Review for Final Exam MATH 1114
6.1: Angles and Their Measure
 Be able to state and apply the definition of radian measure
 Be able to convert from radians to degrees and degrees to radians
 Be able to convert from degrees to degrees/minutes/seconds (DMS) and DMS to degrees
 Be able to apply, define, and provide a measurement of an angle in standard position
 Be able to find the arc length subtended by a given angle
 Be able to calculate the arc length, perimeter and area of a sector of a circle.
 Be able to calculate linear and angular speeds of points on a circle/disk/wheel
6.2: Trigonometric Functions
 Be able to state and apply the definition of the six trigonometric functions
 Know the trigonometric function value and the radian measure of the good angles
o Good Angles: /6 radians (30 degrees) , /4 radians (45 degrees) and /3 radians (60 degrees)
o as well as any angle which has a good angle as its reference angle.
 Know all the trigonometric function values and the radian measures of the quadrantal angles
o 0 radians (0 degrees), /2 radians (90 degrees),  radians (180 degrees), and 3/2 radians (270 degrees)
o as well as any angle which is coterminal (shares a terminal side) with a quadrantal angle
 Know/apply connection of trigonometric functions in terms of circles other than the unit circle
 Know/apply .connection of trigonometric function to ratios of sides in a ride triangle
6.3: Properties of Trig. Functions
 Given the sine or cosine (but not both) of an angle and quadrant for its terminal side, be able to find the exact
values other five trigonometric functions of the angle.
 Given a trigonometric function value of a number and some information to determine quadrant, be able to find the
exact values other five.
 Be able to state and apply the fundamental identities
o the basic trigonometric identities (write in terms of sine and cosine)
o pythagorean identities as well as reciprocal identities
o periodic identities as well as the even/odd identities
 For functions y = cosx and y = sinx, know the domain, range, period amplitude, even/odd
 For functions y = tanx, y = cotx, y = secx, y = cscx,
o know the domain, range, period, the equations of the vertical asymptotes, even/odd
6.4: Graphs of Sine and Cosine
 For functions y = Acos(Bx), y = Asin(Bx), be able to
o graph the function over any subset of its domain
o sketch a graph labeling all maxima, minima, and horizontal intercepts (if any) and vertical intercept (if
any) with their EXACT coordinates over any subset of the domain
o find an equation for a given graph
6.5: Graphs of Tangent, Secant, Cosecant, Cotangent Functions
 For functions y = Acot(Bx+C), y = Atan(Bx+C), y = Acsc(Bx+C), and y = Asec(Bx+C), be able to
o find the domain and range
o sketch a graph labeling all of the horizontal intercepts (if any) and vertical intercept (if any) with their
EXACT coordinates over any subset of the domain
o label all vertical asymptotes with their exact equations over any subset of the domain
6.6: Phase Shift, Sinusoidal Curve Fitting
 For functions y = Acos(Bx+C), y = Asin(Bx+C) be able to find the domain and range as well as sketch a graph
labeling all maxima, minima, and horizontal intercepts (if any) and vertical intercept (if any) with their EXACT
coordinates over any subset of the domain
 Approximation of a curve of the form y = Asin(Bx+C)+D to sinusoidal data and Sinusoidal Regression
7.1/7.2: Inverse Sine, Cosine and Tangent Functions
 Restrictions on sine, cosine and tangent functions to define the corresponding inverse function
 Graphs of inverse sine, inverse cosine and inverse tangent functions
 Finding exact values of inverse sine, inverse cosine and inverse tangent functions and compositions with standard
trig. functions
7.3: Trigonometric Identities
 Definition of an identity, definition of a conditional equation
 The Basic Identities (including Pythagorean, even/odd, and periodic identities)
 Deriving/Establishing identities (providing reasons/justifications for steps in the derivation)
 Guidelines for deriving trigonometric identities
7.4: Sum and Difference Identities
 Sum and Difference identities for cosine
 Sum and Difference identities for sine
 Sum and Difference identities for tangent
 Complementary identities relating sine and cosine
 Using the identities to calculate exact values of trigonometric functions
 Deriving/Establishing identities (providing reasons/justifications for steps in the derivation)
7.5: Double Angle and Half Angle Identities
 Double angle identities are a special case of the sum identities (for sine, cosine, tangent)
 Half angle identities are a rewriting of the double angle identities (for sine, cosine, tangent)
 Using the identities to calculate exact values of trigonometric functions
 Deriving/Establishing identities (providing reasons/justifications for steps in the derivation)
7.7 Solving Trigonometric Equations I
 Solving Equations Involving a Single Trigonometric Function
o based on “good” angles: i.e. sin() = 0.5, cos() = 1/2; tan() = 3
o based on inverse trigonometric functions: i.e. sin() = -.23, cos() = .432, tan() = 5
o always prefer EXACT numeric values of the solutions
o sometimes want only those solutions within the first rotation (i.e. 0 ≤  ≤ 2)
o sometimes want all real solutions to these equations
7.8: Solving Trigonometric Equations II
 Quadratic Form i.e. 3sin2() + 5sin() ─ 2 = 0. Using Identities i.e. cos(2) = cos()
o often rewriting in terms of sine and cosine is an effective strategy; sometimes use factoring
o always prefer EXACT numeric values of the solutions
o sometimes want only those solutions within the first rotation (i.e. 0 ≤  ≤ 2)
o sometimes want all real solutions to these equations
8.1: Applications Involving Right Triangles
 Finding Values of Trigonometric Functions Using Ratios of Sides in Right Triangles (soh cah toa)
 Co- functions where co is a shortened form of the word complement
o A pair of angles are complementary if and only if their measures sum to a right angle measure
o Example: cosine literally translates to complement’s sine leading to cos( ) = sin(90─ )
 Labeling: Vertices A, B, C; Side Lengths a (leg), b (leg), c (hypotenuse); Angles ,  , =90
 Solving Right Triangles using Pythagorean Theorem: a2 + b2 = c2 and  +  = 90 (,  in degrees)
 Solving Applied Problems; Angle of Elevation, Angle of Depression, Bearings
8.2: The Law of Sines
 Triangles are Right (contain a right angle) OR Oblique (do not contain a right angle)
 Labeling: Vertices A, B, C; Side Lengths a, b , c ; Angles ,  ,
 Proving the Law of Sines
o Law of Sines (one form) For any triangle, sin α  sin β  sin γ
a
b
c
o
Law of Sines (another form) For any triangle,
a
b
c


sin α sin β sin γ

Solving Triangles: Apply the Law of Sines in two cases
o AAS two angles and a side (note as soon as we know two angles we know the third, why?)
