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Calculus Volume 1

1.3 Trigonometric Functions

Calculus Volume 11.3 Trigonometric Functions

Learning Objectives

  • 1.3.1 Convert angle measures between degrees and radians.
  • 1.3.2 Recognize the triangular and circular definitions of the basic trigonometric functions.
  • 1.3.3 Write the basic trigonometric identities.
  • 1.3.4 Identify the graphs and periods of the trigonometric functions.
  • 1.3.5 Describe the shift of a sine or cosine graph from the equation of the function.

Trigonometric functions are used to model many phenomena, including sound waves, vibrations of strings, alternating electrical current, and the motion of pendulums. In fact, almost any repetitive, or cyclical, motion can be modeled by some combination of trigonometric functions. In this section, we define the six basic trigonometric functions and look at some of the main identities involving these functions.

Radian Measure

To use trigonometric functions, we first must understand how to measure the angles. Although we can use both radians and degrees, radians are a more natural measurement because they are related directly to the unit circle, a circle with radius 1. The radian measure of an angle is defined as follows. Given an angle θ,θ, let ss be the length of the corresponding arc on the unit circle (Figure 1.30). We say the angle corresponding to the arc of length 1 has radian measure 1.

An image of a circle. At the exact center of the circle there is a point. From this point, there is one line segment that extends horizontally to the right a point on the edge of the circle and another line segment that extends diagonally upwards and to the right to another point on the edge of the circle. These line segments have a length of 1 unit. The curved segment on the edge of the circle that connects the two points at the end of the line segments is labeled “s”. Inside the circle, there is an arrow that points from the horizontal line segment to the diagonal line segment. This arrow has the label “theta = s radians”.
Figure 1.30 The radian measure of an angle θθ is the arc length ss of the associated arc on the unit circle.

Since an angle of 360°360° corresponds to the circumference of a circle, or an arc of length 2π,2π, we conclude that an angle with a degree measure of 360°360° has a radian measure of 2π.2π. Similarly, we see that 180°180° is equivalent to ππ radians. Table 1.8 shows the relationship between common degree and radian values.

Degrees Radians Degrees Radians
0 0 120 2π/32π/3
30 π/6π/6 135 3π/43π/4
45 π/4π/4 150 5π/65π/6
60 π/3π/3 180 ππ
90 π/2π/2
Table 1.8 Common Angles Expressed in Degrees and Radians

Example 1.22

Converting between Radians and Degrees

  1. Express 225°225° using radians.
  2. Express 5π/35π/3 rad using degrees.

Checkpoint 1.17

Express 210°210° using radians. Express 11π/611π/6 rad using degrees.

The Six Basic Trigonometric Functions

Trigonometric functions allow us to use angle measures, in radians or degrees, to find the coordinates of a point on any circle—not only on a unit circle—or to find an angle given a point on a circle. They also define the relationship among the sides and angles of a triangle.

To define the trigonometric functions, first consider the unit circle centered at the origin and a point P=(x,y)P=(x,y) on the unit circle. Let θθ be an angle with an initial side that lies along the positive xx-axis and with a terminal side that is the line segment OP.OP. An angle in this position is said to be in standard position (Figure 1.31). We can then define the values of the six trigonometric functions for θθ in terms of the coordinates xx and y.y.

An image of a graph. The graph has a circle plotted on it, with the center of the circle at the origin, where there is a point. From this point, there is one line segment that extends horizontally along the x axis to the right to a point on the edge of the circle. There is another line segment that extends diagonally upwards and to the right to another point on the edge of the circle. This point is labeled “P = (x, y)”. These line segments have a length of 1 unit. From the point “P”, there is a dotted vertical line that extends downwards until it hits the x axis and thus the horizontal line segment. Inside the circle, there is an arrow that points from the horizontal line segment to the diagonal line segment. This arrow has the label “theta”.
Figure 1.31 The angle θθ is in standard position. The values of the trigonometric functions for θθ are defined in terms of the coordinates xx and y.y.

Definition

Let P=(x,y)P=(x,y) be a point on the unit circle centered at the origin O.O. Let θθ be an angle with an initial side along the positive xx-axis and a terminal side given by the line segment OP.OP. The trigonometric functions are then defined as

sinθ=ycscθ=1ycosθ=xsecθ=1xtanθ=yxcotθ=xysinθ=ycscθ=1ycosθ=xsecθ=1xtanθ=yxcotθ=xy
(1.9)

If x=0,secθx=0,secθ and tanθtanθ are undefined. If y=0,y=0, then cotθcotθ and cscθcscθ are undefined.

