Download Exam 2 Review Questions sec 2.4 to 3.5 rev0912 6th ed Math 170

Exam 2 Review Questions
sec 2.4 to 3.5
rev0912 6th ed
Math 170
Objectives: Use the knowledge of right triangles to solve application problems.
Define reference angles, angles computed with calculator, radian measure of
angles.
Define vectors, addition of vectors, vector components, and bearing, force
vectors, and applications of vectors.
Define circular motion components, arc length and area of a sector, linear and
angular velocity vectors, and applications.
Selected Practice Test Questions
Word Problems: Application of trigonometric functions
a) Applications of right triangles
1. A road up a hill makes an angle of 5.1 o with the horizontal. If the road
from the bottom of the hill to the top of the hill is 2.5mi. long, how high
is the hill?
ans. 0.22 mi
2. If a 73.0 foot flagpole casts a shadow 51.0 ft long, what is the angle of
elevation of the sun?
ans. θ = 55.1 o
3. A boat travels on a course of bearing S 63 o 50’E for 100 miles. How
many miles south and how many miles east has the boat traveled?
ans. x=89.8 mi east, y= 44.1 mi south
4. A person standing at point A notices that the angle of elevation to the
top of the mountain is 47 o 30’ . A second person is standing at point B
33.0 ft. further away from the mountain than the person at A. He finds
the angle of elevation to the top of the mountain to be 42 o 10’. How far
is the person at A from the base of the mountain? ( Assume the base of
the mountain is ⊥ to position)
ans. x= 161 ft.
Vectors:
b) The magnitude of a vector and angle with the positive x axis, find the
vertical and horizontal component of the vector v
1. |v|=17.6, θ = 67.2 o ans. |vx|=6.82, |vy|=16.2
1
2. |v|=4, 8, θ = 120 o
ans. |vx|=24, |vy|=42
c) The magnitude of the vertical and horizontal component of the vector v is
given, find the magnitude of vector v
1. |vx|=45, |vy|=15
ans. |v|=47.4
2. |vx|=2.2, |vy|=5.8
ans. |v|=6.2
d) Vector word problems
1. A boat is crossing a river that runs due north. The heading of the boat is
due east and it is moving through the water at 12.0 mph. If the current of
the river is a constant 3.25 mph., find the true course and true speed of the
boat. ans. N 74.8 o E speed = 12.3 mph
2. A plane has an airspeed of 195 mph and bearing N 30.0 o E . The air
currents are moving at a constant 32.5 mph at 300 o from the North. Find
the ground speed and the true course of the plane.
ans. ground speed 198 mph θ = N 20.5 oE
3. A plane flies for 3 hours at 230 km/hr on a course N 215 o E. How far
west and how far south does it travel in 3 hours?
ans. 396 km west, 565 km south
e) Find the reference angle for the following:
1. 150 o
ans. 30 o
2. 253.8 o ans. 73.8 o
3. -330 o
ans. 30 o
f) Find the exact value of the following using the reference angle
1. cos 135 o
3. cot 480 o
1
2
1
ans. 3
ans. -
2. tan 315 o
ans. -1
4. csc 300 o
ans. -
2
3
2
g) Use a calculator to find the following. Round the answer to 4 decimal
places.
2. csc (-236.7 o ans. 1.1964
1. cot 320 o ans. -1.1918
3. sec (140 o 20’) ans. -1.2991
h) Use a calculator to find θ between 0 o < θ < 360 o
1. cos θ =- 0.7660 and θ in quad III
ans. 220 o
2. cot θ =- 0.1234 and θ in quad IV
ans. 277 o
3. csc θ = 1.43250 and θ in quad II
ans. 135.7 o
4. sec θ =- 3.4159 and θ in quad II
ans. 107.0 o
i) Find θ ,where 0 o < θ < 360 o without using a calculator
1
and θ in quad III
2
1
2. tan θ = and θ in quad III
3
ans. 225 o
1. sin θ =
3. cos θ = -
ans. 210 o
3
/2 and θ in quad III
2
ans. 210 o
ans. 150 o
4. csc θ = 2 and θ in quad II
j) Using the formula s=r θ , find the unknown with given conditions
1. r=6 cm
2. r=
1
cm
4
s=3 cm. find θ
s=
ans. θ =
1
cm. find θ
8
1
radian
2
ans. θ = -
1
radian
2
k) Convert radians to degrees and degrees to radians
π
3π
4
ans. 45 o
2.
