Determine the six trigonometric ratios for a given acute angle of a right triangle.
Determine the trigonometric function values of 30°
Using a calculator, find function values for any acute angle, and given a function value of an acute angle, find the angle.
Given the function values of an acute angle, find the function values of its complement.
We begin our study of trigonometry by considering right triangles and acute angles measured in degrees. An acute angle is an angle with measure greater than 0°
The lengths of the sides of the triangle are used to define the six trigonometric ratios:
The sine of
θ
The ratio depends on the measure of angle θ
The cosine of
θ
The tangent of
θ
The six trigonometric ratios, or trigonometric functions, are defined as follows. Here the domain of each function is the set of acute angles. Later in this chapter, the domain will be extended first to the set of all angles, or rotations, and then to the set of real numbers.
In the right triangle shown at left, find the six trigonometric function values of (a)
θ
We use the definitions.
a)
sin θ=opphyp=1213,csc θ=hypopp=1312,cos θ=adjhyp=513, sec θ=hypadj=135,The references to opposite, adjacent, and hypotenuse are relative to θ.tan θ=oppadj=125, cot θ=adjopp=512Now Try Exercise 1.
In Example 1(a), we note that the value of csc θ, 1312
1213
the values of sec θ
If we know the values of the sine, cosine, and tangent functions of an angle, we can use these reciprocal relationships to find the values of the cosecant, secant, and cotangent functions of that angle.
Given that sin ϕ=45, cos ϕ=35
Using the reciprocal relationships, we have
and
Given that sinβ=√215,cosβ=25
Using the reciprocal relationships, we have
Now Try Exercise 7.
Triangles are said to be similar if their corresponding angles have the same measure. In similar triangles, the lengths of corresponding sides are in the same ratio. The right triangles shown below are similar. Note that the corresponding angles are equal, and the length of each side of the second triangle is four times the length of the corresponding side of the first triangle.
Let’s observe the sine, cosine, and tangent values of β in each triangle. Can we expect corresponding function values to be the same?
First Triangle | Second Triangle |
---|---|
sin β=35 | sin β=1220=35 |
cos β=45 | cos β=1620=45 |
tan β=34 | tan β=1216=34 |
For the two triangles, the corresponding values of sin β, cos β , and tan β are the same. The lengths of the sides are proportional—thus the ratios are the same. This must be the case because in order for the sine, cosine, and tangent to be functions, there must be only one output (the ratio) for each input (the angle β ).
We can find the other five trigonometric function values of an acute angle when one of the function-value ratios is known.
If sin β=67 and β is an acute angle, find the other five trigonometric function values of β .
We know from the definition of the sine function that the ratio
Using this information, let’s consider a right triangle in which the hypotenuse has length 7 and the side opposite β has length 6. To find the length of the side adjacent to β , we use the Pythagorean equation:
We now use the lengths of the three sides to find the other five ratios:
Now Try Exercise 9.
In Examples 1 and 4, we found the trigonometric function values of an acute angle of a right triangle when the lengths of the three sides were known. In most situations, we are asked to find the function values when the measure of the acute angle is given. For certain special angles such as 30° , 45° , and 60° , which are frequently seen in applications, we can use geometry to determine the function values.
A right triangle with a 45° angle actually has two 45° angles. Thus the triangle is isosceles, and the legs are the same length. Let’s consider such a triangle whose legs have length 1. Then we can find the length of its hypotenuse, c , using the Pythagorean equation as follows:
Such a triangle is shown below. From this diagram, we can easily determine the trigonometric function values of 45° .
It is sufficient to find only the function values of the sine, cosine, and tangent, since the other three function values are their reciprocals.
It is also possible to determine the function values of 30° and 60° . A right triangle with 30° and 60° acute angles is half of an equilateral triangle, as shown in the following figure. Thus if we choose an equilateral triangle whose sides have length 2 and take half of it, we obtain a right triangle that has a hypotenuse of length 2 and a leg of length 1. The other leg has length a, which can be found as follows:
We can now determine the function values of 30° and 60° :
Since we will often use the function values of 30°, 45° . and 60° , either the triangles that yield them or the values themselves should be memorized.
30° | 45° | 60° | |
---|---|---|---|
sin | 1/2 | √2/2 | √3/2 |
cos | √3/2 | √2/2 | 1/2 |
tan | √3/3 | 1 | √3 |
Let’s now use what we have learned about trigonometric functions of special angles to solve problems. We will consider such applications in greater detail in Section 6.2.
Trajectories for fireworks involve variables such as the launch angle, the launch velocity, and the size of the shell. Using physics to calculate the trajectory, fireworks technicians know exactly the path of the shell. Launch angles vary from 45° to 90° . For every inch of the diameter of the shell, a firework can travel about 100 ft vertically and 70 ft horizontally. These distances can vary depending on air resistance. (Sources: thinkquest.org, Pyrotechnico; www.dispatch.com, Aaron Harden) A 6-in. shell launched at an angle of 60° travels a horizontal distance of 390 ft. Approximate the height of the fireworks display. Round the answer to the nearest foot.
