Review for Fall Final Exam
1.
Find the degree measure of the angle with the given radian measure.
a.
b.
2.
11
3
2

15
660 degrees
-24 degrees
A woman standing on a hill sees a flagpole that she knows is 60 ft tall. The angle of depression to the
o
o
bottom of the pole is 14 , and the angle of elevation to the top of the pole is 18 . Find her distance from
the pole.
104.5 feet
3.
Find the reference angle of the given angles.
330o
o
b. 199
11
c.
3
5
d.
7
30 degrees
19 degrees
a.
4.
π/3
2π/7
e.
Factor the Trinomial.
a. 6 y 2  11y  21
(y+3)(6y-7)
b.  3x  2  8  3x  2   12
2
5.
6.
Factor out the common factor.
a. y  y  6  9  y  6 
(y-6)(y+9)
b. 2 x2 y  6 xy 2  3xy
xy(2x-6y+3)
Factor expression by grouping terms.
a. x3  4 x2  x  4
b. 2 x  x  6 x  3
Perform the operation and simplify.
3
7.
2
2 x 2  3x  2
x2  1
a.
2 x2  5x  2
x2  x  2
x
2

b.
2
 x  1 x  1
8.
(3x+8)(3x+4)
(x2 +1)(x+4)
(x2 -3)(2x+1)
(x-2)/(x+1)
(3x+2)/(x+1)2
A cork floating in a lake is bobbing in simple harmonic motion. Its displacement above the bottom of the
lake is modeled by y  0.2cos 20 t  8 where y is measured in meters and t is measured in minutes.
a.
Find the frequency of the motion of the cork.
10 seconds
b.
Sketch the graph of y.
y








x

c.
9.




Find the maximum displacement of the cork above the lake bottom.
8.2 m
A mass is suspended on a spring. The spring is compressed so that the mass is located 7 cm above its rest
position. The mass is released at time t=0 and allowed to oscillate. It is observed that the mass reaches its
lowest point ½ seconds after it is released. Find an equation that describes the motion of the mass.
y = 7cos2πt
10. Sketch each triangle and then solve the triangle using the Law of Sines.
a.
A  50o , B  68o , c  230
Angle C = 62 degrees
a = 200
b = 242
b.
A  22o , B  95o , a  420
Angle C = 63 degrees
b = 1116.9
c = 999.0
c.
a  50, b  100, A  50o
No such triangle (no solution)
d.
a  26, c  15, C  29o
Angle A1 = 57.2 degrees
Angle A2 = 122.8 degrees
Angle B1 = 98.8 degrees
Angle B2 = 28.2 degrees
b1=30.9
b2=14.6
11. Solve triangle ABC.
a. a  20, b  25, c  22
Angle A = 49.9 degrees
Angle B = 72.9 degrees Angle C = 57.2 degrees
b.
a  3.0, b  4.0, C  53o
c = 3.2
Angle B = 79 degrees
12. Graph the functions.
a. f ( x)  1  cos x
y




x












b.
f ( x)  3sin x
y




x












c.
1
f ( x)  10sin x
2

y












x
            








      












d.
f ( x)  4 tan x
y




x













e.
f ( x)  2sec x
y




x













Angle A = 48 Degrees
13. Simplify completely the trigonometric expression.
sec x  cos x
=
tan x
b. tan x cos x csc x =
1  csc x
c.
=
cos x  cot x
a.
sin x
1
sec x
14. Prove the following identities.
a.
cot x sec x
1
csc x
(Cos x /sin x )(1/cosx)(sin x) = 1
1 =1
b.
sin B  cos B cot B  csc B
sinB + cosB(CosB/sinB) = cscB
(sin2B + cos2B)/sinB
= cscB
1/sinB
= cscB
cscB = cscB
c.
cos x sin x

1
sec x csc x
Cos2x + sin2x =1
1=1
d.
tan 2 x  sin 2 x  tan 2 x sin 2 x
= tan2x (1-cos2x)
= tan2x – ( sin2x/cos2x)cos2x
= tan2x – sin2x
15. Use addition and subtraction formulas to prove the expression.
a. sin( x   )   sin x
sinxcosπ – cosxsinπ = sinx(-1) – cosx(0) = -sinx
b.




cos  x    sin  x    0
6
3


cosx (cosπ/6) - sinx (sinπ/6) + sinx(cosπ/3) - cosx(sinπ/3) = 0
√3/2 cosx - 1/2 sinx
+ 1/2 sin x
- √3/2 cosx = 0
0=0
c.
sin( x  y)  sin( x  y)  2cos x sin y
Sinx cosy + coxsiny – (sinxcoxy – cosxsiny) = 2cosxsiny
16. Find the exact value of the expression, if it is defined.
a.
cos  cos1 5
not defined
1
b. tan  sin 1 
2


3
c. cos  sin 1 
2 

4
d. tan  sin 1 
5

7
e. csc  cos 1 
25 

√3/3
1/2
4/3
25/24
17. Find all the solutions to the equation.
a. 2sin x  1  0
π/6 + 2πk ; 5π/6 + 2πk
3csc2 x  4  0
b.
π/3 + k π ; 2π/3 + k π
c.
 tan x  3   cos x  2  0
d.
cos x sin x  2cos x  0
e.
2sin x  sin x  1  0
-π/3 + πk
π/2 + πk
2
7π/6 + 2πk ; 11π/6 + 2πk ; π/2 + 2πk
18. Sketch the graph of the polar equation.
a. r  1  2cos 
y




x













r 3
b.
y




x












r  2cos3
c.
y




x













19. Find the modulus of the following complex numbers.
a. 4i
b.
1 
=4
3
3
= (2√3)/3
20. Write z1 and z 2 in polar form and then find the product z1 z2 and the quotients
a.
z1  3  i; z2  1  3i
z1 = 2(cos30 + isin30)
z2 = 2(cos60 + isin60)
z1z2 = 4(cos90 + isin90)
z1/z2 = cos30 - isin30
1/z1 omit
21. Write the complex number in polar form with argument

√2(cos(π/4) + isin (π/4)
= 4(cos0 +isin0)
= 3(cos(3π/2) + isin(3π/2))
a.
1 i =
b.
4
between 0 and 2 .
c. -3i
22. Express the vector with initial point P and terminal point Q in component form.
a. P(3, 2), Q(8,9) <5,7>
b. P(8, 6), Q(1, 1) <7,5>
23. Find 2u, -3v, u+v, and 3u-4v for the given vectors u and v.
a.
u  2,7 , v  3,1
2u = <4,14>
-3v = <-9,-3>
u+v=<5,8>
3u-4v = <-6, 17>
24. Find the indicated quantity, assuming u  i  3 j and v  3i  4 j
a. (u  v)  (u  v) = -15
25. Find u  v and the angle between them.
a. u  2,7 , v  3,1 13; 55.6 degrees
b. u  3, 2 , v  1, 2 -1, 97.1 degrees
26. Determine whether the given vectors are orthogonal.
a.
b.
u  6, 4 , v  2,3
Yes
u  2,6 , v  4, 2 No
z1
z2
and 1
z1
.