Review for Test #5
TOPIC #1:
Analytic Trigonometry
1. Evaluate without using a calculator
8
a. Find sin 𝑥 and cos 𝑥 if tan 𝑥 = 15 and sin 𝑥 < 0
b.
c.
d.
e.
f.
g.
Find csc 𝑥 and cot 𝑥 if sec 𝑥 = 5 and tan 𝑥 > 0
sin 15°
cos 75°
tan 15°
sin 105°
cos 22.5°
2. Find all solutions to the equation in the interval [0,2𝜋)
a. 2 sin 𝑥 cos 𝑥 − cos 𝑥 = 0
b. tan 𝑥𝑠𝑖𝑛2 𝑥 = tan 𝑥
c. √2 tan 𝑥 cos 𝑥 − tan 𝑥 = 0
d. 2𝑠𝑖𝑛2 𝑥 + 3 sin 𝑥 + 1 = 0
e. 4𝑐𝑜𝑠 2 𝑥 − 4 cos 𝑥 + 1 = 0
f. cos 𝑥 − 2𝑠𝑖𝑛2 𝑥 + 1 = 0
g. 2𝑠𝑖𝑛2 𝑥 + 3 sin 𝑥 = 2
h. cos 2𝑥 + cos 𝑥 = 0
i. cos 2𝑥 + sin 𝑥 = 0
3. Prove the Identity
a.
𝑠𝑒𝑐 2 𝑥−1
sin 𝑥
2
=
sin 𝑥
1−𝑠𝑖𝑛2 𝑥
2
b. 𝑠𝑖𝑛 𝑥 − 𝑐𝑜𝑠 𝑥 = 1 − 2𝑐𝑜𝑠 2 𝑥
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
1
+
tan 𝑥
sec 𝑥+1
tan 𝑥
4
tan 𝑥 = sec 𝑥 csc 𝑥
=
sin 𝑥
1−cos 𝑥
2
𝑡𝑎𝑛 𝑥 + 𝑡𝑎𝑛 𝑥 = 𝑠𝑒𝑐 4 𝑥 − 𝑠𝑒𝑐 2 𝑥
sin(𝑥 − 𝑦) + sin(𝑥 + 𝑦) = 2 sin 𝑥 cos 𝑥
cos 3𝑥 = 𝑐𝑜𝑠 3 𝑥 − 3𝑠𝑖𝑛2 𝑥 cos 𝑥
sin 4𝑥 + sin 2𝑥 = 2 sin 3𝑥 cos 𝑥
sin 4𝑥 = 2 sin 2𝑥 cos 2𝑥
cos 6𝑥 = 2𝑐𝑜𝑠 2 3𝑥 − 1
2 cot 2𝑥 = cot 𝑥 − tan 𝑥
sin 3𝑥 = (sin 𝑥)(3 − 4𝑠𝑖𝑛2 𝑥)
tan 𝑥+sin 𝑥
2 tan 𝑥
𝑥
= 𝑐𝑜𝑠 2 (2)
n. 2 sin 𝑥𝑐𝑜𝑠 3 𝑥 + 2𝑠𝑖𝑛3 𝑥 cos 𝑥 = sin 2𝑥
Topic #2:
Vectors
⃗⃗⃗⃗⃗ and 𝑃𝑄
⃗⃗⃗⃗⃗ are equivalent by showing they represent the same vector.
1. Prove 𝑅𝑆
a. R = (-4, 7), S = (-1, 5), P = (0,0) and Q = (3, -2)
b. R = (-2, -1), S = (2, 4), P = (-3, -1) and Q = (1, 4)
2. Let u = ⟨2, −1⟩, 𝑣 = ⟨4,2⟩ 𝑎𝑛𝑑 𝑤 = ⟨1, −3⟩. Find the indicated expressions.
a. u – v
b. 2u – 3w
c. |𝐮 + 𝐯|
d. |𝐰 − 2𝐮|
3. Let A = (2, -1), B = (3, 1), C = (-4, 2) and D = (1, -5). Find the component form and magnitude of
the vector.
⃗⃗⃗⃗⃗
a. 3𝐴𝐵
b. ⃗⃗⃗⃗⃗
𝐴𝐵 + ⃗⃗⃗⃗⃗
𝐶𝐷
c. ⃗⃗⃗⃗⃗
𝐴𝐶 + ⃗⃗⃗⃗⃗⃗
𝐵𝐷
d. ⃗⃗⃗⃗⃗
𝐶𝐷 + ⃗⃗⃗⃗⃗
𝐴𝐵
4. Find 𝐮 ∙ 𝐯
a. u = ⟨5, 3⟩, v = ⟨12,4⟩
b. u = ⟨5, 6⟩, v = ⟨−2,14⟩
c. u = ⟨8, −3⟩, v = ⟨1,7⟩
d. u = ⟨2, −5⟩, v = ⟨22, −4⟩
e. u =-4i – 9j, v = -3i – 2j
f. u =7i , v = -2i + 5j
g. u = 2i – 4j, v = 6j
5. Find (a) a unit vector in the direction of ⃗⃗⃗⃗⃗
𝐴𝐵 and (b) a vector of magnitude 3 in the opposite
direction.
