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Tech Math 2
FINAL EXAM Review
Angle Precision
1
0.1 or 10
0.01 or 1
Angles and Accuracy of Trig Functions
Trig Function Accuracy
Example
2 sig. figs.
sin17  0.29 , tan85  11
3 sig. figs.
sin1.1  0.0192 ,
cos520'  0.996
4 sig. figs.
cos 45.00  0.7071 ,
tan8013'  5.799
Calculating Other Trig Functions
Definition
Example
cos 
1
1
cot  

cot 82.3 
 0.135
sin  tan 
tan 82.3
1
1
sec  
sec 22 
 1.1
cos 
cos 22
1
1
csc  
csc 72.67 
 1.048
sin 
sin 72.67
Section 4-4: The Right Triangle
SOH CAH TOA:
opp
hyp
opp
tan  
adj
1
hyp
sec  

cos 
adj
sin  
Page 1 of 11
Pythagorean Theorem: a2 + b2 = c2
adj
cos 
hyp
1
adj
cot  

tan  opp
1
hyp
csc 

sin  opp
Tech Math 2
FINAL EXAM Review
Page 2 of 11
To solve a right triangle:
1. Make a sketch of the triangle, label sides and angles consistently (a, b, and c for the legs and
hypotenuse; A and B for the complementary angles), and label the given information.
2. Find a way to relate the unknown parts to the given information using a trig function (sine, cosine, or
tangent) or the Pythagorean Theorem (a2 + b2 = c2). Try to use original given information to minimize
rounding errors.
Section 4.5: Applications of Right Triangles
To solve applied right triangle problems:
1. Make a sketch of the situation.
2. Identify/draw right triangles on your sketch that connect given information to unknown information.
3. Solve the right triangle or triangles.
Section 8.4: Applications of Radian Measure
1. A radian is a unit of angle measurement. There are 2 radians in a full circle, as opposed to 360. This
gives us a conversion factor:
 2 radians 
a. To convert from degrees to radians: 57.0 
  0.995 radians
 360 
 360 
b. To convert from radians to degrees: 0.200 radians 
  11.5
 2 radians 
2. Arc Length Formula: s =  r. The angle  must be in radians for this formula to work.
1
3. Area of a Sector of a Circle: A   r 2 . The angle  must be in radians for this formula to work.
2

4. Angular Velocity:  
t
Section 9.2: Components of Vectors
1. Components of a vector are two vectors that, when added together, have a resultant equal to the original
vector. (Usually, the components are perpendicular to each other and along the x and y axes, and are
thus called the x- and y- components of a vector.)
2. Resolving a vector into components is calculating the components of a vector.
3. Steps for resolving a vector into x- and y- components:
a. Place the vector A such that its angle is in standard position (i.e., measured counterclockwise
from the x- axis).
b. Calculate the x- and y- components using right triangle trigonometry (i.e., Ax = Acos , and Ay =
Asin , note: these formulas only work if the angle is in standard position!).
c. Check the components for correct sign and magnitude.
Tech Math 2
FINAL EXAM Review
Page 3 of 11
Section 9.3: Vector Addition by Components
1. To add vectors by components:
a. Resolve all vectors into components.
b. Add all x-components to get the x-component of the resultant vector (Rx).
c. Add all y-components to get the y-component of the resultant vector (Ry).
d. Find the magnitude of the resultant vector R using the formula R  Rx2  Ry2 .
 Ry 
 , and then
e. Find the angle of the resultant vector R by first using the formula   tan 1 
 Rx 


using the signs of Rx and Ry to convert the angle into the correct quadrant.
Section 9.4: Applications of Vectors
1.
2.
3.
4.
In this section, translate the word problem into vectors, then do vector addition.
Physics: sum of forces = 0 for a body at rest or moving at constant velocity.
Physics: sum of forces = mass times acceleration for accelerating bodies.
To find the displacement between two vectors, subtract vector components instead of adding them!
Section 9.5: Oblique Triangles, the Law of Sines
The Law of Sines
a
b
c


sin A sin B sin C
OR
sin A sin B sin C


a
b
c
For the ambiguous case, the side opposite the given angle must be less that the side adjacent to the given angle:
B
sin B sin 30