We can see that for a point P=(x,y)P=(x,y) on a circle of radius rr with a corresponding angle θ,θ, the coordinates xx and yy satisfy

cosθ=xrx=rcosθcosθ=xrx=rcosθ
sinθ=yry=rsinθ.sinθ=yry=rsinθ.

The values of the other trigonometric functions can be expressed in terms of x,y,x,y, and rr (Figure 1.32).

An image of a graph. The graph has a circle plotted on it, with the center of the circle at the origin, where there is a point. From this point, there is one blue line segment that extends horizontally along the x axis to the right to a point on the edge of the circle. There is another blue line segment that extends diagonally upwards and to the right to another point on the edge of the circle. This point is labeled “P = (x, y)”. These line segments have a length of “r” units. Between these line segments within the circle is the label “theta”, representing the angle between the segments. From the point “P”, there is a blue vertical line that extends downwards until it hits the x axis and thus hits the horizontal line segment, at a point labeled “x”. At the intersection horizontal line segment and vertical line segment at the point x, there is a right triangle symbol. From the point “P”, there is a dotted horizontal line segment that extends left until it hits the y axis at a point labeled “y”.
Figure 1.32 For a point P=(x,y)P=(x,y) on a circle of radius r,r, the coordinates xx and yy satisfy x=rcosθx=rcosθ and y=rsinθ.y=rsinθ.

Table 1.9 shows the values of sine and cosine at the major angles in the first quadrant. From this table, we can determine the values of sine and cosine at the corresponding angles in the other quadrants. The values of the other trigonometric functions are calculated easily from the values of sinθsinθ and cosθ.cosθ.

θθ sinθsinθ cosθcosθ
00 00 11
π6π6 1212 3232
π4π4 2222 2222
π3π3 3232 1212
π2π2 11 00
Table 1.9 Values of sin θ sin θ and cos θ cos θ at Major Angles θ θ in the First Quadrant

Example 1.23

Evaluating Trigonometric Functions

Evaluate each of the following expressions.

  1. sin(2π3)sin(2π3)
  2. cos(5π6)cos(5π6)
  3. tan(15π4)tan(15π4)

Checkpoint 1.18

Evaluate cos(3π/4)cos(3π/4) and sin(π/6).sin(π/6).

As mentioned earlier, the ratios of the side lengths of a right triangle can be expressed in terms of the trigonometric functions evaluated at either of the acute angles of the triangle. Let θθ be one of the acute angles. Let AA be the length of the adjacent leg, OO be the length of the opposite leg, and HH be the length of the hypotenuse. By inscribing the triangle into a circle of radius H,H, as shown in Figure 1.33, we see that A,H,A,H, and OO satisfy the following relationships with θ:θ:

sinθ=OHcscθ=HOcosθ=AHsecθ=HAtanθ=OAcotθ=AOsinθ=OHcscθ=HOcosθ=AHsecθ=HAtanθ=OAcotθ=AO
An image of a graph. The graph has a circle plotted on it, with the center of the circle at the origin, where there is a point. From this point, there is one line segment that extends horizontally along the x axis to the right to a point on the edge of the circle. There is another line segment with length labeled “H” that extends diagonally upwards and to the right to another point on the edge of the circle. From the point, there is vertical line with a length labeled “O” that extends downwards until it hits the x axis and thus the horizontal line segment at a point with a right triangle symbol. The distance from this point to the center of the circle is labeled “A”. Inside the circle, there is an arrow that points from the horizontal line segment to the diagonal line segment. This arrow has the label “theta”.
Figure 1.33 By inscribing a right triangle in a circle, we can express the ratios of the side lengths in terms of the trigonometric functions evaluated at θ.θ.

Example 1.24

Constructing a Wooden Ramp

A wooden ramp is to be built with one end on the ground and the other end at the top of a short staircase. If the top of the staircase is 44 ft from the ground and the angle between the ground and the ramp is to be 10°,10°, how long does the ramp need to be?

