Download Honors Math 3 Name: Date: Final Exam Review Problems Part 2

Honors Math 3
Name:
Date:
Final Exam Review Problems Part 2
Chapter 4 and 5: Trigonometry
22. Without a calculator, evaluate sin(cos–1(3/5)).
23. For any angle , how are cos( p2 – ) and sin( – ) related? Justify your answer using
a circle diagram.
cos3x = 4cos3 x - 3cos x
24. Prove the identity:
25. Suppose that sin  =
3
5
and sin  =
24
25
, where 0 <  <
p
2
<  < . Find cos( + ).
26. Find the general solution to each equation (in radians):
a.
27.
cos x = -0.82
b. sin x = 0.812
Find the general solution to each equation (in degrees):
a. cos x = 0.756
b. sin x = -0.155
28. Using non-graphical methods, find all solutions to the equations in the interval 0 ≤ x
< 2. Check your answer graphically.
a. sin(3x) = –0.5
b. sin2 x – sin x = cos2 x
c. 3sin ( x + 2 ) -10 = -11
d.
1 æp ö
cosç x ÷ - 3 = 4
2 è4 ø
29. Sketch two full periods of the graphs of the following functions on graph paper:
a. f (x) = -2cos( p2 x ) -1
6
( ( x - ))
b. g(x) = 3sin
p
1
2
2
30. Find 2 equation of the sinusoid, one4 using the sine function and one using the cosine
function.
2
2π
π
π
2π
3π
2
4
6
8
10
31. a. Write a function formula for a sinusoidal function f(x) having the following
properties:
o Two adjacent maximum points of f(x) are located at (3, 5) and (7, 5).
o The graph of f(x) is tangent to the x-axis.
b. Suppose that the graph of g(x) is formed by shrinking f(x) horizontally by a factor
of 8. Write a function formula for g(x).
32. Assume that you are aboard a submarine, submerged in the Pacific Ocean. At time t
= 0 minutes you spot an enemy destroyer. Immediately, you start diving lower to
avoid detection. At t = 4 minutes, you are at your greatest depth of -1000 miles. At
time t = 9 minutes, you have ascended to your minimum depth of -200 miles.
Assume that the path of the submarine varies sinusoidally with respect to t for t ³ 0.
a. Sketch two of these dives for the submarine (2 cycles).
b. Find a possible formula for f(t) which represents the depth of the submarine.
c. Your submarine is safe when it is below a depth of -300 miles. At time t = 0
minutes, was your submarine safe? Explain your answer.
d. Between what two (non-negative) times is your submarine first safe?
33.
Solve the following triangles below (find all possible missing angles and sides).
Not draw to scale.
a.
b.
B
E
10
7
60°
40°
A
12
c.
C
D
12
Y
6
32°
X
34.
10
Z
Find the area of quadrilateral ABCD below. (Hint: Start by connecting points A
and C)
D
55º
A
8
10.8
B
120º C
10
F
Chapter 6: Complex Numbers and Polynomials
35. Find a cubic polynomial, C(x), with real coefficients has the following properties:
 1- 2i is a complex root.
36.
 the graph of C(x) has an x-intercept at
x = 2.

C(1) = -12
Prove that the quotient of two complex numbers is complex. In other words,
show that the quotient can be expressed in the form a + bi. State any restrictions that
are necessary.
37. Solve each equation. Find all solutions.
a. x 3 = x 2 + 23x + 42
b. x 3 = -8
38. You are given this information about F(x):


