Download Test 3 Review 2412 Sp15.tst

Test Three Review
Math 2412
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Find the exact value of the expression.
2π π
- 1) cos 18
9
1)
Identify α and β in the following expression which is the right side of the formula for cos ( α - β).
5π
π
5π
π
2) cos cos + sin sin 18
9
18
9
Complete the identity.
3π
3) cos ( - x) = ?
2
2)
3)
Use the given information to find the exact value of the expression.
4
2
4) sin α = , α lies in quadrant II, and cos β = , β lies in quadrant I
5
5
Find cos (α - β).
Find the exact value by using a sum or difference identity.
5) sin 165°
4)
5)
Complete the identity.
cos (α - β)
6)
= ?
cos (α + β)
6)
Verify the identity.
7) sin(α - β) cos(α + β) = sin α cos α - sin β cos β
7)
Use the given information to find the exact value of the expression.
5
3
8) tan α = , α lies in quadrant III, and cos β = - , β lies in quadrant II
12
5
Find the exact value by using a difference identity.
π
9) tan 12
Find sin (α + β).
8)
9)
Use trigonometric identities to find the exact value.
tan 100° - tan (-20°)
10)
1 + tan 100° tan (-20°)
10)
Find the exact value under the given conditions.
7
3π
8 π
11) tan α = , π < α < ; cos β = - , < β < π
24
2
17 2
Find tan (α + β).
1
11)
Use the graph to complete the identity.
3
3
12) cos 3x cos x + sin 3x sin x = ? 2
2
12)
y
3
-2

-
2
x
-3
Use the figure to find the exact value of the trigonometric function.
13) Find sin 2θ.
13)
5
4 3
Use the given information to find the exact value of the expression.
5
14) cos θ = , θ lies in quadrant IV
Find sin 2θ.
13
Complete the identity.
sin 2x
15)
= ?
1 - cos 2x
14)
15)
Verify the identity.
16) tan 2 x (1 + cos 2x) = 1 - cos 2x
16)
17) cos 4θ = 2 cos2 (2θ) - 1
17)
Rewrite the expression as an equivalent expression that does not contain powers of trigonometric functions greater than
1.
18) sin3 x
18)
Use a half-angle formula to find the exact value of the expression.
3π
19) cos 8
2
19)
Use the given information to find the exact value of the trigonometric function.
θ
20) sec θ = 4, θ lies in quadrant I
Find cos .
2
6
21) csc θ = - , tan θ > 0
5
Use a graph in a [-2π, 2π, 22)
θ
Find cos .
2
20)
21)
π
] by [-3, 3, 1] viewing rectangle to complete the identity.
2
1 - 2 cos 2x
= ?
2 sin x - 1
22)
Express the product as a sum or difference.
23) sin 4x sin 7x
23)
Complete the identity.
24) sin x (sin 2x + sin 4x) = ?
24)
Express the sum or difference as a product.
25) sin 6x - sin 2x
25)
Complete the identity.
sin x + sin y
26)
= ?
cos x + cos y
26)
Verify the identity.
sin α - sin β
α - β
α + β
27)
= tan cot sin α + sin β
2
2
27)
Use substitution to determine whether the given x -value is a solution of the equation.
5π
28) cos x + 1 = sin x, x = 4
Find all solutions of the equation.
29) 9 cos x + 6 2 = 7 cos x+ 5 2
28)
29)
Solve the equation on the interval [0, 2 π).
3
30) cos 2x = 2
30)
31) cos x = sin x
31)
32) csc5 x - 4 csc x = 0
32)
3
Solve the equation on the interval [0, 2 π).
33) -tan2 x sin x = -tan2 x
33)
Solve the equation on the interval [0, 2 π).
34) sin 2x + sin x = 0
34)
Solve the triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.
35) A = 41°
B = 44°
a = 19.8
35)
Two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle,
two triangles, or no triangle at all. Solve each triangle that results. Round lengths to the nearest tenth and angle
measures to the nearest degree.