3.875
3.125
Angle B for this triangle is 180 - B from previous
triangle.
Tech Math 2
FINAL EXAM Review
Page 4 of 11
Section 9.6: The Law of Cosines
The Law of Cosines
a 2  b 2  c 2  2bc cos A  cos A 
b2  c 2  a 2
2bc
OR
a 2  c2  b2
b  c  a  2ca cos B  cos B 
2ac
2
2
2
OR
c 2  a 2  b 2  2ab cos C  cos C 
a 2  b2  c2
2ab
Section 20.1: Fundamental Trigonometric Identities
Basic Identities:
sec 
1
cos 
cos  
1
sec
Reciprocal Identities
1
csc 
sin 
sin  
cos  sec  1
sin  csc   1
tan  
sin 
cos 
sin   cos   1
2
1
csc
2
sin 2   1  cos 2 
cos 2   1  sin 2 
cot  
1
tan 
tan  
1
cot 
tan  cot   1
Tangent/Cotangent Identities
cot  
Pythagorean Identities
1  tan 2   sec 2 
tan 2   sec 2   1
cos 
sin 
1  cot 2   csc 2 
cot 2   csc2   1
Tech Math 2
FINAL EXAM Review
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Tips for Proving Trigonometric Identities:
 Sometimes it helps to write everything as sines and cosines.
 Look for pieces that can be simplified using the Pythagorean Identities.
 Sometimes it helps to multiply things out.
 Sometimes it helps to factor things.
 Sometimes you’ll have to find common denominators to add fractions.
Section 20.6: The Inverse Trigonometric Functions
1. An inverse trigonometric function is the function that “undoes” a trigonometric function.
2. That means that inverse trig functions are “angle-finders”, because they return the angle for which a trig
function takes on a given value.
3. Example: the inverse sine function is written as sin-1, and is used to find an angle that gives a certain
number. For example, if  = sin-1( ½ ), then  = 30, because sin(30) = ½.
4. We also want the inverse trig function to return only one value, so by definition we restrict what angles
it can return. For example, the sin-1 function is restricted to returning angles between -90 and 90,
which is why we don’t also say that sin-1( ½ ) = 150, because even though sin(150) = ½, 150 is out of
the restricted range of values that the function is allowed to return.
5. The inverse trig functions on a calculator are useful for solving trigonometric equations that aren’t
“nice”, like sin x = 0.85  x = sin-1(0.85)  58.21.
Inverse trig function
 = sin-1( x )
 = cos-1( x )
 = tan-1( x )
 = cot-1( x )
 = sec-1( x )
 = csc-1( x )
Meaning of the function
Theta is the angle whose sine is x.
Theta is the angle whose cosine is x.
Theta is the angle whose tangent is x.
Theta is the angle whose cotangent is x.
Theta is the angle whose secant is x.
Theta is the angle whose cosecant is x.
Restricted range of return values
-90  x  90
0  x  180
-90  x  90
0  x  180
0  x  180
-90  x  90
Tech Math 2
a a  a
m
n
a 
m n
FINAL EXAM Review
Page 6 of 11
Exponent Formulas:
am
 a mn
an
a 0  1 for a  0
mn
 a mn
ab n
 a nb n
an
a
   n
b
b
a n 
1
for a  0
an
a
 
b
a
n
1
n
n a
an 
m n
n
 a
n
n
a
n
b
 
a
n
a  mn a
1
1
a b
a b



a b
a b
a b a b
 b  b 2  4ac
2a
n
n
a
n
b
n
 n am 
 a  b  
a
ax 2  bx  c  0  x 
m
n
n
n
 a
n
m
ab
a
b
1
1
a b
a b



a b
a b
a b a b
a(f(x))2 + b(f(x)) + c = 0  f ( x) 
b  b2  4ac
2a
Section 11-1: Simplifying Expressions with Integral Exponents
Big Idea: Integer exponents represent repeated multiplication. This leads to formulas for simplifying
expressions with exponents whose basis lie in the concept of repeated multiplication or cancellation.
Section 11-2: Fractional Exponents
Big Idea: A fractional exponent is another way to represent a root.
Section 11-3: Simplest Radical Form
Big Idea: A radical is in simplest form when:
1. As many powers as possible are pulled out of the radical.
2. The order of the radical is as low as possible.
3. There are no radicals in the denominator.
Vocabulary:
1. Radicand is the number under the radical.
2. The order (or index) of a radical is the number indicating the root being taken.
Tech Math 2
FINAL EXAM Review
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Section 11-4: Addition and Subtraction of Radicals
Big Idea: Radicals can only be added or subtracted when they are similar. Similar radicals have the same
order and same radicand.
 If the radicals are not similar, then they can not be combined by addition or subtraction.
 Example of similar radicals that can be added:
43 7  53 7  93 7
 Example of radicals that are not similar because of different orders and thus can not be added:
3
757
 Example of radicals that are not similar because of different radicands and thus can not be added:
11  17
 Radicals that do not look similar may in fact be similar if they are simplified first:
8  2  42  2  2 2  2  3 2
Section 11-5: Multiplication and Division of Radicals
Big Ideas:
 Radicals that are multiplied can only be combined under the same radical if the orders of the two
radicals are the same.
 If the orders are not the same, write each radical using fractional exponents, convert each fraction to a
common denominator, and then combine them with multiplication.
 If a
 appears in the denominator of an expression, multiply the numerator and denominator by
to rationalize the denominator.
Section 7-3: Quadratic Equations: The Quadratic Formula
Big Idea: One way to solve the quadratic equation ax 2  bx  c  0 is to plug its coefficients into the quadratic
formula:


 b  b 2  4ac
2a
2
b – 4ac is called the discriminant. The discriminant is important because it determines the character of
the roots of the quadratic equation.
The Quadratic Formula: x 
Tech Math 2
FINAL EXAM Review
Page 8 of 11
Section 14.3: Equations in Quadratic Form
Big Idea: An equation in quadratic form is an equation that follows the pattern of a quadratic equation (a
times “something” squared plus b times “something” plus c), but has a function in place of the unknown
variable. It can be solved using the quadratic formula, but the result from the quadratic formula is what the
“something” is equal to, not the variable itself.
 An equation in Quadratic Form – “unknown” is x2.5: 3x5 + 2x2.5 – 5 = 0
Notice that (x2.5)2 = x5
x 2.5 
2  22  4  3  (5)
23
2  64
6
5
2  8
x2
6
5

