Download HOMEWORK 1 (Chapter 6.1-6.2) 1. Convert each angle in degrees

HOMEWORK 1 (Chapter 6.1-6.2)
1. Convert each angle in degrees to radians.
(A) 240°
(B) −135°
(C) −40°
2. Convert each angle in radian to degrees:
(A) −
5𝜋
4
(B) 2
3. Let 𝑠 denotes the length of the arc of a circle of radius 𝑟 subtended by the central angle 𝜃. Let 𝐴 denoted the area
of the section of a circle of radius 𝑟 subtended by the central angle 𝜃. Find the missing quantity. Round answers to
three decimal places
1
(A) 𝑟 = 2 inches, 𝜃 = 30°, 𝑠 =?
(B) 𝜃 = 3 radian, 𝐴 = 2 square feet, 𝑟 =?
4. An object is traveling around a circle with a radius of 5 centimeters. If in 20 seconds a central angle of
1
3
radian is
swept out, what is the angular speed of the object? What is its linear speed?
5. 𝑃(𝑥, 𝑦) is the point on the unit circle that corresponds to a real number 𝜃. Find the exact values of the six
trigonometric function of 𝜃.
2 √21
)
5 5
(A) (− ,
(B) (3, −4)
(C) (−5, −12)
6. Find the exact value of each expression. Do not use a calculator.
(A) tan(6𝜋)
(B) sec(−𝜋)
(C) 4 sin(90°) − 3 tan(180°)
7. Find the exact values of the six trigonometric functions of the given angle. If any are not defined, say “not
defined”. Do not use a calculator.
(A)
8𝜋
3
(B) −135°
HOMEWORK 2 (Chapter 6.4-6.6)
1. Use the fact that the trigonometric functions are periodic to find the exact value of each expression. Do not use a
calculator.
19𝜋
)
6
(A) sin(405°)
𝜋
(B) tan (
𝜋
(C) cos (− 4 )
(E) sin(80°) csc(80°)
(D) sec (− 6 )
2. Find the exact value of each of the remaining trigonometric function of 𝜃
4
(A) sin 𝜃 = − 5 , 𝜃 in quadrant III
1
3
(C) cos 𝜃 = − ,
𝜋
2
<𝜃<𝜋
4
(B) cos 𝜃 = − 5 , 𝜃 in quadrant III
(D) tan 𝜃 =
5
,
12
sin 𝜃 < 0
3. Determine the amplitude, period, and phase shift of each function without graphing.
5
(A) 𝑦 = 2 sin 𝑥
(B) 𝑦 = 3 cos (−
𝜋
(C) 𝑦 = 2 cos (3𝑥 + 2 )
2𝜋
𝑥)
3
𝜋
(D) 𝑦 = −3 sin (−2𝑥 + 2 )
4. Write the equation of a sine function that has the given characteristics.
(A) Amplitude: 2 and period: 4𝜋
(B) Amplitude: 2 and period: 𝜋; phase shift:
5. Find the domain, range, and period of the given functions.
𝜋
(A) 𝑦 = tan ( 2 𝑥)
1
(B) 𝑦 = 4 sec (2 𝑥)
1
2
HOMEWORK 3 (Chapter 7.1-7.4)
1. Find the exact value of each expression. Do not use a calculator.
√2
(A) sin−1 ( 2 )
(D) cos−1 (−
√3
)
2
(B) sin−1(−1)
(C) tan−1 (√3)
(E) cot −1 (√3)
(F) csc −1 (−1)
2. Find the exact value of each expression. Do not use a calculator.
1
2
(C) sin(tan−1 (3))
7𝜋
))
6
(F) cos−1 (cos ( ))
√2
(B) sec (sin−1 (− ))
5𝜋
4
(E) sin−1 (cos (−
(A) cos (sin−1 ( 2 ))
(D) cos−1 (sin ( ))
4𝜋
5
3. Use a calculator to find the value for each expression rounded to two decimal places
√2
(A) sin−1(0.1)
(B) cos−1 ( 3 )
(D) sec −1(4)
(E) cot −1 (2)
(C) cos(cos −1 (1.2))
4. Establish each identity
(A) (sec 𝑥 − 1)(sec 𝑥 + 1) = tan2 𝑥
1−sin 𝑥
(C) 1+sin 𝑥 = (sec 𝑥 − tan 𝑥)2
(B) 3 sin2 𝑥 + 4 cos2 𝑥 = 3 + cos 2 𝑥
(D)
sec 𝑥−cos 𝑥
sec 𝑥 cos 𝑥
= sec 𝑥 − cos 𝑥
HOMEWORK 4 (Chapter 7.5-7.7)