Checkpoint 1.19

A house painter wants to lean a 2020-ft ladder against a house. If the angle between the base of the ladder and the ground is to be 60°,60°, how far from the house should she place the base of the ladder?

Trigonometric Identities

A trigonometric identity is an equation involving trigonometric functions that is true for all angles θθ for which the functions are defined. We can use the identities to help us solve or simplify equations. The main trigonometric identities are listed next.

Rule: Trigonometric Identities

Reciprocal identities

tanθ=sinθcosθcotθ=cosθsinθcscθ=1sinθsecθ=1cosθtanθ=sinθcosθcotθ=cosθsinθcscθ=1sinθsecθ=1cosθ

Pythagorean identities

sin2θ+cos2θ=11+tan2θ=sec2θ1+cot2θ=csc2θsin2θ+cos2θ=11+tan2θ=sec2θ1+cot2θ=csc2θ

Addition and subtraction formulas

sin(α±β)=sinαcosβ±cosαsinβsin(α±β)=sinαcosβ±cosαsinβ
cos(α±β)=cosαcosβsinαsinβcos(α±β)=cosαcosβsinαsinβ

Double-angle formulas

sin(2θ)=2sinθcosθsin(2θ)=2sinθcosθ
cos(2θ)=2cos2θ1=12sin2θ=cos2θsin2θcos(2θ)=2cos2θ1=12sin2θ=cos2θsin2θ

Example 1.25

Solving Trigonometric Equations

For each of the following equations, use a trigonometric identity to find all solutions.

  1. 1+cos(2θ)=cosθ1+cos(2θ)=cosθ
  2. sin(2θ)=tanθsin(2θ)=tanθ

Checkpoint 1.20

Find all solutions to the equation cos(2θ)=sinθ.cos(2θ)=sinθ.

Example 1.26

Proving a Trigonometric Identity

Prove the trigonometric identity 1+tan2θ=sec2θ.1+tan2θ=sec2θ.

Checkpoint 1.21

Prove the trigonometric identity 1+cot2θ=csc2θ.1+cot2θ=csc2θ.

Graphs and Periods of the Trigonometric Functions

We have seen that as we travel around the unit circle, the values of the trigonometric functions repeat. We can see this pattern in the graphs of the functions. Let P=(x,y)P=(x,y) be a point on the unit circle and let θθ be the corresponding angle .. Since the angle θθ and θ+2πθ+2π correspond to the same point P,P, the values of the trigonometric functions at θθ and at θ+2πθ+2π are the same. Consequently, the trigonometric functions are periodic functions. The period of a function ff is defined to be the smallest positive value pp such that f(x+p)=f(x)f(x+p)=f(x) for all values xx in the domain of f.f. The sine, cosine, secant, and cosecant functions have a period of 2π.2π. Since the tangent and cotangent functions repeat on an interval of length π,π, their period is ππ (Figure 1.34).

