7.5: The Other Trigonometric Functions
- Page ID
- 115093
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In this section you will:
- Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent of π3,π4,π3,π4, and π6.π6.
- Use reference angles to evaluate the trigonometric functions secant, tangent, and cotangent.
- Use properties of even and odd trigonometric functions.
- Recognize and use fundamental identities.
- Evaluate trigonometric functions with a calculator.
A wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with the ground whose tangent is 112112 or less, regardless of its length. A tangent represents a ratio, so this means that for every 1 inch of rise, the ramp must have 12 inches of run. Trigonometric functions allow us to specify the shapes and proportions of objects independent of exact dimensions. We have already defined the sine and cosine functions of an angle. Though sine and cosine are the trigonometric functions most often used, there are four others. Together they make up the set of six trigonometric functions. In this section, we will investigate the remaining functions.
Finding Exact Values of the Trigonometric Functions Secant, Cosecant, Tangent, and Cotangent
We can also define the remaining functions in terms of the unit circle with a point (x,y)(x,y) corresponding to an angle of t,t, as shown in Figure 1. As with the sine and cosine, we can use the (x,y)(x,y) coordinates to find the other functions.
Figure 1
The first function we will define is the tangent. The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the unit circle. In Figure 1, the tangent of angle tt is equal to yx,x≠0.yx,x≠0. Because the y-value is equal to the sine of t,t, and the x-value is equal to the cosine of t,t, the tangent of angle tt can also be defined as sintcost,cost≠0.sintcost,cost≠0. The tangent function is abbreviated as tan.tan. The remaining three functions can all be expressed as reciprocals of functions we have already defined.
- The secant function is the reciprocal of the cosine function. In Figure 1, the secant of angle tt is equal to 1cost=1x,x≠0.1cost=1x,x≠0. The secant function is abbreviated as sec.sec.
- The cotangent function is the reciprocal of the tangent function. In Figure 1, the cotangent of angle tt is equal to costsint=xy,y≠0.costsint=xy,y≠0. The cotangent function is abbreviated as cot.cot.
- The cosecant function is the reciprocal of the sine function. In Figure 1, the cosecant of angle tt is equal to 1sint=1y,y≠0.1sint=1y,y≠0. The cosecant function is abbreviated as csc.csc.
If tt is a real number and (x,y)(x,y) is a point where the terminal side of an angle of tt radians intercepts the unit circle, then
tan tsec tcsc tcot t====yx,x≠01x,x≠01y,y≠0xy,y≠0tan t=yx,x≠0sec t=1x,x≠0csc t=1y,y≠0cot t=xy,y≠0
EXAMPLE 1
Finding Trigonometric Functions from a Point on the Unit Circle
The point (−3√2,12)(−32,12) is on the unit circle, as shown in Figure 2. Find sint,cost,tant,sect,csct,sint,cost,tant,sect,csct, and cott.cott.
Figure 2
- Answer
-
The point (2√2,−2√2)(22,−22) is on the unit circle, as shown in Figure 3. Find sint,cost,tant,sect,csct,sint,cost,tant,sect,csct, and cott.cott.
Figure 3
EXAMPLE 2
Finding the Trigonometric Functions of an Angle
Find sint,cost,tant,sect,csct,sint,cost,tant,sect,csct, and cott.cott. when t=π6.t=π6.
- Answer
-
Find sint,cost,tant,sect,csct,sint,cost,tant,sect,csct, and cott.cott. when t=π3.t=π3.
Because we know the sine and cosine values for the common first-quadrant angles, we can find the other function values for those angles as well by setting xx equal to the cosine and yy equal to the sine and then using the definitions of tangent, secant, cosecant, and cotangent. The results are shown in Table 1.