1. Find the exact value of each expression.
𝜋
(A) cos(70°) cos(20°) − sin(70°) sin(20°)
(C)
5𝜋
tan(20°)+tan(25°)
1−tan(20°) tan(25°)
(D) cos(255°) − cos(195°)
2. Express each product as a sum containing only sines or cosines.
(A) sin(4𝑥) cos(2𝑥)
(B) cos(3𝑥) cos(5𝑥)
3. Express each sum or difference as a product of sines and/or cosines.
(A) sin(4𝑥) − sin(2𝑥)
(B) sin 𝑥 + sin(3𝑥)
4. Find the exact value of each expression.
7𝜋
17𝜋
(A) cos ( 12 )
(B) sin ( 12 )
3
5. Find sin(𝑥 + 𝑦) when sin 𝑥 = 5 , 0 < 𝑥 <
𝑥
2
3
5
𝜋
2
4
3
7. Find cos(2𝑥) if tan 𝑥 = , 𝜋 < 𝑥 <
12
𝜋
and cos 𝑦 = 13 , − 2 < 𝑦 < 0.
6. Find sin(2𝑥) and cos ( ) if sin 𝑥 = , 0 < 𝑥 <
𝜋
5𝜋
(B) cos (12) sin ( 12 ) + sin (12) cos ( 12 )
𝜋
2
3𝜋
.
2
8. Use the half-angle formulas to find the exact value of sin(22.5°)
HOMEWORK 5 (Chapter7.3, 8.3-8.4)
1. Solve the triangle:
(A) 𝑏 = 1, 𝑐 = 3, 𝐴 = 80°
(B) 𝑎 = 5, 𝑏 = 8, 𝑐 = 9
2. A cruise ship maintains an average speed of 15 knots in going from San Juan, Puerto Rico, to Barbados, West
Indies, a distance of 600 nautical miles. To avoid a tropical storm, the captain heads out of San Juan in a direction
of 20° off a direct heading to Barbados. The captain maintains the 15-knot speed for 10 hours, after which time
the path to Barbados becomes clear of storms.
San Juan
Barbados
20°
ship
(A) Through what angle should the captain turn to head directly to Barbados?
(B) Once the turn is made, how long will it be before the ship reaches Barbados if the same 15-knot speed is
maintained?
3. Find the area of each triangle. Round answers to two decimal places.
(A) 𝑎 = 5, 𝑏 = 8, 𝑐 = 9
(B) 𝑎 = 6, 𝑏 = 4, 𝐶 = 60°
4. Solve each equation on the interval 0 ≤ 𝜃 < 2𝜋
(A) 2 sin 𝜃 + 3 = 2
1
(C) cos(2𝜃) = − 2
(B) 4 cos2 𝜃 = 1
𝜃
𝜋
(D) tan (2 + 3 ) = 1
HOMEWORK 6 (Chapter 7.3, 8.1-8.2)
1. Find 𝑎, 𝑐, and B when 𝑏 = 4, 𝐴 = 10° in the right triangle shown below.
2. A 22-foot extension ladder leaning against a building makes a 70o angle with the ground. How far up the building
does the ladder touch?
3. Two sides and an angle are given. Determine whether the given information results in one triangle; two triangles;
or no triangle at all. Solve each triangle that results
(A) 𝑎 = 3, 𝑏 = 2, 𝐴 = 50°
(B) 𝑎 = 2, 𝑏 = 1, 𝐵 = 100°
(C) 𝑎 = 2, 𝑏 = 1, 𝐵 = 25°
4. To find the distance across a canyon, a surveying team locates points A and B on one side of the canyon and point
C on the other side of the canyon. The distance between A and B is found to be 92 yards. The angle CAB is 67°,
and angle CBA is 89°. Find the distance across the canyon. Round to the nearest yard.
