Review for Quiz 18
Math 143
page 1
Review Problems
1. Compute the exact value of cos 15 .
2. Simplify each of the following.
16
a) 3log9 (a )
1
log3 10
b) log10 50 + log10 60
c) 5log25 (x+y)
log5 (x+y)
3. Solve the system of equation
4log2 (x+y)
log4 (x+y)
x
2
= 3
= 16
y
4
4. Solve each of the following.
a) 4x
2x+1 = 8
b) log2 x + logx 2 =
29
10
c)
3x + 1
2x 7
5
3
5. Prove each of the following identities.
a) sin 3x =
4 sin3 x + 3 sin x
b) cos 3x = 4 cos3 x
3 cos x
c) tan2 x + 1 = sec2 x
6. Derive the sum-formula for cotangent.
2
7. Let (an ) be a geometric sequence with a1 = 50 and r = .
3
a) Find an approximate value (up to 4 decimal places) of a7 .
b) Find an approximate value (up to 4 decimal places) of s10 .
8. Find the exact value of the in…nite sum of the geometric sequence 180; 60; 20; :::
9. Find the exact value of sin x if cos 2x =
10. Find the exact value of cos x if cos 2x =
119
:
169
7
9
11. Let (an ) be an arithmetic sequence with …rst element a1 = 4: Find the common di¤erence in the
sequence if a3 ; a5 ; and a13 ; in this order, form consecutive elements of a non-constant geometric
sequence.
5
12. Let l be the line y = x: Let k be the line that bisects the angle formed by l and the positive
12
part of the x axis. Find an equation of k.
Review for Quiz 18
Math 143
page 2
13. Consider a square with sides 1 unit long. To the inside of each side, we draw an isosceles triangle
with its greatest angle, opposite the unit long base, measures 150 . Cosider all vertices of these
triangles that are not on the square. If we connect these vertices, we obtain a square. Compute
the exact value of the area of this square.
14. Graph each of the following functions.
p
a) f (x) = x + 3 2 b) f (x) = log2 (x + 2) + 1 c) f (x) =
1
x+1
3 d) f (x) = jx
2j
3
2j
3
Answers
p
1.)
2+
4
p
6
4.) a) 2
b)
8.) 270
p
5
c)
p
4; 4 2
12
13
p
-2
-1
y
0
1
2
3
4
1
3
3.) x = 2; y = 1
[ [38; 1)
2
3
2
3
5.) see solutions
= 147: 398 770 5
1
12.) y = x
5
11.) 3
y
5
x
5
-1
6.) see solutions
10
2 b) f (x) = log2 (x + 2) + 1 c) f (x) =
3
0
7
2
b) 50
10.)
x+3
1
-3
1
x+y
1
2
-4
1;
= 4: 389 574 76
14.) a) f (x) =
-5
c) p
1
9.)
y
b) 3
6
2
3
7.) a) 50
2.) a) a8
p
13.) 2
1
x+1
3
3 d) f (x) = jx
y
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
0
0
-2
-2
-1
0
1
2
3
4
5
6
7
8
x
-5
-4
-3
-2
-1
0
-1
1
2
3
4
5
x
-4
-3
-2
-1
0
-3
-1
-1
-4
-2
-2
-2
-5
-3
-3
-3
-6
-4
-4
-4
-7
-5
-5
-5
1
2
3
4
5
6
x
Review for Quiz 18
Math 143
page 3
Solutions:
1. c)
log (x+y)
log25 (x+y) log5 (x+y)
5
251=2 25
5log25 (x+y)
=
=
5log5 (x+y)
5log5 (x+y)
p
x+y
1
=p
=
x+y
x+y
25log25 (x+y)
=
5log5 (x+y)
1=2
(x + y)1=2
=
x+y
4 sin3 x + 3 sin x
5.) a) sin 3x =
sin 3x = sin (x + 2x) = sin x cos 2x + cos x sin 2x = sin x cos2 x sin2 x + cos x (2 sin x cos x)
= sin x cos2 x sin2 x + 2 sin x cos2 x = sin x 1 2 sin2 x + 2 sin x 1 sin2 x
= sin x 2 sin3 x + 2 sin x 2 sin3 x = 4 sin3 x + 3 sin x
b) cos 3x = 4 cos3 x
cos 3x =
=
=
=
3 cos x
cos (x + 2x) = cos x cos 2x sin x sin 2x = cos x cos 2x sin x sin 2x
cos x cos2 x sin2 x
sin x (2 sin x cos x) = cos x 2 cos2 x 1
2 sin2 x cos x
cos x 2 cos2 x 1
2 1 cos2 x cos x =
2 cos3 x cos x 2 cos x + 2 cos3 x = 4 cos3 x 3 cos x
c) tan2 x + 1 = sec2 x
sin2 x
sin2 x cos2 x
sin2 x + cos2 x
1
tan x + 1 =
+
1
=
+
=
=
= sec2 x
2
2
2
2
2
cos x
cos x cos x
cos x
cos x
2
6.) Derive the sum-formula for cotangent.
cot (x + y) =
cos (x + y)
cos x cos y sin x sin y
=
sin (x + y)
sin x cos y + cos x sin y
divide both numerator and denominator by sin x sin y
cos x cos y sin x sin y
cos x cos y sin x sin y
sin x sin y
cot (x + y) =
=
sin x cos y + cos x sin y
sin x cos y + cos x sin y
sin x sin y
cos x cos y sin x sin y
cot x cot y 1
sin x sin y
sin x sin y
=
=
sin x cos y cos x sin y
cot x + cot y
+
sin x sin y
sin x sin y
Math 143
Review for Quiz 18
page 4
13.)
]AF B = 150 =) ]F AB = ]DAE = 15 =) ]EAF = 60
]EAF = 60 =) triangle AEF is regular =) EF = AF
We compute AF in the right triangle AF M where M is the midpoint of side AB.
1
2
cos 15 =
=)
AF
1
1
2
AF =
=
=
cos 15
2 cos 15
1
2
p
p
p
=p
p !=
2+ 6
6+ 2
2+ 6
2
2
4
p
p
p
p
p
p
p
p
2 6
2
2 6
2
6
2
6
2
2
p p
p =
=
=
= p
4
2
6 2
6+ 2
6
2
p
p !2
p
6
2
2
Then the area of the square is EF =
=2
3
2
p
1