We begin with a diagram of the situation. We know the measure of an acute angle and the length of the adjacent side.
Since we want to determine the length of the opposite side, we can use either the tangent ratio or the cotangent ratio. In this case, we use the tangent ratio:
The fireworks display is approximately 675 ft high.
Now Try Exercise 29.
Historically, the measure of an angle has been expressed in degrees, minutes, and seconds. One minute, denoted 1′ , is such that 60′=1° , or 1′=160⋅(1°) . One second, denoted 1" , is such that 60"=1′ , or 1"=160⋅(1′) . Then 61 degrees, 27 minutes, 4 seconds could be written as 61°27′4″ . This D°M'S" form was common before the widespread use of calculators. Now the preferred notation is to express fraction parts of degrees in decimal degree form. For example, 61°27′4″≈61.45° in decimal degree form. Although the D°M′S″ notation is still widely used in navigation, we will most often use the decimal form in this text.
Most calculators can convert 5°42′30″ notation to decimal degree notation and vice versa. Procedures among calculators vary.
Convert 5°42′30″ . to decimal degree notation.
We enter 5°42′30″ . The calculator gives us
rounded to the nearest hundredth of a degree.
Without a calculator, we can convert as follows:
Now Try Exercise 37.
Convert 72.18° to D°M′S″ notation.
On a calculator, we enter 72.18. The result is
Without a calculator, we can convert as follows:
Now Try Exercise 45.
So far we have measured angles using degrees. Another useful unit for angle measure is the radian, which we will study in
Section 6.4. Most calculators work with either degrees or radians. Be sure to use whichever mode is appropriate. In this section, we use the DEGREE
mode.
Keep in mind the difference between an exact answer and an approximation. For example,
But using a calculator, you get an answer like
Calculators generally provide values only of the sine, cosine, and tangent functions. You can find values of the cosecant, secant, and cotangent by taking reciprocals of the sine, cosine, and tangent functions, respectively.
Using a calculator, find the trigonometric function value, rounded to four decimal places, of each of the following.
a) tan 29.7°
b) sec 48°
c) sin 84°10′39″
a) We check to be sure that the calculator is in DEGREE
mode. The function value is
b) The secant function value can be found by taking the reciprocal of the cosine function value:
c) We enter sin 84°10′39″ . The result is
Now Try Exercises 61 and 69.
We can use a calculator to find an angle for which we know a trigonometric function value.
Find the acute angle, to the nearest tenth of a degree, whose sine value is approximately 0.20113—that is, given sinθ=0.20113, find θ .
The quickest way to find the angle with a calculator is to use an inverse function key. (We first studied inverse functions in Section 5.1 and will consider inverse trigonometric functions in Section 7.4.) First check to be sure that your calculator is in DEGREE
mode. Usually two keys must be pressed in sequence. For this example, if we press
we find that the acute angle whose sine is 0.20113 is approximately 11.60304613°, or 11.6° .
Now Try Exercise 75.
A window-washing crew has purchased new 30-ft extension ladders. The manufacturer states that the safest placement on a wall is to extend the ladder to 25 ft and to position the base 6.5 ft from the wall (Source: R. D. Werner Co., Inc.). What angle does the ladder make with the ground in this position?
We make a drawing and then use the most convenient trigonometric function. Because we know the length of the side adjacent to θ and the length of the hypotenuse, we choose the cosine function.
From the definition of the cosine function, we have
Using a calculator, we find the acute angle whose cosine is 0.26:
Thus when the ladder is in its safest position, it makes an angle of about 75° with the ground.
Two angles are complementary whenever the sum of their measures is 90° . Each is the complement of the other. In a right triangle, the acute angles are complementary, because the sum of all three angle measures is 180° and the right angle accounts for 90° of this total. Thus if one acute angle of a right triangle is θ, the other is 90°-θ .
The six trigonometric function values of each of the acute angles in the right triangle below are listed on the right. Note that 53° and 37° are complementary angles because 53°+37°=90° .
For these angles, we note that
The sine of an angle is also the cosine of the angle’s complement. Similarly, the tangent of an angle is the cotangent of the angle’s complement, and the secant of an angle is the cosecant of the angle’s complement. These pairs of functions are called cofunctions. A list of cofunction identities follows.
Given that sin 18°≈0.3090, cos 18°≈0.9511, and tan 18°≈0.3249, find the six trigonometric function values of 72° .
Using reciprocal relationships, we know that
Since 72° and 18° are complementary, we have
Now Try Exercise 97.
In Exercises 1–6, find the six trigonometric function values of the specified angle.
1.
2.
3.
4.
5.
6.
7. Given that sin α=√53, cos α=23, and tan α=√52, find csc α, secα, and cotα.
8. Given that sin β=2√23, cos β=13, and tan β=2√2, find csc β, secβ, and cotβ.
Given a function value of an acute angle, find the other five trigonometric function values.