An image of six graphs. Each graph has an x axis that runs from -2 pi to 2 pi and a y axis that runs from -2 to 2. The first graph is of the function “f(x) = sin(x)”, which is a curved wave function. The graph of the function starts at the point (-2 pi, 0) and increases until the point (-((3 pi)/2), 1). After this point, the function decreases until the point (-(pi/2), -1). After this point, the function increases until the point ((pi/2), 1). After this point, the function decreases until the point (((3 pi)/2), -1). After this point, the function begins to increase again. The x intercepts shown on the graph are at the points (-2 pi, 0), (-pi, 0), (0, 0), (pi, 0), and (2 pi, 0). The y intercept is at the origin. The second graph is of the function “f(x) = cos(x)”, which is a curved wave function. The graph of the function starts at the point (-2 pi, 1) and decreases until the point (-pi, -1). After this point, the function increases until the point (0, 1). After this point, the function decreases until the point (pi, -1). After this point, the function increases again. The x intercepts shown on the graph are at the points (-((3 pi)/2), 0), (-(pi/2), 0), ((pi/2), 0), and (((3 pi)/2), 0). The y intercept is at the point (0, 1). The graph of cos(x) is the same as the graph of sin(x), except it is shifted to the left by a distance of (pi/2). On the next four graphs there are dotted vertical lines which are not a part of the function, but act as boundaries for the function, boundaries the function will never touch. They are known as vertical asymptotes. There are infinite vertical asymptotes for all of these functions, but these graphs only show a few. The third graph is of the function “f(x) = csc(x)”. The vertical asymptotes for “f(x) = csc(x)” on this graph occur at “x = -2 pi”, “x = -pi”, “x = 0”, “x = pi”, and “x = 2 pi”. Between the “x = -2 pi” and “x = -pi” asymptotes, the function looks like an upward facing “U”, with a minimum at the point (-((3 pi)/2), 1). Between the “x = -pi” and “x = 0” asymptotes, the function looks like an downward facing “U”, with a maximum at the point (-(pi/2), -1). Between the “x = 0” and “x = pi” asymptotes, the function looks like an upward facing “U”, with a minimum at the point ((pi/2), 1). Between the “x = pi” and “x = 2 pi” asymptotes, the function looks like an downward facing “U”, with a maximum at the point (((3 pi)/2), -1). The fourth graph is of the function “f(x) = sec(x)”. The vertical asymptotes for this function on this graph are at “x = -((3 pi)/2)”, “x = -(pi/2)”, “x = (pi/2)”, and “x = ((3 pi)/2)”. Between the “x = -((3 pi)/2)” and “x = -(pi/2)” asymptotes, the function looks like an downward facing “U”, with a maximum at the point (-pi, -1). Between the “x = -(pi/2)” and “x = (pi/2)” asymptotes, the function looks like an upward facing “U”, with a minimum at the point (0, 1). Between the “x = (pi/2)” and “x = (3pi/2)” asymptotes, the function looks like an downward facing “U”, with a maximum at the point (pi, -1). The graph of sec(x) is the same as the graph of csc(x), except it is shifted to the left by a distance of (pi/2). The fifth graph is of the function “f(x) = tan(x)”. The vertical asymptotes of this function on this graph occur at “x = -((3 pi)/2)”, “x = -(pi/2)”, “x = (pi/2)”, and “x = ((3 pi)/2)”. In between all of the vertical asymptotes, the function is always increasing but it never touches the asymptotes. The x intercepts on this graph occur at the points (-2 pi, 0), (-pi, 0), (0, 0), (pi, 0), and (2 pi, 0). The y intercept is at the origin. The sixth graph is of the function “f(x) = cot(x)”. The vertical asymptotes of this function on this graph occur at “x = -2 pi”, “x = -pi”, “x = 0”, “x = pi”, and “x = 2 pi”. In between all of the vertical asymptotes, the function is always decreasing but it never touches the asymptotes. The x intercepts on this graph occur at the points (-((3 pi)/2), 0), (-(pi/2), 0), ((pi/2), 0), and (((3 pi)/2), 0) and there is no y intercept.
Figure 1.34 The six trigonometric functions are periodic.

Just as with algebraic functions, we can apply transformations to trigonometric functions. In particular, consider the following function:

f(x)=Acos(B(xα))+C.f(x)=Acos(B(xα))+C.
(1.10)

In Figure 1.35, the constant αα causes a horizontal or phase shift. The factor BB changes the period. This transformed sine function will have a period 2π/|B|.2π/|B|. The factor AA results in a vertical stretch by a factor of |A|.|A|. We say |A||A| is the “amplitude of f.f.” The constant CC causes a vertical shift.

An image of a graph. The graph is of the function “f(x) = Acos(B(x - alpha)) + C”. Along the y axis, there are 3 hash marks: starting from the bottom and moving up, the hash marks are at the values “C - A”, “C”, and “C + A”. The distance from the origin to “C” is labeled “vertical shift”. The distance from “C - A” to “A” and the distance from “A” to “C + A” is “A”, which is labeled “amplitude”. On the x axis is a hash mark at the value “alpha” and the distance between the origin and “alpha” is labeled “horizontal shift”. The distance between two successive minimum values of the function (in other words, the distance between two bottom parts of the wave that are next to each other) is “(2 pi)/(absolute value of B)” is labeled the period. The period is also the distance between two successive maximum values of the function.
Figure 1.35 A graph of a general cosine function.

Notice in Figure 1.34 that the graph of y=cosxy=cosx is the graph of y=sinxy=sinx shifted to the left π/2π/2 units. Therefore, we can write cosx=sin(x+π/2).cosx=sin(x+π/2). Similarly, we can view the graph of y=sinxy=sinx as the graph of y=cosxy=cosx shifted right π/2π/2 units, and state that sinx=cos(xπ/2).sinx=cos(xπ/2).