o SSA two sides and an angle opposite one of these sides (this is the ambiguous case)
 Ambiguous case as possibilities include no triangle, one triangle or two triangles
 Solving Applied Problems; Angle of Elevation, Angle of Depression, Bearings
8.3: The Law of Cosines
 Proving the Law of Cosines
o Law of Cosines (one form)
For any triangle, a 2  b 2  c2  2bc cos
o Law of Cosines (another form) For any triangle, b 2  a 2  c 2  2ac cos
o Law of Cosines (another form) For any triangle, c 2  a 2  b 2  2ab cos
 Double angle identities are a special case of the sum identities (for sine, cosine, tangent)
 Solving Triangles Apply the Law of Cosines in two cases:
o SAS two sides and the included angle (the included angle is the one formed by these sides)
o SSS all three sides
 Solving Applied Problems; Angle of Elevation, Angle of Depression, Bearings
6.1 in text: 11, 23, 29, 35, 47, 61, 71, 79, 97, 101
others: 13, 15, 17,19, 21, 25, 27, 31, 33, 37, 39, 41, 43, 45, 49, 51, 53, 55, 57, 59, 63, 65, 67, 69, 73, 75, 77, 81, 83, 87 , 89, 91, 93, 95,
99, 107
6.2 in text: 11, 19, 33, 39, 49, 53, 57, 63, 67, 83
others: 13, 17, 21, 23, 25, 27, 29, 31, 35, 37, 41, 43, 45, 47, 51, 55, 59, 61, 65, 69, 77, 85, 89, 91, 93, 123
6.3 in text: 11, 27, 35, 43, 59, 79, 97
others: 13, 19, 29, 31, 33, 37, 41, 45, 47, 53, 55, 57, 61, 69, 77, 81, 85, 89, 91, 93, 94, 115
6.4 in text: 9, 11, 13, 23, 45, 49, 53, 59, 67, 71
others: 15, 17, 19, 25, 27, 33, 35, 51, 61, 65, 73, 75, 77, 79, 89, 91
6.5 in text: 7, 15, 21, 23, 29
others: 9, 11, 13, 17, 27, 35, 37, 43, 45, 47, 49
6.6 in text: 3, 29, 35
others: 3, 5, 7, 9, 11, 13, 19, 25, 31
7.1 in text: 13, 17, 19, 23, 25, 37, 41, 43, 45, 49, 55, 61
others: 15, 47, 51, 59, 63, 65, 67
7.2 in text: 9, 27, 39, 45, 57
others: 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 41, 43, 59, 63
7.3 in text: 9, 11, 13, 19, 23, 27, 49, 53, 69
others: 21, 37, 39, 47, 55, 65, 73, 77, 81, 87, 93, 95, 97
7.4 in text: 11, 17, 23, 27, 31, 45, 57, 73, 83
others: 9, 13, 15, 19, 21, 25, 29, 33, 37, 39, 43, 51, 59, 67, 71, 75, 79, 87, 95
7.5 in text: 7, 19, 41, 65
others: 9, 11, 13, 15, 21, 23, 25, 27, 33, 35, 37, 71, 73, 77
7.7 in text: 7, 13, 31, 41
others: 9, 11, 21, 25, 29, 33, 37, 39, 53
7.8 in text: 7, 23, 41, 53
others: 5, 11, 13, 19, 21, 31, 33, 35
8.1 in text: 9, 19, 29, 39, 49, 51, 55, 63, 71
others: 15, 17, 21, 23, 25, 37, 41, 47, 57, 59, 69, 75, 79
8.2 in text: 9, 23, 25, 31, 37, 39
others: 11, 19, 27, 29, 35, 41, 47, 53, 57
8.3 in text: 9, 15, 35
others: 11, 13, 19, 27, 29, 33, 37, 39, 43