3. 540 o
ans. 3 π radians
4. 15 o
5. -43 o
ans. -0.75 radians
6. 5 rad ans. 286.5 o
1.
4
ans. 135 o
ans.
π
12
radians
3
ans. 114.6 o
7. 2 radians
l) Find the six trigonometric functions for each angle
1. Using the unit circle θ = 270 o
ans. sin θ = -1, cos θ = 0 ,
tan θ = undefined,
csc θ = -1, sec θ = undefined, cot θ = 0
2. Using the reference angle
ans. sin θ = -
θ=
5π
4
1
1
, cos θ = , tan θ = 1,
2
2
csc θ = - 2 , sec θ = - 2 ,
cot θ = 1
m) Use the unit circle and reference angles to evaluate
ans. - 1
1. sin(-90 o)
2. sin (
7π
)
4
1
2
3. If sin θ = , what is sin (- θ )
4
5
4. If sin (- θ )= , what is sin θ
π
5. cos ( 2 π + )
2
=-
2
2
1
2
4
ans. 5
ans. -
ans. 0
2
7π
)
3
13π
7. sin
6
1
2
1
ans.
2
6. cos (
8. cos
1
ans. -
ans.
3π
4
ans. -
2
1
=2
2
9. If angle θ is in standard position and intersects the unit circle at the point
(-
1
3
,) find sin θ , cos θ , tan θ
10
10
ans. sin =-
3
1
, cos θ = , tan θ = 3
10
10
4
n) Find the arc length and area of a sector
1. θ =2, r=3 in., find s
2. θ =315 o, r=5 in, find s
π
ans. 6 in
ans. 27.5 in
3. θ = , r=12 cm, find s
ans. 12.6 cm
4. θ =1, s=2 ft, find r
5. θ =150 o s=5 km, find r
ans. 2 ft
ans. 1.91 km
6. θ =15 o, r=10 m, find s
ans.
3
7. θ =
2π
, r=3 m, find area
5
8. θ =3, r=2 cm, find area
5π
m
6
ans. A=5.65m2
ans. A= 6 m2
o) Find the linear velocity v, where s=distance traveled and t = time.
(Use distance = rate x time, s=vt)
1
ft/sec
4
1. s=10 ft and t = 2 min
ans. v=
2. s=100 mi and t = 4 hr
ans. v=25 mph
p) Find the distance traveled given v and t
1. v = 10 ft/s and t = 4 sec
ans. s=40 ft
2. v = 63 mph and t = 10 sec
ans. s=0.175 mi
q) Find the angular velocity, ω , given the central angle, θ , and time t
1. θ =
3π
and t = 5 sec
4
2. θ = 24 π and t = 1.8 hr
ans. ω =
3π
rad/sec
20
ans. ω =41.9 rad/hr
r) Given the angular velocity, ω , radius r, and time t, find s, the distance
traveled
1. ω = 2 rad/sec, r=4 in and t = 5 sec ans. s=80 in
2. ω =
4π
, r= 8 m, and t = 20 sec
3
ans. s=
640π
m
3
5
s) Find the angular velocity, ω , associated with the given rpm (revolutions
per minute)
1. 20 rpm
2. 16
2
rpm
3
ans. ω = 40π rad/min , ≈ 126 rad/min
ans. ω =
100π
rad/min, ≈ 105 rad/min
3
t) Given the angular velocity, ω , radius r, find linear velocity v
1. ω = 4 rad/sec and r=8 in
2. ω = 20 rpm and r=1 ft
3. Find ω given r=6 cm and v=3 cm/sec
Forces
t)Force Vector word problems
ans. v=32 in/sec
ans. v=126 ft/min
ans. ω =0.5 rad/sec
1. Danny weighs 42.0 pounds and is sitting on a swing when his sister
Stacy pushes him horizontally on the swing at an angle of 30.0 o and stops.
Find the tension in the ropes of the swing and the magnitude and force
exerted by Stacey. (See figure in text) ans. T= 48.5 lbs, H = 24.2 lbs
2. A 10 pound weight is lying on a sit up bench. If the bench is inclined at
15 o there are 3 forces acting: The normal force N ⊥ to the bench, the
force F due to friction which keeps the weight from sliding, and the weight
W. If the weight does not move, the sum of the forces (x and y
components) = 0. Find the magnitudes of N and F.
ans. |N|=9.7 lbs |F| = 2.6 lbs.
6