F(x) is a polynomial with real coefficients.
The graph of F(x) for real numbers x is given on the
grid. All zeros and extrema can be seen from this
graph. The intercepts are at (2, 0), (4, 0), and (0, 5).
 F(x) has the smallest degree that it could possibly have
based on the number of zeros and the number of
extrema shown on the graph.
 In the complex numbers, F(–3 + i) = 0.
Find the factorization of F(x) in the real number system.
39.
Given: z = 3+ 4i and w =1+ 2i. Find the magnitude and argument of zw.
Answers
22. A right triangle with angle cos–1( 35 ) may have sides of 3, 4, and 5. This gives
sin(cos–1( 35 )) = 45 .
23. cos( p2 – ) = sin  because the x-coordinate at angle p2 –  is the same as the ycoordinate at angle . sin( – ) = sin  because the y-coordinates at the same at
angle  –  and angle . Therefore, cos( p2 – ) = sin( – ).
24.
cos3x = cos(2x + x) = cos2x cos x - sin2x sin x = (cos2 x - sin2 x )× cos x - (2sin x cos x )× sin x =
cos3 x - sin2 x× cos x - 2sin2 x cos x = cos3 x - 3sin2 x cos x =
cos3 x - 3(1- cos2 x)× cos x = cos3 x - 3cos x + 3cos3 x = 4cos3 x - 3cos x
3 24
-100
-4
25. cos( + ) = cos  cos  – sin  sin  = 45 × -7
25 - 5 × 25 = 125 = 5 .
26. a. ±2.5322 + 2pk
b. 0.9476 + 2pk and 2.194 + 2pk
27.
a. x = ±40.89 + 360k
b. x =188.92 + 360k and x = 351+ 360k
7p
11p
28. a. 3x = 6 + 2pk or 3x = 6 + 2pk
7p
7p 11p 19p 23p 31p 35p
+ 23 pk or x = 1118p + 23 pk
, 18 , 18 , 18 , 18 , 18 }.
x = 18
In the specified interval: { 18
b. sin2 x – sin x = 1 – sin2 x
2 sin2 x – sin x – 1 = 0
(2 sin x + 1)(sin x – 1) = 0
sin x = - 12 or sin x = 1
c. 1.48, 3.9
d. no solution
In the specified interval: { p2 , 76p , 116p }.
29. check graphs on your calculator
2æ
pö
30. possible answers: f (x) = -4 + 5sin ç x + ÷
3è
4ø
31. a.
2æ
pö
f (x) = -4 + 5cos ç x - ÷
3è
2ø
f (x) = 2.5sin( 24p (x - 2)) + 2.5 or - 2.5cos( 24p (x -1)) + 2.5
b. g(x) = 2.5sin( 24p (8x - 2)) + 2.5
32. b. f (t) = -600 - 400cos
d.
Between
p
c. not safe since f (0) = -276.39 ft
( t - 4)
5
.18 minutes and 7.8 minutes
33. a. ASA – one triangle
Use triangle sum to find last angle ÐB = 80°
Use Law of Sines to find each of the other two sides
12
a
c
=
=
c =10.553
a = 7.832
sin80 sin 40 sin 60
b. SSS – one triangle
Use Law of Cosines to find two of the angles
ÐE = 87.95°
122 =102 + 72 - 2(10)(7)cosE
2
2
2
ÐD = 56.39°
10 =12 + 7 - 2(12)(7)cosD
Use triangle sum to find the third angle ÐF = 35.66°
c. AAS – could be 0, 1, or 2 triangles possible
Can use either LOS or LOC to start…
If use LOS to find another angle
6
10
=
ÐY = 62.03° OR ÐY =117.67°
sin32 sinY
By triangle sum…
ÐX = 85.97° OR ÐX = 30.33°
Using LOS or LOC x =11.294 OR x = 5.718
34.
Area = 108.32 sq units
35.
C must have 3 zeros (it is a cubic). Since 1- 2i is a zero, 1+ 2i must also be a
zero (real coefficients, complex conjugates theorem). C must have the form:
C(x) = a(x - 2)(x - (1- 2i))(x - (1+ 2i)), where a will be determined by the condition
f (1) = -12
C(x) = a(x 3 - 4x 2 + 9x -10)
36.
a +bi
c +di
×
c -di
c -di
=
(ac +bd )+(cb -ad )i
c 2 +d 2
=
(
C(1) = a(1- 4 + 9 -10) = -12 Þ a = 3
ac +db
c 2 +d 2
)+(
cb -ad
c 2 +d 2
)i
the last expression can be thought of as: (real number) + (real number)i .
37. a. x 3 - x 2 - 23x - 42 = 0Graph on calc and find that that x = 6 is a rational zero.
Use division to get: (x - 6)(x 2 + 5x + 7) = 0 use the quadratic formula to find the other
-5 ± 3i
two solutions. Solutions: x = 6,
.
2
b. x 3 = -8; same method can be used as in 36a. solutions are: x = -2, 1 ± 3i .
-1
2
38. f (x) = ( x - 2)( x - 4 ) (x 2 + 6x +10)
64
39. magnitude of zw = zw = z × w = 5 5
arg(zw) = arg(z)+arg(w) =