SECTION 6.3 Properties of the Trigonometric Functions 405 104. What is the range of the cotangent function? 105. What is the range of the secant function? 106. What is the range of the cosecant function? 107. Is the sine function even, odd, or neither? Is its graph symmetric? With respect to what? 108. Is the cosine function even, odd, or neither? Is its graph 109. Is the tangent function even, odd, or neither? Is its graph symmetric? With respect to what? 110. Is the cotangent function even, odd, or neither? Is its graph symmetric? With respect to what? 111. Is the secant function even, odd, or neither? Is its graph symmetric? With respect to what? symmetric? With respect to what? 112. Is the cosecant function even, odd, or neither? Is its graph symmetric? With respect to what? Applications and Extensions In Problems 113–118, use the periodic and even-odd properties. 120. Calculating the Time of a Trip Two oceanfront homes are located 8 miles apart on a straight stretch of beach, each a distance of 1 mile from a paved path that parallels the ocean. Sally can jog 8 miles per hour on the paved path, but only 3 miles per hour in the sand on the beach. Because a river flows directly between the two houses, it is necessary to jog in the sand to the road, continue on the path, and then jog directly back in the sand to get from one house to the other. See the figure. The time T to get from one house to the other as a function of the angle 0 shown in the figure is = sin 0 and f(a) 113. If f(0) find the exact value of: 3' (a) f(-a) (b) f(a) + f(a + 27) + f(a + 47) 1 find the exact value of: = cos 0 and f(a) 114. If f(0) (a) f(-a) (b) f(a) + f(a + 27) + f(a - 27) = tan 0 and f(a) = 2, find the exact value of: 115. If f(0) (a) f(-a) 16. If f(0) = cot 0 and f(a) = -3, find the exact value of: (b) f(a) + f(a + T) + f(a + 27) 2. 2 T(0) = 1 + 3 sin 0 4 tan 0

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
109,115
108. Is the cosine function even, odd, or neither? Is its graph
107 Is the sine function even, odd, or neither? Is its graph
106. What is the range of the cosecant function?
SECTION 6.3 Properties of the Trigonometric Functions 405
What is the range of the cotangent function?
109. Is the tangent function even, odd, or neither? Is its graph
symmetric? With respect to what?
110. Is the cotangent function even, odd, or neither? Is its graph
symmetric? With respect to what?
111. Is the secant function even, odd, or neither? Is its graph
symmetric? With respect to what?
112. Is the cosecant function even, odd, or neither? Is its graph
symmetric? With respect to what?
Symmetric? With respect to what?
symmetric? With respect to what?
Applications and Extensions
Doblems 113-118, use the periodic and even-odd properties.
120. Calculating the Time of a Trip Two oceanfront homes are
located 8 miles apart on a straight stretch of beach, each a
distance of 1 mile from a paved path that parallels the ocean.
Sally can jog 8 miles per hour on the paved path, but only
3 miles per hour in the sand on the beach. Because a
flows directly between the two houses, it is necessary to jog
in the sand to the road, continue on the path, and then jog
directly back in the sand to get from one house to the other.
See the figure. The time T to get from one house to the other
as a function of the angle 0 shown in the figure is
113. If f(0) = sin 0 and f(a) = !
find the exact value of:
(a) f(-a)
(b) f(a) + f(a + 27) + f(a + 4)
river
1
cos 0 and f(a)
find the exact value of:
4"
%3D
114. If f(8)
(b) f(a) + f(a + 27) + f(a – 2)
(a) f(-a)
15. If f(0) = tan 0 and f(a) = 2, find the exact value of:
(b) f(a) + f(a + 7) + f(a + 27)
(a) f(-a)
116. If f(8) = cot 0 and f(a) = -3, find the exact value of:
T(0) = 1 -
0 <0 <
3 sin 0
4 tan 0
(a) f(-a)
(b) f(a) + f(a + ™) + f(a + 47)
117. If f(0) = sec 0 and f(a) = -4, find the exact value of:
(b) f(a) + f(a + 27) + f(a + 47)
(a) Calculate the time T for tan 0 =
4'
(a) f(-a)
(b) Describe the path taken.
118. If f(0) = csc 0 and f(a) = 2, find the exact value of:
(c) Explain why 0 must be larger than 14°.
(a) f(-a)
119. Calculating the Time of a Trip From a parking lot, you
want to walk to a house on the beach. The house is located
1500 feet down a paved path that parallels the ocean, which
is 500 feet away. See the figure. Along the path you can walk
300 feet per minute, but in the sand on the beach you can
only walk 100 feet per minute.