Angle | 00 | π6,or 30°π6,or 30° | π4,or 45°π4,or 45° | π3,or 60°π3,or 60° | π2,or 90°π2,or 90° |
---|---|---|---|---|---|
Cosine | 1 | 3√232 | 2√222 | 1212 | 0 |
Sine | 0 | 1212 | 2√222 | 3√232 | 1 |
Tangent | 0 | 3√333 | 1 | 3–√3 | Undefined |
Secant | 1 | 23√3233 | 2–√2 | 2 | Undefined |
Cosecant | Undefined | 2 | 2–√2 | 23√3233 | 1 |
Cotangent | Undefined | 3–√3 | 1 | 3√333 | 0 |
Table 1
Using Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent
We can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions. The procedure is the same: Find the reference angle formed by the terminal side of the given angle with the horizontal axis. The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x- and y-values in the original quadrant. Figure 4 shows which functions are positive in which quadrant.
To help remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase “A Smart Trig Class.” Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is “A,” all of the six trigonometric functions are positive. In quadrant II, “Smart,” only sine and its reciprocal function, cosecant, are positive. In quadrant III, “Trig,” only tangent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, “Class,” only cosine and its reciprocal function, secant, are positive.
Figure 4 The trigonometric functions are each listed in the quadrants in which they are positive.
Given an angle not in the first quadrant, use reference angles to find all six trigonometric functions.
- Measure the angle formed by the terminal side of the given angle and the horizontal axis. This is the reference angle.
- Evaluate the function at the reference angle.
- Observe the quadrant where the terminal side of the original angle is located. Based on the quadrant, determine whether the output is positive or negative.
EXAMPLE 3
Using Reference Angles to Find Trigonometric Functions
Use reference angles to find all six trigonometric functions of −5π6.−5π6.
- Answer
-
Use reference angles to find all six trigonometric functions of −7π4.−7π4.
Using Even and Odd Trigonometric Functions
To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.
Consider the function f(x)=x2,f(x)=x2, shown in Figure 5. The graph of the function is symmetrical about the y-axis. All along the curve, any two points with opposite x-values have the same function value. This matches the result of calculation: (4)2=(−4)2,(−5)2=(5)2,(4)2=(−4)2,(−5)2=(5)2, and so on. So f(x)=x2f(x)=x2 is an even function, a function such that two inputs that are opposites have the same output. That means f(−x)=f(x).f(−x)=f(x).
Figure 5 The function f(x)=x2f(x)=x2 is an even function.
Now consider the function f(x)=x3,f(x)=x3, shown in Figure 6. The graph is not symmetrical about the y-axis. All along the graph, any two points with opposite x-values also have opposite y-values. So f(x)=x3f(x)=x3 is an odd function, one such that two inputs that are opposites have outputs that are also opposites. That means f(−x)=−f(x).f(−x)=−f(x).
Figure 6 The function f(x)=x3f(x)=x3 is an odd function.
We can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle, as in Figure 7. The sine of the positive angle is y.y. The sine of the negative angle is −y.−y. The sine function, then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in Table 2.
Figure 7
sin tsin(−t)sin t==≠y−ysin(−t)sin t=ysin(−t)=−ysin t≠sin(−t) |
cos tcos(−t)cos t===xxcos(−t)cos t=xcos(−t)=xcos t=cos(−t) |
tan(t)tan(−t)tan t==≠yx−yxtan(−t)tan(t)=yxtan(−t)=−yxtan t≠tan(−t) |
sec tsec(−t)sec t===1x1xsec(−t)sec t=1xsec(−t)=1xsec t=sec(−t) |
csc tcsc(−t)csc t==≠1y1−ycsc(−t)csc t=1ycsc(−t)=1−ycsc t≠csc(−t) |
cot tcot(−t)cot t==≠xyx−ycot(−t)cot t=xycot(−t)=x−ycot t≠cot(−t) |
Table 2
An even function is one in which f(−x)=f(x).f(−x)=f(x).
An odd function is one in which f(−x)=−f(x).f(−x)=−f(x).
Cosine and secant are even:
cos(−t)sec(−t)==cos tsec tcos(−t)=cos tsec(−t)=sec t
Sine, tangent, cosecant, and cotangent are odd:
sin(−t)tan(−t)csc(−t)cot(−t)====−sin t−tan t−csc t−cot tsin(−t)=−sin ttan(−t)=−tan tcsc(−t)=−csc tcot(−t)=−cot t
EXAMPLE 4
Using Even and Odd Properties of Trigonometric Functions
If the secant of angle tt is 2, what is the secant of −t?−t?