A shifted sine curve arises naturally when graphing the number of hours of daylight in a given location as a function of the day of the year. For example, suppose a city reports that June 21 is the longest day of the year with 15.715.7 hours and December 21 is the shortest day of the year with 8.38.3 hours. It can be shown that the function

h(t)=3.7sin(2π365(t80.5))+12h(t)=3.7sin(2π365(t80.5))+12

is a model for the number of hours of daylight hh as a function of day of the year tt (Figure 1.36).

An image of a graph. The x axis runs from 0 to 365 and is labeled “t, day of the year”. The y axis runs from 0 to 20 and is labeled “h, number of daylight hours”. The graph is of the function “h(t) = 3.7sin(((2 pi)/365)(t - 80.5)) + 12”, which is a curved wave function. The function starts at the approximate point (0, 8.4) and begins increasing until the approximate point (171.8, 15.7). After this point, the function decreases until the approximate point (354.3, 8.3). After this point, the function begins increasing again.
Figure 1.36 The hours of daylight as a function of day of the year can be modeled by a shifted sine curve.

Example 1.27

Sketching the Graph of a Transformed Sine Curve

Sketch a graph of f(x)=3sin(2(xπ4))+1.f(x)=3sin(2(xπ4))+1.

Checkpoint 1.22

Describe the relationship between the graph of f(x)=3sin(4x)5f(x)=3sin(4x)5 and the graph of y=sin(x).y=sin(x).

Section 1.3 Exercises

For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of π.π.

113.

240 ° 240 °

114.

15 ° 15 °

115.

−60 ° −60 °

116.

−225 ° −225 °

117.

330 ° 330 °

For the following exercises, convert each angle in radians to degrees.

118.

π 2 rad π 2 rad

119.

7 π 6 rad 7 π 6 rad

120.

11 π 2 rad 11 π 2 rad

121.

−3 π rad −3 π rad

122.

5 π 12 rad 5 π 12 rad

Evaluate the following functional values.

123.

cos ( 4 π 3 ) cos ( 4 π 3 )

124.

tan ( 19 π 4 ) tan ( 19 π 4 )

125.

sin ( 3 π 4 ) sin ( 3 π 4 )

126.

sec ( π 6 ) sec ( π 6 )

127.

sin ( π 12 ) sin ( π 12 )

128.

cos ( 5 π 12 ) cos ( 5 π 12 )

For the following exercises, consider triangle ABC, a right triangle with a right angle at C. a. Find the missing side of the triangle. b. Find the six trigonometric function values for the angle at A. Where necessary, simplify to a fraction or round to three decimal places.

An image of a triangle. The three corners of the triangle are labeled “A”, “B”, and “C”. Between the corner A and corner C is the side b. Between corner C and corner B is the side a. Between corner B and corner A is the side c. The angle of corner C is marked with a right triangle symbol. The angle of corner A is marked with an angle symbol.
129.

a = 4 , c = 7 a = 4 , c = 7

130.

a = 21 , c = 29 a = 21 , c = 29

131.

a = 85.3 , b = 125.5 a = 85.3 , b = 125.5

132.

b = 40 , c = 41 b = 40 , c = 41

133.

a = 84 , b = 13 a = 84 , b = 13

134.

b = 28 , c = 35 b = 28 , c = 35

For the following exercises, PP is a point on the unit circle. a. Find the (exact) missing coordinate value of each point and b. find the values of the six trigonometric functions for the angle θθ with a terminal side that passes through point P.P. Rationalize denominators.

135.

P ( 7 25 , y ) , y > 0 P ( 7 25 , y ) , y > 0

136.

P ( −15 17 , y ) , y < 0 P ( −15 17 , y ) , y < 0

137.

P ( x , 7 3 ) , x < 0 P ( x , 7 3 ) , x < 0

138.

P ( x , 15 4 ) , x > 0 P ( x , 15 4 ) , x > 0

For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

139.

tan 2 x + sin x csc x tan 2 x + sin x csc x

140.

sec x sin x cot x sec x sin x cot x

141.

tan 2 x sec 2 x tan 2 x sec 2 x

142.

sec x cos x sec x cos x

143.