36) A = 65°, a = 6, b = 10
36)
Find the area of the triangle having the given measurements. Round to the nearest square unit.
37) A = 38°, b = 14 meters, c = 12 meters
37)
Solve the triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.
38) a = 3, c = 1, B = 90°
38)
Use Heronʹs formula to find the area of the triangle. Round to the nearest square unit.
39) a = 8 inches, b = 14 inches, c = 8 inches
39)
Use a polar coordinate system to plot the point with the given polar coordinates.
9π
40) 2, 4
40)
5
-5
5
-5
Find another representation, (r, θ), for the point under the given conditions.
π
41) 8, , r > 0 and -2π < θ < 0
6
4
41)
Polar coordinates of a point are given. Find the rectangular coordinates of the point.
2π
42) 7, 3
42)
The rectangular coordinates of a point are given. Find polar coordinates of the point. Express θ in radians.
43)
43) (0, - 3)
Convert the rectangular equation to a polar equation that expresses r in terms of θ.
44) (x - 13)2 + y2 = 169
44)
Convert the polar equation to a rectangular equation.
45) r = -5 cos θ
45)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
The graph of a polar equation is given. Select the polar equation for the graph.
46)
46)
5
4
3
2
1
-5 -4 -3 -2 -1
-1
1
2
3
4
5
r
-2
-3
-4
-5
A) r = 4 + cos θ
C) r = 4 + sin θ
B) r = 8 cos θ
D) r = 8 sin θ
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Test the equation for symmetry with respect to the given axis, line, or pole.
π
47) r = 4 + 4 cos θ; the line θ = 2
48) r cos θ = 2; the polar axis
47)
48)
5
Graph the polar equation.
49) r = 3 - cos θ
49)
10
5
-10
-5
10 r
5
-5
-10
Use a graphing utility to graph the polar equation.
50) r = cos 3θ
50)
1
1
2
-1
-1
2
1
2
1
r
-1
2
-1
Plot the complex number.
51) -4 + 3i
10
51)
i
5
-10
-5
5
R
-5
-10
Find the absolute value of the complex number.
52) z = -15 + 4i
52)
Write the complex number in polar form. Express the argument in degrees.
53) 3 - 4i
53)
6
Write the complex number in polar form. Express the argument in radians.
54) - 5 + 5 3i
Write the complex number in rectangular form.
2π
2π
55) -5(cos + i sin )
3
3
55)
Find the product of the complex numbers. Leave answer in polar form.
π
π
56) z 1 = 8 cos + i sin 6
6
z 2 = 3 cos Find the quotient z1
z2
54)
56)
π
π
+ i sin 2
2
of the complex numbers. Leave answer in polar form.
57) z 1 = 5(cos 200° + i sin 200°)
57)
z 2 = 4(cos 50° + i sin 50°)
Use DeMoivreʹs Theorem to find the indicated power of the complex number. Write the answer in rectangular form.
3π
3π 3
58) 10 (cos + i sin )
58)
4
4
Solve the problem.
59) Let vector u have initial point P1 = (0, 2) and terminal point P2 = (3, 0). Let vector v have
initial point Q1 = (3, 0) and terminal point Q2 = (6, -2). u and v have the same direction.
Find u and v . Is u = v?
7
59)
Use the vectors v, u, w, and z to draw the indicated vector.
60) z - v
60)
Sketch the vector as a position vector and find its magnitude.
61) v = -2i + 5j
61)
y
6
4
2
-6
-4
-2
2
4
6
x
-2
-4
-6
Let v be the vector from initial point P1 to terminal point P2 . Write v in terms of i and j.
62) P1 = (0, 0); P2 = (3, -4)
62)
Find the specified vector or scalar.
63) u = -9i - 2j, v = 5i + 7j; Find u - v.
63)
Find the unit vector that has the same direction as the vector v.
64) v = 5i + 12j
64)
8
Write the vector v in terms of i and j whose magnitude v
65) v = 8, θ = 30°
and direction angle θ are given.
65)
Use the given vectors to find the specified scalar.
66) v = 6i + 2j; Find v · v.
