, the radian measure of an arc angle in a circle equals the length of the arc that the sector cuts from the unit circle

Radian Measure

• For a circle of radius r other than 1, the formula is

• Note that, though we call this a radian measure, it actually has no units

• Solving for gives , and both s and r have the same units

• When we start to use trigonometric functions in calculus, the angles must be in radians! Always use your calculator in radian mode.

Example 1: Finding Arc Length

Find the length of an arc subtended on a circle of radius 3 by a cental angle of measure .The formula is , so the arc length is

Periodicity

DEFINITION:

A function is periodic if there is a positive number p such that

for every value of x. The smallest such value of p is the period of f.

• As the name suggests, periodic functions have equal function values once x values become values, where n is an integer

• The value of p, however, is the smallest of these other than zero (i.e., when )

Periodicity

What is the period of the function ?In order to answer this question we will make use of the following identity

By definition of periodicity, . We wish to find the smallest positive value of p for which this is true. Using the above identity

This equation will hold true if and . The first value for which this is true is , but that’s the trivial case since is an identity. The next value (think of one full revolution of a circle) is . Since this is the smallest value for which , then the sine function has period 2. Using a similar argument, the period of the cosine function is also .

Periodicity

What is the period of the function ?Since , with being the smallest value for which this is true, then

So the cosecant function has periodicity , as does the secant function.

Periodicity

What is the period of the function ?We will again use the angle addition identity for sine and also the angle addition identity for cosine, which is

If , what is the value of p?

Periodicity

What is the period of the function ?The smallest value for which is . Hence, the period of the tangent function is , as is the period of the cotangent function.

Note that we still have . But the period is because .

Even & Odd Trigonometric Functions

Is the sine function even or odd?Although we can refer to the graph of the function, it must be possible to answer this question analytically. This time we will make use of the angle subtraction identity for sine

Proceed as follows

So the sine function is odd.

Even & Odd Trigonometric Functions

Is the cosine function even or odd?As an exercise, try to determine analytically whether the cosine function is even or odd. Use the angle subtraction identity for cosine

Can you determine whether the tangent function is even or odd?

Standard Position of an Angle

DEFINITION:

An angle is in standard position in the x-y plane if the vertex of the angle is at the origin and the initial side of the angle is the positive x-axis. By convention, we take in the counterclockwise direction and in the clockwise direction

With an angle in standard position in a circle of radius r, we can define the six trigonometric functions as follows

Standard Position of an Angle

Standard Position of an Angle

Example 3: Finding Trigonometric Values

Find all the trigonometric values of is and .Since sine is negative in Q III and Q IV, and tangent is negative in Q II and Q IV, the angle is in quadrant IV. So we have

Transformation of Trigonometric Graphs

• The general form of a parent function can be written as

• The value a causes the graph to stretch or shrink vertically, or to reflect over the x-axis

• The value b causes the graph to stretch or shrink horizontally, or to reflect over the y-axis

• The value c causes the graph to translate horizontally

• The value d causes the graph to translate vertically

Transformation of Trigonometric Graphs

• The general sine function can be written in the form

• Here, is the amplitude, is the period, C is the horizontal translation, and D is the vertical translation

Example 4: Graphing a Trigonometric FunctionDetermine (a) period, (b) domain, (c) range, and (d) draw the graph of the function .We should first rewrite the function in the form . The key is to factor out 2 from and then find B: . The period is B when so that . The domain for cosine is and this remains unchanged. The range for the parent function is . But multiplying by 3 changes the range to and adding 1 translates the graph vertically by 1. So the range becomes . The graph is shown on the next slide, with the graph of the parent function for comparison.

Example 4: Graphing a Trigonometric Function

Example 5: Finding the Frequency of a Musical NoteConsider the tuning fork data in Table 1.18 (Finney, page 49).

(a) Find a sinusoidal regression equation (general sine curve) for the data and superimpose its graph on a scatter plot of the data

(b) The frequency of a musical note, or wave, is measured in cycles per second, or hertz (1 Hz = 1 cycle per second). The frequency is the reciprocal of the period of the wave, which is measured in seconds per cycle. Estimate the frequency of the note produced by the tuning fork.

NOTE: there are many data points, so rather than do this in class, I will simply reprint what is in your textbook

Example 5: Finding the Frequency of a Musical Note(a) The sinusoidal regression equation produced by our calculators is approximately

Figure 1.46 (Finney page 49) shows its graph together with a scatter plot of the tuning fork data.

(b) The period is sec, so the frequency is Hz.

The tuning fork is vibrating at a frequency of about 396 Hz. On the pure tone scale, this is the note G above middle C. It is a few cycles per second different from the frequency of the G we hear on a piano’s tempered scale, 392 Hz.

Inverse Trigonometric Functions

• None of the trigonometric functions is one-to-one

• We can define inverse trigonometric functions by appropriately restricting their domains

• Recall that, for the inverse of a function, the domain and range switch roles

• Hence, in order to appropriately restrict the domains of the trigonometric functions, we must consider their range; an inverse function must cover the entire range (which becomes the domain)

Inverse Trigonometric Functions

• Switching the x and y values for the sine function produces a graph like that shown at left

• The domain of our function must be

• What values of y will ensure the proper domain?

• The values we choose will tell us how to restrict the sine function

• What this means in practice (i.e., values you see in a calculator) is that the inverse sine function will only produce values in the restricted range

Inverse Trigonometric Functions

DEFINITIONS:Function Domain Range

Example 7: Finding Angles in Degrees & RadiansFind the measure of in degrees and radians.To find the measure in degrees, set your calculator to degree mode, then evaluate

To find the measure in radians, set your calculator to radian mode, then evaluate

Graphs of all inverse trigonometric functions can be found in Finney page 51.

Example 8: Using the Inverse Trigonometric FunctionsSolve for x.

(a) (b) (a) The sine function is positive in quadrants I and II, so there are two answers. The calculator gives

This is the result in the range . The other answer is in quadrant II:

(b) Since the domain is all real numbers, we will have an infinite number of solutions. We can represent these by adding , where n is an integer, to the calculator value.

Hence, the entire solution set is , where n is an integer.

Exercise 1.6

Finney page 52, #1-14, 17-22, 24, 31-35 odds, 41-49