( 1 + tan θ ) 2 2 tan θ ( 1 + tan θ ) 2 2 tan θ

144.

sin x ( csc x sin x ) sin x ( csc x sin x )

145.

cos t sin t + sin t 1 + cos t cos t sin t + sin t 1 + cos t

146.

1 + tan 2 α 1 + cot 2 α 1 + tan 2 α 1 + cot 2 α

For the following exercises, verify that each equation is an identity.

147.

tan θ cot θ csc θ = sin θ tan θ cot θ csc θ = sin θ

148.

sec 2 θ tan θ = sec θ csc θ sec 2 θ tan θ = sec θ csc θ

149.

sin t csc t + cos t sec t = 1 sin t csc t + cos t sec t = 1

150.

sin x cos x + 1 + cos x 1 sin x = 0 sin x cos x + 1 + cos x 1 sin x = 0

151.

cot γ + tan γ = sec γ csc γ cot γ + tan γ = sec γ csc γ

152.

sin 2 β + tan 2 β + cos 2 β = sec 2 β sin 2 β + tan 2 β + cos 2 β = sec 2 β

153.

1 1 sin α + 1 1 + sin α = 2 sec 2 α 1 1 sin α + 1 1 + sin α = 2 sec 2 α

154.

tan θ cot θ sin θ cos θ = sec 2 θ csc 2 θ tan θ cot θ sin θ cos θ = sec 2 θ csc 2 θ

For the following exercises, solve the trigonometric equations on the interval 0θ<2π.0θ<2π.

155.

2 sin θ 1 = 0 2 sin θ 1 = 0

156.

1 + cos θ = 1 2 1 + cos θ = 1 2

157.

2 tan 2 θ = 2 2 tan 2 θ = 2

158.

4 sin 2 θ 2 = 0 4 sin 2 θ 2 = 0

159.

3 cot θ + 1 = 0 3 cot θ + 1 = 0

160.

3 sec θ 2 3 = 0 3 sec θ 2 3 = 0

161.

2 cos θ sin θ = sin θ 2 cos θ sin θ = sin θ

162.

csc 2 θ + 2 csc θ + 1 = 0 csc 2 θ + 2 csc θ + 1 = 0

For the following exercises, each graph is of the form y=AsinBxy=AsinBx or y=AcosBx,y=AcosBx, where B>0.B>0. Write the equation of the graph.

163.
An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -5 to 5. The graph is of a curved wave function that starts at the point (-4, 0) and decreases until the point (-2, 4). After this point the function begins increasing until it hits the point (2, 4). After this point the function begins decreasing again. The x intercepts of the function on this graph are at (-4, 0), (0, 0), and (4, 0). The y intercept is at the origin.
164.
An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -5 to 5. The graph is of a curved wave function that starts at the point (-4, -2) and increases until the point (-3, 2). After this point the function decreases until it hits the point (-2, -2). After this point the function increases until it hits the point (-1, 2). After this point the function decreases until it hits the point (0, -2). After this point the function increases until it hits the point (1, 2). After this point the function decreases until it hits the point (2, -2). After this point the function increases until it hits the point (3, 2). After this point the function begins decreasing again. The x intercepts of the function on this graph are at (-3.5, 0), (-2.5, 0), (-1.5, 0), (-0.5, 0), (0.5, 0), (1.5, 0), (2.5, 0), and (3.5, 0). The y intercept is at the (0, -2).
165.
An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -5 to 5. The graph is of a curved wave function. There are many periods and only a few will be explained. The function begins decreasing at the point (-1, 1) and decreases until the point (-0.5, -1). After this point the function increases until it hits the point (0, 1). After this point the function decreases until it hits the point (0.5, -1). After this point the function increases until it hits the point (1, 1). After this point the function decreases again. The x intercepts of the function on this graph are at (-0.75, 0), (-0.25, 0), (0.25, 0), and (0.75, 0). The y intercept is at (0, 1).
166.
An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -5 to 5. The graph is of a curved wave function. There are many periods and only a few will be explained. The function begins decreasing at the point (-1.25, 0.75) and decreases until the point (-0.75, -0.75). After this point the function increases until it hits the point (0.25, 0.75). After this point the function decreases until it hits the point (0.25, -0.75). After this point the function increases until it hits the point (0.75, 0.75). After this point the function decreases again. The x intercepts of the function on this graph are at (-1, 0), (-0.5, 0), (0, 0), and (0.5, 0). The y intercept is at the origin.