- Answer
-
If the cotangent of angle tt is 3–√,3, what is the cotangent of −t?−t?
Recognizing and Using Fundamental Identities
We have explored a number of properties of trigonometric functions. Now, we can take the relationships a step further, and derive some fundamental identities. Identities are statements that are true for all values of the input on which they are defined. Usually, identities can be derived from definitions and relationships we already know. For example, the Pythagorean Identity we learned earlier was derived from the Pythagorean Theorem and the definitions of sine and cosine.
We can derive some useful identities from the six trigonometric functions. The other four trigonometric functions can be related back to the sine and cosine functions using these basic relationships:
tant=sintcosttant=sintcost
sect=1costsect=1cost
csct=1sintcsct=1sint
cott=1tant=costsintcott=1tant=costsint
EXAMPLE 5
Using Identities to Evaluate Trigonometric Functions
- ⓐ Given sin(45°)=2√2,cos(45°)=2√2,sin(45°)=22,cos(45°)=22, evaluate tan(45°).tan(45°).
- ⓑ Given sin(5π6)=12,cos(5π6)=−3√2,sin(5π6)=12,cos(5π6)=−32, evaluate sec(5π6).sec(5π6).
- Answer
-
Evaluate csc(7π6).csc(7π6).
EXAMPLE 6
Using Identities to Simplify Trigonometric Expressions
Simplify secttant.secttant.
- Answer
-
Simplify (tant)(cost).(tant)(cost).
Alternate Forms of the Pythagorean Identity
We can use these fundamental identities to derive alternate forms of the Pythagorean Identity, cos2t+sin2t=1.cos2t+sin2t=1. One form is obtained by dividing both sides by cos2t.cos2t.
cos2tcos2t+sin2tcos2t1+tan2t==1cos2tsec2tcos2tcos2t+sin2tcos2t=1cos2t1+tan2t=sec2t
The other form is obtained by dividing both sides by sin2t.sin2t.
cos2tsin2t+sin2tsin2tcot2t+1==1sin2tcsc2tcos2tsin2t+sin2tsin2t=1sin2tcot2t+1=csc2t
1+tan2t=sec2t1+tan2t=sec2t
cot2t+1=csc2tcot2t+1=csc2t
EXAMPLE 7
Using Identities to Relate Trigonometric Functions
If cos(t)=1213cos(t)=1213 and tt is in quadrant IV, as shown in Figure 8, find the values of the other five trigonometric functions.
Figure 8
- Answer
-
If sec(t)=−178sec(t)=−178 and 0<t<π,0<t<π, find the values of the other five functions.
As we discussed at the beginning of the chapter, a function that repeats its values in regular intervals is known as a periodic function. The trigonometric functions are periodic. For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or 2π,2π, will result in the same outputs for these functions. And for tangent and cotangent, only a half a revolution will result in the same outputs.
Other functions can also be periodic. For example, the lengths of months repeat every four years. If xx represents the length time, measured in years, and f(x)f(x) represents the number of days in February, then f(x+4)=f(x).f(x+4)=f(x). This pattern repeats over and over through time. In other words, every four years, February is guaranteed to have the same number of days as it did 4 years earlier. The positive number 4 is the smallest positive number that satisfies this condition and is called the period. A period is the shortest interval over which a function completes one full cycle—in this example, the period is 4 and represents the time it takes for us to be certain February has the same number of days.
The period PP of a repeating function ff is the number representing the interval such that f(x+P)=f(x)f(x+P)=f(x) for any value of x.x.
The period of the cosine, sine, secant, and cosecant functions is 2π.2π.
The period of the tangent and cotangent functions is π.π.
EXAMPLE 8
Finding the Values of Trigonometric Functions
Find the values of the six trigonometric functions of angle tt based on Figure 9.
Figure 9
- Answer
-
Find the values of the six trigonometric functions of angle tt based on Figure 10.
