Calculus_Volume_1_-_WEB_68M1Z5W-17

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1.22 Describe the relationship between the graph of f (x) = 3sin(4x) − 5 and the graph of y = sin(x).
Chapter 1 | Functions and Graphs 73
1.3 EXERCISES
For the following exercises, convert each angle in degrees
to radians. Write the answer as a multiple of π.
113. 240°
114. 15°
115. −60°
116. −225°
117. 330°
For the following exercises, convert each angle in radians
to degrees.
118. π
2 rad
119. 7π
6 rad
120. 11π
2 rad
121. −3π rad
122. 5π
12 rad
Evaluate the following functional values.
123. cos⎛⎝4π3
⎞
⎠
124. tan⎛⎝19π
4
⎞
⎠
125. sin⎛⎝−3π
4
⎞
⎠
126. sec⎛⎝π6
⎞
⎠
127. sin⎛⎝ π12
⎞
⎠
128. cos⎛⎝5π12
⎞
⎠
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, round to one decimal
place.
129. a = 4, c = 7
130. a = 21, c = 29
131. a = 85.3, b = 125.5
132. b = 40, c = 41
133. a = 84, b = 13
134. b = 28, c = 35
For the following exercises, P 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 θ with a terminal side that passes through point
P. Rationalize denominators.
135. P⎛⎝ 7
25, y⎞⎠, y > 0
136. P⎛⎝−15
17 , y⎞⎠, y < 0
137. P⎛⎝x, 7
3
⎞
⎠, x < 0
138. 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. tan2 x + sinxcscx
140. secxsinxcotx
141. tan2 x
sec2 x
142. secx − cosx
143. (1 + tanθ)2 − 2tanθ
74 Chapter 1 | Functions and Graphs
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144. sinx(cscx − sinx)
145. cos t
sin t + sin t
1 + cos t
146. 1 + tan2α
1 + cot2α
For the following exercises, verify that each equation is an
identity.
147. tanθcotθ
cscθ = sinθ
148. sec2 θ
tanθ = secθcscθ
149. sin t
csc t + cos t
sec t = 1
150. sinx
cosx + 1 + cosx − 1
sinx = 0
151. cotγ + tanγ = secγcscγ
152. sin2 β + tan2 β + cos2 β = sec2 β
153. 1
1 − sinα + 1
1 + sinα = 2sec2α
154. tanθ − cotθ
sinθcosθ = sec2 θ − csc2 θ
For the following exercises, solve the trigonometric
equations on the interval 0 ≤ θ < 2π.
155. 2sinθ − 1 = 0
156. 1 + cosθ = 1
2
157. 2tan2 θ = 2
158. 4sin2 θ − 2 = 0
159. 3cotθ + 1 = 0
160. 3secθ − 2 3 = 0
161. 2cosθsinθ = sinθ
162. csc2 θ + 2cscθ + 1 = 0
For the following exercises, each graph is of the form
y = AsinBx or y = AcosBx, where B > 0. Write the
equation of the graph.
163.
164.
165.
Chapter 1 | Functions and Graphs 75
166.
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
⎞
⎠
168. y = 3cos(2x + 3)
169. y = −1
2 sin⎛⎝14x
⎞
⎠
170. y = 2cos⎛⎝x − π
3
⎞
⎠
171. y = −3sin(πx + 2)
172. y = 4cos⎛⎝2x − 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°,
how many inches does it move? Approximate to the nearest
whole inch.
174. [T] Find the length of the arc intercepted by central
angle θ in a circle of radius r. Round to the nearest
hundredth. a. r = 12.8 cm, θ = 5π
6 rad b. r = 4.378 cm,
θ = 7π
6 rad c. r = 0.964 cm, θ = 50° d. r = 8.55 cm,
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, and is given by
ω = θ/t, where θ is in radians and t is time. Find the
angular speed for the given data. Round to the nearest
thousandth. a. θ = 7π
4 rad, t = 10 sec b.
θ = 3π
5 rad, t = 8 sec c. θ = 2π
9 rad, t = 1 min d.
θ = 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.
a. Find the radius of the circular land area.
b. If the land area is to form a 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
2x
2 sinθ, where θ 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/12 rad.
178. [T] A particle travels in a circular path at a constant
angular speed ω. The angular speed is modeled by the
function ω = 9|cos(πt − π/12)|. Determine the angular
speed at t = 9 sec.
179. [T] An alternating current for outlets in a home has
voltage given by the function V(t) = 150cos368t, where
V is the voltage in volts at time t in seconds.
a. Find the period of the function and interpret its
meaning.
b. 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 + 3sin⎡⎣ 2π
365(t − 79)⎤⎦,
where t is the number of days after January 1.
a. Find the amplitude and period.
b. Determine the number of hours of daylight on the
longest day of the year.
c. Determine the number of hours of daylight on the
shortest day of the year.
d. Determine the number of hours of daylight 90 days
after January 1.
e. Sketch the graph of the function for one period
starting on January 1.
76 Chapter 1 | Functions and Graphs
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181. [T] Suppose that T = 50 + 10sin⎡⎣ π12(t − 8)⎤⎦ is a
mathematical model of the temperature (in degrees
Fahrenheit) at t hours after midnight on a certain day of the
week.
a. Determine the amplitude and period.
b. Find the temperature 7 hours after midnight.
c. At what time does T = 60°?
d. Sketch the graph of T over 0 ≤ t ≤ 24.
182. [T] The function H(t) = 8sin⎛⎝π6 t
⎞
⎠ models the height
H (in feet) of the tide t hours after midnight. Assume that
t = 0 is midnight.
a. Find the amplitude and period.
b. Graph the function over one period.
c. What is the height of the tide at 4:30 a.m.?
Chapter 1 | Functions and Graphs 77