For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.

167.

y = sin ( x π 4 ) y = sin ( x π 4 )

168.

y = 3 cos ( 2 x + 3 ) y = 3 cos ( 2 x + 3 )

169.

y = −1 2 sin ( 1 4 x ) y = −1 2 sin ( 1 4 x )

170.

y = 2 cos ( x π 3 ) y = 2 cos ( x π 3 )

171.

y = −3 sin ( π x + 2 ) y = −3 sin ( π x + 2 )

172.

y = 4 cos ( 2 x π 2 ) y = 4 cos ( 2 x π 2 )

173.

[T] The diameter of a wheel rolling on the ground is 40 in. If the wheel rotates through an angle of 120°,120°, how many inches does it move? Approximate to the nearest whole inch.

174.

[T] Find the length of the arc intercepted by central angle θθ in a circle of radius r. Round to the nearest hundredth.

a. r=12.8r=12.8 cm, θ=5π6θ=5π6 rad b. r=4.378r=4.378 cm, θ=7π6θ=7π6 rad c. r=0.964r=0.964 cm, θ=50°θ=50° d. r=8.55r=8.55 cm, θ=325°θ=325°

175.

[T] As a point P moves around a circle, the measure of the angle changes. The measure of how fast the angle is changing is called angular speed, ω,ω, and is given by ω=θ/t,ω=θ/t, where θθ is in radians and t is time. Find the angular speed for the given data. Round to the nearest thousandth.

a. θ=7π4rad,t=10θ=7π4rad,t=10 sec b. θ=3π5rad,t=8θ=3π5rad,t=8 sec c. θ=2π9rad,t=1θ=2π9rad,t=1 min d. θ=23.76rad,t=14θ=23.76rad,t=14 min

176.

[T] A total of 250,000 m2 of land is needed to build a nuclear power plant. Suppose it is decided that the area on which the power plant is to be built should be circular.

  1. Find the radius of the circular land area.
  2. If the land area is to form a 45°45° sector of a circle instead of a whole circle, find the length of the curved side.
177.

[T] The area of an isosceles triangle with equal sides of length x is

1 2 x 2 sin θ , 1 2 x 2 sin θ ,

where θθ is the angle formed by the two sides. Find the area of an isosceles triangle with equal sides of length 8 in. and angle θ=5π/12θ=5π/12 rad.

178.

[T] A particle travels in a circular path at a constant angular speed ω.ω. The angular speed is modeled by the function ω=9|cos(πtπ/12)|.ω=9|cos(πtπ/12)|. Determine the angular speed at t=9t=9 sec.

179.

[T] An alternating current for outlets in a home has voltage given by the function

V ( t ) = 150 cos 368 t , V ( t ) = 150 cos 368 t ,

where V is the voltage in volts at time t in seconds.

  1. Find the period of the function and interpret its meaning.
  2. Determine the number of periods that occur when 1 sec has passed.
180.

[T] The number of hours of daylight in a northeast city is modeled by the function

N ( t ) = 12 + 3 sin [ 2 π 365 ( t 79 ) ] , N ( t ) = 12 + 3 sin [ 2 π 365 ( t 79 ) ] ,

where t is the number of days after January 1.

  1. Find the amplitude and period.
  2. Determine the number of hours of daylight on the longest day of the year.
  3. Determine the number of hours of daylight on the shortest day of the year.
  4. Determine the number of hours of daylight 90 days after January 1.
  5. Sketch the graph of the function for one period starting on January 1.
181.

[T] Suppose that T=50+10sin[π12(t8)]T=50+10sin[π12(t8)] is a mathematical model of the temperature (in degrees Fahrenheit) at t hours after midnight on a certain day of the week.

  1. Determine the amplitude and period.
  2. Find the temperature 7 hours after midnight.
  3. At what time does T=60°?T=60°?
  4. Sketch the graph of TT over 0t24.0t24.
182.

[T] The function H(t)=8sin(π6t)H(t)=8sin(π6t) models the height H (in feet) of the tide t hours after midnight. Assume that t=0t=0 is midnight.

  1. Find the amplitude and period.
  2. Graph the function over one period.
  3. What is the height of the tide at 4:30 a.m.?
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