Figure 10
EXAMPLE 9
Finding the Value of Trigonometric Functions
If sin(t)=−3√2andcos(t)=12,findsec(t),csc(t),tan(t),cot(t).sin(t)=−32andcos(t)=12,findsec(t),csc(t),tan(t),cot(t).
- Answer
-
sin(t)=2√2andcos(t)=2√2,findsec(t),csc(t),tan(t),andcot(t)sin(t)=22andcos(t)=22,findsec(t),csc(t),tan(t),andcot(t)
Evaluating Trigonometric Functions with a Calculator
We have learned how to evaluate the six trigonometric functions for the common first-quadrant angles and to use them as reference angles for angles in other quadrants. To evaluate trigonometric functions of other angles, we use a scientific or graphing calculator or computer software. If the calculator has a degree mode and a radian mode, confirm the correct mode is chosen before making a calculation.
Evaluating a tangent function with a scientific calculator as opposed to a graphing calculator or computer algebra system is like evaluating a sine or cosine: Enter the value and press the TAN key. For the reciprocal functions, there may not be any dedicated keys that say CSC, SEC, or COT. In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent.
If we need to work with degrees and our calculator or software does not have a degree mode, we can enter the degrees multiplied by the conversion factor π180π180 to convert the degrees to radians. To find the secant of 30°,30°, we could press
(for a scientific calculator):130×π180COSor(for a graphing calculator):1cos(30π180)(for a scientific calculator):130×π180COSor(for a graphing calculator):1cos(30π180)
Given an angle measure in radians, use a scientific calculator to find the cosecant.
- If the calculator has degree mode and radian mode, set it to radian mode.
- Enter: 1/1/
- Enter the value of the angle inside parentheses.
- Press the SIN key.
- Press the = key.
Given an angle measure in radians, use a graphing utility/calculator to find the cosecant.
- If the graphing utility has degree mode and radian mode, set it to radian mode.
- Enter: 1/1/
- Press the SIN key.
- Enter the value of the angle inside parentheses.
- Press the ENTER key.
EXAMPLE 10
Evaluating the Cosecant Using Technology
Evaluate the cosecant of 5π7.5π7.
- Answer
-
Evaluate the cotangent of −π8.−π8.
Access these online resources for additional instruction and practice with other trigonometric functions.
7.4 Section Exercises
Verbal
1.
On an interval of [0,2π),[ 0,2π ), can the sine and cosine values of a radian measure ever be equal? If so, where?
2.
What would you estimate the cosine of ππ degrees to be? Explain your reasoning.
3.
For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?
4.
Describe the secant function.
5.
Tangent and cotangent have a period of π.π. What does this tell us about the output of these functions?
Algebraic
For the following exercises, find the exact value of each expression.
6.
tanπ6tanπ6
7.
secπ6secπ6
8.
cscπ6cscπ6
9.
cotπ6cotπ6
10.
tanπ4tanπ4
11.
secπ4secπ4
12.
cscπ4cscπ4
13.
cotπ4cotπ4
14.
tanπ3tanπ3
15.
secπ3secπ3
16.
cscπ3cscπ3
17.
cotπ3cotπ3
For the following exercises, use reference angles to evaluate the expression.
18.
tan5π6tan5π6
19.
sec7π6sec7π6
20.
csc11π6csc11π6
21.
cot13π6cot13π6
22.
tan7π4tan7π4
23.
sec3π4sec3π4
24.
csc5π4csc5π4
25.
cot11π4cot11π4
26.
tan8π3tan8π3
27.
sec4π3sec4π3
28.
csc2π3csc2π3
29.
cot5π3cot5π3
30.
tan225°tan225°
31.
sec300°sec300°
32.
csc150°csc150°
33.
cot240°cot240°
34.
tan330°tan330°
35.
sec120°sec120°
36.
csc210°csc210°
37.
cot315°cot315°
38.
If sint=34,sint=34, and tt is in quadrant II, find cost,sect,csct,tant,cost,sect,csct,tant, and cott.cott.
39.