66)
Find the angle between the given vectors. Round to the nearest tenth of a degree.
67) u = i - j, v = 3i + 4j
67)
Use the dot product to determine whether the vectors are parallel, orthogonal, or neither.
68) v = 2i + 4j, w = 4i + 8j
68)
Find proj wv.
69) v = 2i + 3j; w = 8i - 6j
69)
9
Answer Key
Testname: TEST 3 REVIEW 2412 SP15
1)
3
2
2) α = 5π
π
, β = 18
9
3) -sin x
-6 + 4 21
4)
25
5)
2( 3 - 1)
4
6)
1 + tan α tan β
1 - tan α tan β
7) sin(α - β) cos (α + β) = (sin α cos β - cos α sin β)(cos α cos β - sin α sin β)
= sin α cos α cos2 β - sin2 α cos β sin β - cos2 α sin β cos β + cos α sin α sin2 β
= sin α cos α(cos2 β + sin2 β) - sin β cos β(sin2 α + cos2 α ) = sin α cos α - sin β cos β
33
8) - 65
9) 2 - 3
10) - 3
304
11) - 297
3
12) y = cos x
2
13)
24
25
14) - 120
169
15) cot x
16) tan 2 x (1 + cos 2x) = 1 - cos 2x
(1 + cos 2x) = 1 - cos 2x
1 + cos 2x
17) cos 4θ = cos[2(2θ)] = 2 cos2 (2θ) - 1
3
1
18) sin x - sin 3x
4
4
19)
20)
1
2
2 - 2
10
4
21) - 18 - 3 11
6
22) 2 sin x + 1
1
23) (cos 3x - cos 11x)
2
24) cos x (cos 2x - cos 4x)
25) 2 sin 2x cos 4x
10
Answer Key
Testname: TEST 3 REVIEW 2412 SP15
26) tan 27)
x + y
2
sin α - sin β
= sin α + sin β
2 sin α + β
α - β
cos 2
2
α + β
α - β
2 sin cos 2
2
sin = α - β
2
α - β
cos 2
cos · α + β
2
α + β
sin 2
28) No
29) x = 3π
5π
+ 2nπ or x = + 2nπ
4
4
30)
π 11π 13π 23π
, , , 12
12
12 12
31)
π 5π
, 4 4
32)
π 3π 5π 7π
, , , 4 4
4
4
33) 0, π
4π
2π
, π, 34) 0, 3
3
35) C = 95°, b = 21, c = 30.1
36) no triangle
37) 52 square meters
38) b = 3.2, A = 70°, C = 20°
39) 30 square inches
40)
5
-5
5
-5
41) 8, - 11
π
6
7 7 3
42) - , 2
2
43) (- 3, 90°)
44) r = 26 cos θ
25
5 2
+ y2 = 45) x + 4
2
46) A
11
= tan α - β
α + β
cot 2
2
Answer Key
Testname: TEST 3 REVIEW 2412 SP15
47) may or may not have symmetry with respect to the line θ = 48) has symmetry with respect to polar axis
49)
10
5
-10
-5
10 r
5
-5
-10
50)
1
1
2
-1
-1
2
1
2
1
r
-1
2
-1
51)
10
i
5
-10
-5
5
R
-5
-10
52) 241
53) 5(cos 306.9° + i sin 306.9°)
2π
2π
+ i sin 54) 10 cos 3
3
55)
5 -5 3
+ i
2
2
12
π
2
Answer Key
Testname: TEST 3 REVIEW 2412 SP15
56) 24 cos 57)
2π
2π
+ i sin 3
3
5
(cos 150° + i sin 150°)
4
58) -500 2 + 500 2i
59) u = 13, v = 13; yes
60)
61) v = 29
y
6
4
2
-6
-4
-2
2
4
6
x
-2
-4
-6
62) v = 3i - 4j
63) -14i - 9j
12
5
64) u = i + j
13
13
65) v = 4 3i + 4j
66) 40
67) 98.1°
68) parallel
1
69) - (4i - 3j)
25
13