a. A = (4, 0), B = (2, 1)
b. A = (-1, -4), B = (5, 2)
c. A = (3, 1), B = (-2, 3)
6. Find (a) the direction angles of u and v and (b) the angle between u and v.
a. u = ⟨4,3⟩, v = ⟨2,5⟩
b. u = ⟨2, −3⟩, v = ⟨1,6⟩
c. u = ⟨−2,4⟩, v = ⟨6,4⟩
Topic #3:
Parametric Equations
1. Use an algebraic method to eliminate the parameter and identify the graph of the parametric
curve.
a. 𝑥 = 3 − 5𝑡, 𝑦 = 4 + 3𝑡
b. 𝑥 = 4 + 𝑡, 𝑦 = −8 − 5𝑡
c. 𝑥 = 2𝑡 2 + 3, 𝑦 = 𝑡 − 1
d. 𝑥 = 3 cos 𝑡, 𝑦 = 3 sin 𝑡
e. 𝑥 = 𝑒 2𝑡 − 1, 𝑦 = 𝑒 𝑡
f. 𝑥 = 𝑡 3 , 𝑦 = ln 𝑥
Topic #4:
Polar Equations
1. Use an algebraic method to find the rectangular coordinates of the point with given polar
coordinates.
a. (1.5,
7𝜋
)
3
b. (2.5,
17𝜋
)
4
c. (−2, 𝜋)
𝜋
2
e. (2,270°)
d. (1, )
2. Rectangular coordinates of point P are given. Find 3 polar points that represent the same point.
a. (1, 1)
b. (3, 5)
c. (-2, 3)
d. (-1, -3)
e. (4, -1)
3. Convert the polar equation to rectangular form.
a. r = -2
b. r = −2 sin 𝜃
c. 𝑟 = −3 cos 𝜃 − 2 sin 𝜃
4. Convert the rectangular equation to polar form.
a. 𝑦 = −4
b. 𝑥 = 5
c. 2𝑥 − 3𝑦 = 4
d. 𝑟 = sec 𝜃
d. (𝑥 − 3)2 + (𝑦 + 1)2 = 10
5. Write the equation of each polar graph. Window is [-5, 5] by [-5, 5]
a.
b.
c.
d.
e.
Topic #5:
Complex Numbers
1. Write the complex number in standard form
a. 6(cos 30° + 𝑖 sin 30°)
c.
4𝜋
2.5 (cos 3
+
4𝜋
𝑖 sin 3 )
f.
b. 3(cos 150° + 𝑖 sin 150°)
d. 4(cos 2.5 + 𝑖 sin 2.5)
Topic #6:
Applications
1. An airplane is flying on a bearing of 80° at 540 mph. A wind is blowing with a bearing of 100° at
55 mph.
a. Find the component form of the velocity of the airplane.
b. Find the actual speed and direction of the airplane.
2. An airplane is flying on a bearing of 285° at 480 mph. A wind is blowing with a bearing of 265°
at 30 mph.
a. Find the component form of the velocity of the airplane.
b. Find the actual speed and direction of the airplane.
3. A force of 120 lb acts on an object at an angle of 20°. A second force of 300 lb acts on the
object at an angle of −5°. Find the direction and magnitude of the resultant force.
4. Stewart shoots an arrow straight up from the top of a building with an initial velocity of 245
ft/sec. The arrow leaves from a point 200 ft above level ground.
a. Write an equation that models the height of the arrow as a function of time t.
b. How high is the arrow after 4 sec?
c. What is the maximum height of the arrow? When does it reach its maximum height.
d. How long will it be before the arrow hits the ground?
5. Lucinda is on a Ferris wheel of radius 35 ft that turns at the rate of one revolution every 20 sec.
The lowest point of the Ferris wheel (6 o’clock) is 15 ft above ground level at the point (0, 15) of
a rectangular coordinate system. Find parametric equations for the position of Lucinda as a
function of time t in seconds if Lucinda starts (t = 0) at the point (35, 50)
6. The lowest point of a Ferris wheel (6 o’clock) of radius 40 ft is 10 ft above the ground and the
center is on the y – axis. Find parametric equations for Henry’s position as a function of time t in
seconds if his starting position (t = 0) is the point (0, 10) and the wheel turns at the rate of one
revolution every 15 sec.
7. Sharon releases a baseball 4 ft above the ground with an initial velocity of 66 ft/sec at an angle
of 5° with the horizontal. How many seconds after the ball is thrown will it hit the ground?
How far from Sharon will the ball be when it hits the ground?
8. Brian hits a baseball straight toward a 15 ft-high fence that is 400 ft from home plate. The ball is
hit when it is 2.5 ft above the ground and leaves the bat at an angle of 30° with the horizontal.
Find the initial velocity needed for the ball to clear the fence.
9. Spencer practices kicking field goals 40 yd from a goal post with a crossbar 10 ft high. If he kicks
the ball with an initial velocity of 70 ft/sec at a 45° angle with the horizontal, will Spencer make
the field goal if the kick sails “true”?
10. An NFL place-kicker kicks a football downfield with an initial velocity of 85 ft/sec. The ball leaves
his foot at the 15 yd line at an angle of 56° with the horizontal. What is the maximum height
the football gets above the field? What is the total time the ball is in the air?