If cost=−13,cost=−13, and tt is in quadrant III, find sint,sect,csct,tant,sint,sect,csct,tant, and cott.cott.
40.
If tant=125tant=125, and 0≤t<π20≤t<π2, find sint,cost,sect,csct,andcott.sint,cost,sect,csct,andcott.
41.
If sint=3√2sint=32 and cost=12,cost=12, find sect,csct,tant,sect,csct,tant, and cott.cott.
42.
If sin40°≈0.643sin40°≈0.643 and cos40°≈0.766,cos40°≈0.766, find sec40°,csc40°,tan40°,sec40°,csc40°,tan40°, and cot40°.cot40°.
43.
If sint=2√2,sint=22, what is the sin(−t)?sin(−t)?
44.
If cost=12,cost=12, what is the cos(−t)?cos(−t)?
45.
If sect=3.1,sect=3.1, what is the sec(−t)?sec(−t)?
46.
If csct=0.34,csct=0.34, what is the csc(−t)?csc(−t)?
47.
If tant=−1.4,tant=−1.4, what is the tan(−t)?tan(−t)?
48.
If cott=9.23,cott=9.23, what is the cot(−t)?cot(−t)?
Graphical
For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions.
49.
50.
51.
Technology
For the following exercises, use a graphing calculator to evaluate to three decimal places.
52.
csc5π9csc5π9
53.
cot4π7cot4π7
54.
secπ10secπ10
55.
tan5π8tan5π8
56.
sec3π4sec3π4
57.
cscπ4cscπ4
58.
tan98°tan98°
59.
cot33°cot33°
60.
cot140°cot140°
61.
sec310°sec310°
Extensions
For the following exercises, use identities to evaluate the expression.
62.
If tan(t)≈2.7,tan(t)≈2.7, and sin(t)≈0.94,sin(t)≈0.94, find cos(t).cos(t).
63.
If tan(t)≈1.3,tan(t)≈1.3, and cos(t)≈0.61,cos(t)≈0.61, find sin(t).sin(t).
64.
If csc(t)≈3.2,csc(t)≈3.2, and cos(t)≈0.95,cos(t)≈0.95, find tan(t).tan(t).
65.
If cot(t)≈0.58,cot(t)≈0.58, and cos(t)≈0.5,cos(t)≈0.5, find csc(t).csc(t).
66.
Determine whether the function f(x)=2sinxcosxf(x)=2sinxcosx is even, odd, or neither.
67.
Determine whether the function f(x)=3sin2xcosx+secxf(x)=3sin2xcosx+secx is even, odd, or neither.
68.
Determine whether the function f(x)=sinx−2cos2xf(x)=sinx−2cos2x is even, odd, or neither.
69.
Determine whether the function f(x)=csc2x+secxf(x)=csc2x+secx is even, odd, or neither.
For the following exercises, use identities to simplify the expression.
70.
cscttantcscttant
71.
sectcsctsectcsct
Real-World Applications
72.
The amount of sunlight in a certain city can be modeled by the function h=15cos(1600d),h=15cos(1600d), where hh represents the hours of sunlight, and dd is the day of the year. Use the equation to find how many hours of sunlight there are on February 11, the 42nd day of the year. State the period of the function.
73.
The amount of sunlight in a certain city can be modeled by the function h=16cos(1500d),h=16cos(1500d), where hh represents the hours of sunlight, and dd is the day of the year. Use the equation to find how many hours of sunlight there are on September 24, the 267th day of the year. State the period of the function.
74.
The equation P=20sin(2πt)+100P=20sin(2πt)+100 models the blood pressure, P,P, where tt represents time in seconds. (a) Find the blood pressure after 15 seconds. (b) What are the maximum and minimum blood pressures?
75.
The height of a piston, h,h, in inches, can be modeled by the equation y=3sinx+1,y=3sinx+1, where xx represents the crank angle. Find the height of the piston when the crank angle is 55°.55°.
76.
The height of a piston, h,h, in inches, can be modeled by the equation y=2cosx+5,y=2cosx+5, where xx represents the crank angle. Find the height of the piston when the crank angle is 55°.