5. An aircraft is spotted by two observers who are 1000 feet apart. As the airplane passes over the line joining them,
each observer takes a sighting of the angle of elevation to the plane, as indicated in the figure. How high is the
airplane?
6. Find the approximate value of h in the following diagram. Round to the nearest tenth.
HOMEWORK 7 (Chapter 9.1-9.2)
1. The polar coordinates of a point are given. Find the rectangular coordinates of each point.
(A) (−2,
3𝜋
)
4
𝜋
(B) (−1, − 3 )
2. The rectangular coordinates of a point are given. Find the polar coordinates of each point.
(A) (1, −1)
(B) (√3, 1)
3. Transform each rectangular equation to an equation in polar coordinates.
(A) 2𝑥 2 + 2𝑦 2 = 3
(B) 𝑥 2 = 4𝑦
4. Transform each polar equation to an equation in rectangular coordinates.
(A) 𝑟 = 4
(B) 𝑟 sin 𝜃 = 4
(C) 𝑟 = 2 cos 𝜃
5. Plot each complex number in the complex plane and write it in polar form. Express the argument in degrees.
(A) 1 + 𝑖
(B) −3𝑖
6. Write each complex number in rectangular form
7𝜋
4
(A) 2(cos(120°) + 𝑖 sin(120°))
𝑧
7. Find 𝑧𝑤 and 𝑤. Leave your answer in polar form.
(A) 𝑧 = 2(cos(40°) + 𝑖 sin(40°)); 𝑤 = 6(cos(20°) + 𝑖 sin(20°))
𝜋
8
𝜋
8
9𝜋
16
9𝜋
16
(B) 𝑧 = 2 (cos ( ) + 𝑖 sin ( )) ; 𝑤 = 8 (cos ( ) + 𝑖 sin ( ))
𝜋
7𝜋
4
(B) 4 (cos ( ) + 𝑖 sin ( ))
𝜋
5
8. Write the expression [2 (cos (10) + 𝑖 sin (10))] in the standard form 𝑎 + 𝑏𝑖
HOMEWORK 8 (Chapter 9.3-9.4)
1. The vector 𝐯 has initial point P and terminal point Q. Write 𝐯 in the form 𝑎𝐢 + 𝑏𝐣.
(A) 𝑃 = (−2, −1); 𝑄 = (6, −2)
(B) 𝑃 = (−3,8); 𝑄 = (5, −2)
2. Find each quantity if 𝐯 = 3𝐢 − 5𝐣 and 𝐰 = −2𝐢 + 3𝐣.
(A) ‖𝐯‖
(B) 𝟐𝐯 + 𝟑𝐰
(C) ‖𝐯 − 𝐰‖
(D) ‖𝐯‖ − ‖𝐰‖
3. Find the unit vector in the same direction 𝐯 when 𝐯 = 3𝐢 − 4𝐣.
4. Write the vector 𝐯 in the form 𝑎𝐢 + 𝑏𝐣 , given its magnitude ‖𝐯‖ and the angle 𝜃 it makes the positive 𝑥-axis if
‖𝐯‖ = 14, 𝜃 = 120°.
5. A Boeing 747 jumbo jet maintains a constant airspeed of 550 miles per hour heading due north. The jet stream is
100 miles per hour in the northeasterly direction. Find the ground speed of the jet.
6. An airplane is traveling due north at 500 mph. The wind is blowing northwest at 50 mph. What is the airplane’s
speed as measured from the ground? Round to the nearest mile per hour.
Problem 7-9: (A) find the dot product 𝐯 ∙ 𝐰; (B) find the angle between 𝐯 and 𝐰; (C) state whether the vectors are
parallel, orthogonal, or neither.
7. 𝐯 = 𝐢 − 𝐣 and 𝐰 = 𝐢 + 𝐣
9. 𝐯 = 4𝐢 and 𝐰 = 𝐣
8. 𝐯 = √3𝐢 − 𝐣 and 𝐰 = 𝐢 + 𝐣