9. sin θ=2425
10. cos σ=0.7
11. tan ϕ=2
12. cotθ=13
13. csc θ=1.5
14. secβ=√17
15. cos β=√55
16. sin σ=1011
Find the exact function value.
17. cos 45°
18. tan 30°
19. sec 60°
20. sin 45°
21. cot 60°
22. csc 45°
23. sin 30°
24. cos 60°
25. tan 45°
26. sec 30°
27. csc 30°
28. tan 60°
29. Four Square. The game Four Square is making a comeback on college campuses. The game is played on a 16-ft square court divided into four smaller squares that meet in the center (Source: www.squarefour.org/
If a line is drawn diagonally from one corner to another corner, then a right triangle QTS is formed, where ∠QTS is 45°. Using the cosecant function, find the length of the diagonal. Round the answer to the nearest tenth of a foot.
30. Distance to a Fire Cave. Massive trees can survive wildfires that leave large caves in them (Source: National Geographic, October 2009, p. 32). A hiker observes scientists measuring a fire cave in a redwood tree in Prairie Creek Redwoods State Park. He estimates that he is 80 ft from the tree and that the angle between the ground and the line of sight to the scientists is 60°. Approximate how high the fire cave is. Round the answer to the nearest foot.
Convert to decimal degree notation. Round to two decimal places.
31. 9°43′
32. 52°15′
33. 35°50″
34. 64°53′
35. 3°2′
36. 19°47′23″
37. 49°38′46″
38. 76°11′34″
39. 15′5″
40. 68°2″
41. 5°53″
42. 44′10″
Convert to D°M′S″ notation. Round to the nearest second.
43. 17.6°
44. 20.14°
45. 83.025°
46. 67.84°
47. 11.75°
48. 29.8°
49. 47.8268°
50. 0.253°
51. 0.9°
52. 30.2505°
53. 39.45°
54. 2.4°
Find the function value. Round to four decimal places.
55. cos 51°
56. cot 17°
57. tan 4°13′
58. sin 26.1°
59. sec 38.43°
60. cos 74°10′40″
61. cos 40.35°
62. csc 45.2°
63. sin 69°
64. tan 63°48′
65. tan 85.4°
66. cos 4°
67. csc 89.5°
68. sec 35.28°
69. cot 30°25′6″
70. sin 59.2°
Using a calculator, find the acute angle θ , to the nearest tenth of a degree, for the given function value.
71. sinθ=0.5125
72. tanθ=2.032
73. tanθ=0.2226
74. cos θ=0.3842
75. sinθ=0.9022
76. tanθ=3.056
77. cos θ=0.6879
78. sinθ=0.4005
79. cotθ=2.127
(Hint:tanθ=1cotθ.)
80. csc θ=1.147
81. sec θ=1.279
82. cot θ=1.351
Find the exact acute angle θ for the given function value.
83. sin θ=√22
84. cotθ=√33
85. cos θ=12
86. sinθ=12
87. tanθ=1
88. cos θ=√32
89. csc θ=2√33
90. tanθ=√3
91. cotθ=√3
92. secθ=√2
Use the cofunction and reciprocal identities to complete each of the following.
93. cos 20°=______ 70°=1______ 20°
94. sin 64°=______ 26°=1______ 64°
95. tan 52°=cot ______ =1______ 52°
96. sec 13°=csc ______ =1______ 13°
97. Given that
sin 65°≈0.9063,cos 65°≈0.4226,tan 65°≈2.1445,cot 65°≈0.4663,sec 65°≈2.3662,csc 65°≈1.1034,
find the six function values of 25°.
98. Given that
sin 8°≈0.1392,cos 8°≈0.9903,tan 8°≈0.1405,cot 8°≈7.1154,sec 8°≈1.0098,csc 8°≈7.1853,
find the six function values of 82°.
99. Given that sin 71°10′5″≈0.9465, cos 71°10′5″≈0.3228, and tan 71°10′5″≈2.9321, find the six function values of 18°49′55″.
100. Given that sin 38.7°≈0.6252, cos 38.7°≈0.7804, and tan 38.7°≈0.8012, find the six function values of 51.3°.
101. Given that sin 82°=p, cos 82°=q, and tan 82°=r, find the six function values of 8° in terms of p, q, and r.
Graph the function.
102. f(x)=2-x [5.2]
103. f(x)=ex/2 [5.2]
104. g(x)=log2x [5.3]
105. h(x)=lnx [5.3]
Solve.
106. et=10,000 [5.5]
107. 5x=625 [5.5]
108. log (3x+1)-log (x-1)=2 [5.5]
109. log7 x=3 [5.5]
110. Given that cos θ=0.9651, find csc (90°-θ).
111. Given that secβ=1.5304, find sin(90°-β).
112. Find the six trigonometric function values of α.
113. Show that the area of this triangle is 12absinθ.