MAT 170
Hao Liu
Precalculus
Review 3
1
Exponential and Logarithmic Functions (Continued)
Objectives [§3.3, §3.4] The base e; logarithmic functions; natural logarithm and
common logarithm; properties of logarithmic functions: exponential and logarithmic identities; product rule; quotient rule; power rule; change-of-base formula; exponential and logarithmic equations.
Skills know the natural base e; definition of logarithmic functions; definition of natural logarithm and common logarithm; use logarithmic and exponential identities
to evaluate a logarithmic or exponential expression; use product rule and quotient rule to rewrite a logarithmic expression as a sum or a difference; use power
rule to rewrite a logarithmic expression as a product; use change-of-base formula; Solve exponential equations by ea = eb ⇐⇒ a = b; solve logarithmic
equations by loga M = loga N ⇐⇒ M = N ; solve logarithmic equations by the
definition of logarithmic equations; solve exponential equations by logarithms.
Review Problems
√
1. Write loga (x2 x2 + 1) as a sum.
2. Write ln
ex
as a difference.
x2
3. Write each of the following expression as a single logarithm and simplify your
answer:
(a) log2 1.6 + log2 10
(c)
2
log 8 + log 25
3
(b) loga 7 − loga x, provided a > 0, a 6= 1.
3
(d) − ln x2 + 3 ln x − 2 ln 5 + ln x
2
4. Use the change-of-base formula to calculate:
(a) log 1 3
9
(b) log√e e2.5
5. Solve each of the following equation by ea = eb ⇐⇒ a = b:
School of Mathematical & Statistical Sciences - Arizona State University
July 28, 2010
MAT 170
Hao Liu
Precalculus
2 +1
(a) ex = e2
(b) 4x
(c) e2x + 6ex − 7 = 0
(d) 9x − 8 · 3x − 9 = 0
= 64
6. Solve each of the following equation by the definition of logarithm (rewriting the
equation in the exponential form):
(a) log2 x = 5
(b) loge x2 = 10
(c) log(x + 1) = 2
(d) log 1 x = −2
3
7. Solve each of the following equation by loga M = loga N ⇐⇒ M = N .
(a) log2 2x = log2 (x2 + x)
(b) log4 x = log2 x + 1
(c) ln x2 = ln x
(d) log(x + 1) = ln e2
Hint for (b): use the change-of-base formula to rewrite log4 x as a logarithm with
base 2.
8. Solve each of the following exponential equation by logarithms:
(a) 4x = 12
2
(b) ex
2 +1
=5
Trigonometry
Objectives [§4.1, §4.3, §4.4, §6.1, §6.2] Use degree measure; use radian measure;
convert between degrees and radians; find coterminal angles; find the length of
a circular arc; right triangle trigonometric ratios; trigonometric ratio for special
angles; cofunction of complements; trigonometric ratios of any angle; signs of
trigonometric ratios; reference angles; the law of sines; the law of cosines.
Skills Use degree measure; use radian measure; convert between degrees and radians;
find coterminal angles; find the length of a circular arc; find right triangle
trigonometric ratios; find trigonometric ratios for special angles; find cofunction
of complements; find signs of trigonometric ratios; find reference angles; use law
of sines to solve oblique triangles; use law of cosines to solve oblique triangles.
Review Problems
1. Convert each angle in radians to degrees.
School of Mathematical & Statistical Sciences - Arizona State University
July 28, 2010
MAT 170
Hao Liu
Precalculus
(1)
π
2
(2)
π
8
(4)
9π
5
(5) −2π
(3)
5π
6
2. Convert each angle in degrees to radians.
(1) 15◦
(2) 75◦
(3) 60◦
(4) −90◦
(5) −120◦
3. Find a positive angle less than 360◦ or 2π that is coterminal with the given angle.
(1) −135◦
(2) −700◦
(3)
5π
3
(4)
16π
5
4. Find the length of the arc on a circle of radius 8 inches intercepted by a central
angle 60◦ .
Use the accompanying figure for Exercise 5 through 8 and give each trigonometric
ratio.
B
13
A
5. tan A
5
12
6. sec B
C
7. csc A
8. cot B
9. Evaluate the following expressions.
(1) cos 30◦
(2) cot
π
3
(3) tan 45◦
(4) csc
π
6
π
1
10. If θ is an acute angle and sin θ = , find cos( − θ).
6
2
11. A tower is 120 feet tall and it casts a shadow 150 feet long. Find the angle of
elevation of the sun. Round off your answer to the first decimal place.
12. Let θ be an angle in standard position. Name the quadrant in which θ lies.
School of Mathematical & Statistical Sciences - Arizona State University
July 28, 2010
MAT 170
Hao Liu
Precalculus
(1) sin θ > 0, cos θ < 0
(2) sin θ < 0, sec θ < 0
(3) sin θ < 0, tan θ < 0
(4) cot θ > 0, cos θ > 0
13. Find the reference angle for each angle.
(1) 130◦
(2) 260◦
(3)
5π
4
(4) −
π
3
14. Use reference angles to find the exact value of each expression. Do not use a
calculator.
(1) cos 300◦
(4) sin
2π
3
(2) tan 240◦
(5) csc
3π
4
(3) sin(−225◦ )
(6) cot
9π
2
15. Solve the triangle, given A = 45◦ , B = 26◦ , a = 8. Round lengths to the nearest
tenth and angle measures to the nearest degree.
16. Two sides and an angle of a triangle are given, that is, a = 10, b = 20, A = 25◦ .
Determine whether the given measurements produce one triangle, two triangles
or no triangle at all.
17. Find the area of the triangle having the given measurements. A = 45◦ , b =
20, c = 40.
18. Solve the triangle. a = 5, b = 6, C = 40◦ . Round lengths to the nearest tenth
and angle measures to the nearest degree.
School of Mathematical & Statistical Sciences - Arizona State University
July 28, 2010