1
4
x2
3
x2
5
 
x
5
2
2
5
 1  4 


 3 
2
5
 1  4 
x5

 3 
2
 1  4 
 3
5
x 
  5    1 1
 3 
 3
2
2
5
25
 1  4 
 5 
x 
 1.2267
 5  5
9
 3 
 3 
2
2
5
Section 14.4: Equations with Radicals
Big Idea: Equations with radicals are solved by raising both sides of the equation to an appropriate power to
eliminate the radicals. Some strategies:
 If there is just a single radical in the equation, isolate the radical on one side, then raise both sides to an
appropriate power.
 If there are two separate radicals in the equation, isolate one of the radicals on one side, raise both sides
to an appropriate power to eliminate that radical, then isolate the other radical and raise both sides to an
appropriate power again.
 If there are two nested radicals in the equation, isolate the nested radical on one side, then raise both
sides to an appropriate power, then isolate the remaining inner radical, then raise both sides to an
appropriate power again.
 DONT FORGET TO CHECK ALL YOUR ANSWERS!
Tech Math 2
FINAL EXAM Review
Page 9 of 11
Section 13.1: Exponential Functions
Facts about the exponential function:
1. The exponential function is a constant number raised to an exponent that is a variable: y = bx
2. The constant number b is called the base.
3. The base must be greater than zero (b > 0).
4. Some common bases are 2, 10, and e.
5. The base can’t be equal to one (b  1).
6. The variable x, which is the exponent, can be any real number (i.e., the domain of the function is all real
numbers).
7. The dependent variable y will take on all values greater than zero (i.e., the range is y > 0).
8. The graph always passes through the point (0, 1), because any number raised to the power of zero equals
one.
9. The graph of the function increases if b > 1, while the graph decreases if 0 < b < 1, as shown below:
This is for b > 1
This is for 0 < b < 1
Section 13.2: Logarithmic Functions
Facts about the logarithmic function for b > 1:
1. The logarithmic function “undoes” the exponential function: If y = bx, then x = logb y.
2. The first formula is in “exponential form,” while the second equation is in “logarithmic form.”
3. The logarithm is just an exponent. In fact, it helps sometimes to not to say the word logarithm, but
instead “the exponent that gives y”
4. The constant number b is called the base.
5. Some common bases are 2, 10, and e.
6. The variable y ends up being any real number (i.e., the range of the function is all real numbers).
7. The variable x can only take on values greater than zero (i.e., the domain is x > 0).
8. The graph always passes through the point (1, 0), because any number raised to the power of zero equals
one.
9. The logarithm of a fraction is negative.
10. The logarithm of a number greater than one is positive.
11. The graph of the function increases, as shown below:
Tech Math 2
FINAL EXAM Review
Page 10 of 11
Section 13.3: Properties of Logarithms
Big Idea: The fact that a logarithm is an exponent means that there are rules of logarithms that correspond to
rules of exponents.
Exponent Rule
b0  1
Corresponding Logarithm Rule
logb 1  0
b1  b
logb b  1
bn  bn
log b b n  n
bu b v  bu  v
logb xy  logb x  logb y
bu
 bu  v
v
b
 x
logb    logb x  logb y
 y
log b  x n   n log b x
b 
m n
 b mn
Section 13.4: Logarithms with a Base of 10
Big Idea: A common base for logarithms is 10. In fact, logarithms with a base of 10 are called common
logarithms. When calculating with common logarithms, it is customary to just write log instead of log10. Large
numbers can be calculated by equating the large number to a variable, taking the log of both sides, simplifying
the log, then writing the answer in exponential form using scientific notation.
Tech Math 2
FINAL EXAM Review
Page 11 of 11
Section 13.5: Natural Logarithms
Big Idea: Another widely used base for logarithms is e. Logarithms with a base of e are called natural
logarithms. When writing the natural logarithm function, it is customary to just write ln instead of loge.
Cool Formula: To calculate a logarithm in a base of b in terms of a logarithm of base a:
log a x
log b x 
log a b
Section 13-6: Exponential and Logarithmic Equations
Big Ideas:
 An exponential equation has exponents that are variables.
o To solve exponential equations, take the logarithm of both sides of the equation.
 A logarithmic equation involves the logarithm of variables.
o To solve logarithmic equations, combine all logarithms into a single, isolated logarithm, then
write that resulting logarithmic equation in exponential form.
 Notice the symmetry: to solve exponential equations, use logarithms, to solve logarithmic equations, use
exponential form.
Compound Angles:
c  L2  W 2  H 2
sin  
tan  
H
L W 2  H 2
2
W
L