The time T to get from the parking lot to the beach
house expressed as a function of the angle 0 shown in the
figure is
(b) f(a) + f(a + 27) + f(a + 4 )
Осean
4 mi
4 mi -
Beach
1 mi
Paved path ----
River
T(0)
: 5
0 < 0 <
3 tan 0
sin 0
121. Predator Population In predator-prey relationships, th.
populations of the predator and prey are often cyclical. I
a conservation area, rangers monitor the red fox populatic
and have determined that the population can be modeled
the function
Calculate the time Tif you walk directly from the parking lot
to the house.
500
(Hint: tan e =
1500
Ocean
P(t) = 40 cos-
+ 110 ba1RI
where t is the number of months
from the time monitoring began.
Use the model to estimate the
Beach
500 ft
Paved path
population of red foxes in
the conservation area after
10 months, 20 months, and
30 months.
Forest
1500 ft
Parking lot
Transcribed Image Text:108. Is the cosine function even, odd, or neither? Is its graph 107 Is the sine function even, odd, or neither? Is its graph 106. What is the range of the cosecant function? SECTION 6.3 Properties of the Trigonometric Functions 405 What is the range of the cotangent function? 109. Is the tangent function even, odd, or neither? Is its graph symmetric? With respect to what? 110. Is the cotangent function even, odd, or neither? Is its graph symmetric? With respect to what? 111. Is the secant function even, odd, or neither? Is its graph symmetric? With respect to what? 112. Is the cosecant function even, odd, or neither? Is its graph symmetric? With respect to what? Symmetric? With respect to what? symmetric? With respect to what? Applications and Extensions Doblems 113-118, use the periodic and even-odd properties. 120. Calculating the Time of a Trip Two oceanfront homes are located 8 miles apart on a straight stretch of beach, each a distance of 1 mile from a paved path that parallels the ocean. Sally can jog 8 miles per hour on the paved path, but only 3 miles per hour in the sand on the beach. Because a flows directly between the two houses, it is necessary to jog in the sand to the road, continue on the path, and then jog directly back in the sand to get from one house to the other. See the figure. The time T to get from one house to the other as a function of the angle 0 shown in the figure is 113. If f(0) = sin 0 and f(a) = ! find the exact value of: (a) f(-a) (b) f(a) + f(a + 27) + f(a + 4) river 1 cos 0 and f(a) find the exact value of: 4" %3D 114. If f(8) (b) f(a) + f(a + 27) + f(a – 2) (a) f(-a) 15. If f(0) = tan 0 and f(a) = 2, find the exact value of: (b) f(a) + f(a + 7) + f(a + 27) (a) f(-a) 116. If f(8) = cot 0 and f(a) = -3, find the exact value of: T(0) = 1 - 0 <0 < 3 sin 0 4 tan 0 (a) f(-a) (b) f(a) + f(a + ™) + f(a + 47) 117. If f(0) = sec 0 and f(a) = -4, find the exact value of: (b) f(a) + f(a + 27) + f(a + 47) (a) Calculate the time T for tan 0 = 4' (a) f(-a) (b) Describe the path taken. 118. If f(0) = csc 0 and f(a) = 2, find the exact value of: (c) Explain why 0 must be larger than 14°. (a) f(-a) 119. Calculating the Time of a Trip From a parking lot, you want to walk to a house on the beach. The house is located 1500 feet down a paved path that parallels the ocean, which is 500 feet away. See the figure. Along the path you can walk 300 feet per minute, but in the sand on the beach you can only walk 100 feet per minute. The time T to get from the parking lot to the beach house expressed as a function of the angle 0 shown in the figure is (b) f(a) + f(a + 27) + f(a + 4 ) Осean 4 mi 4 mi - Beach 1 mi Paved path ---- River T(0) : 5 0 < 0 < 3 tan 0 sin 0 121. Predator Population In predator-prey relationships, th. populations of the predator and prey are often cyclical. I a conservation area, rangers monitor the red fox populatic and have determined that the population can be modeled the function Calculate the time Tif you walk directly from the parking lot to the house. 500 (Hint: tan e = 1500 Ocean P(t) = 40 cos- + 110 ba1RI where t is the number of months from the time monitoring began. Use the model to estimate the Beach 500 ft Paved path population of red foxes in the conservation area after 10 months, 20 months, and 30 months. Forest 1500 ft Parking lot
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