Unit 5 Review Answers

  Unit 5 Review Answers

Unit 4 Review answers

  Unit 4 Review Answers

Unit 7 Review answers

  Unit 7 Review Answers

Unit 6 Review answers

  Unit 6 Review Answers

8.4 answers

  8.4 Answers

Unit 3 Review answers

  Unit 3 Review Answers

Unit 2 Review answers

  Unit 2 Review Answers

Unit 1 Review answers

  Unit 1 Review Answers

Unit 7 Jeopardy Review

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7.4a: The Dot Product

Objectives:

• Find the dot product of two vectors and use the Properties of the Dot Product.

• Find the angle between two vectors and determine whether two vectors are orthogonal.

Definitions & Formulas:

• Dot Product

• Orthogonal

• Properties of the Dot Product

• Angle Between Two Vectors

Examples:

Ex1. Finding the Dot Product (+ student example 1)


Ex2. Properties of the Dot Product (+student example 2)


Ex3. Angle Between Two Vectors


student example 3


Ex4. Orthogonal Vectors


student example 4

Problem set 7.4a:

2-14 even

26-30 even

36, 48

7.3a: Vectors

Objectives:  

• Represent vectors as directed line segments.

• Write the component forms of vectors.

• Perform basic vector operations and represent them graphically.

Vector Properties:

Examples:

Example 1. Vector Representation by Directed Line Segments


Example 2. Component Form of a Vector


Example 3. Vector Operations


Student Example


Example 4. Finding a Unit Vector


Example 5. Linear Combination of Unit Vectors


Example 6. Vector Operations


Student Example


Problem Set 7.3a:
# 4-10 even, 17-19, 22, 30, 32, 34

7.2: The Law of Cosines

Objectives:

• Use the Law of Cosines to solve oblique triangles (SSS or SAS).

• Use the Law of Cosines to model and solve real-life problems.

Examples:

Example 1. The Law of Cosines (SSS)


Example 2. The Law of Cosines (SAS)



Student example 1




Example 3. Using Heron's area formula to find the area of a triangle


Student Example




7.2 Problem Set
#2-10 even, 25-28, 31, 39,

7.1: The Law of Sines

Objectives:

• Use the Law of Sines to solve oblique triangles (AAS, ASA, SSA).

• Find the areas of oblique triangles.

• Use the Law of Sines to model and solve real-life problems.

Examples

Example 1. Solving an oblique triangle


Example 2. Solving an oblique triangle 2


Student Example


Ex3. Area of an oblique triangle


Student Example




Example 4. The ambiguous case 1


Example 5. The ambiguous case 2


Example 6. The ambiguous case 3


Student Example


6.1 Problem set
#2-8 even, 20-24, 29, 30, 31, 35, 36

6.5b: Product to Sum and Sum to Product Formulas

Objectives:

• Use product to sum and sum to product formulas to, evaluate trig functions, verify trig identities, and solve trig equations.

Example 1

Example 2
*MISTAKE
Thank you Harrison for pointing out that (195-105)/2=45 and not 90. This changes the final answer. Come up with the answer and we will discuss it in class.

Example 3

Problem set 6.5b:

# 3, 5, 64, 68, 78, 79, 80, 84, 86, 88, 90

6.5a: Multiple Angle Formulas

Objectives:

• Use the multiple angle and half angle formulas to evaluate, verify, and solve trig equations.

Example 1

Student example 1

Example 2

Student example 2

Example 3

Student example 3

Example 4

Example 5

Problem set 6.5a:

4, 6, 11, 12, 14, 24, 36, 41, 50

6.4: Sum and Difference Formulas

Objectives:

• Use Sum and Difference Formulas to evaluate trig functions, verify identities, and solve trig equations.

Example 1a

Example 1b

Student example 1

Example 2

Student example 2

Example 3

Student example 3

Problem set 6.4

# 4, 6, 12, 17, 24, 26, 37, 38, 70

6.3b: Solving Trig Equations

Objectives:

• Solve trig equations involving multiple angles.
• Solve trig equations using inverse trig functions.

Example 1

Student example 1

Example 2

Student example 2

Problem set 6.3b:

# 13, 17,  35, 36, 38, 60-62

6.3a: Solving Trig Equations

Objectives:

• Solve trig equations using algebraic techniques.
• Solve trig equations of the quadratic type.

Example 1
*MISTAKE: one of the initial solutions is incorrect. 3pi/4 is not a solution. It should be changed to 7pi/4. 
(Thanks Rachael B.)

Example 2

Example 3

Example 4

Example 5

Student example 1

Student example 2

Problem set 6.3a:

#2, 5, 7, 12, 15, 17, 21, 24, 26, 29

6.2 solutions

These are the worked out solutions to the 6.2 problem sets. There are definitely other ways to verify these identities, but the methods used here are generally the shortest paths.

Note* the title says "5.2" however it is 6.2 for us.

-Tolar

  6.2 solutions by  

6.1: The Fundamental Trig Identities

Objectives:

• Recognize and write fundamental trig identities.
• Use trig identities to evaluate trig functions and simplify trig expressions.

Example 1

Student example 1

Example 2 & Student example 2

Example 3

Example 4

Student Example 3

Problem set 6.1

# 2, 6, 8, 28-38 even, 45, 46, 48, 52

5.8: Applications and Models

Objectives:

• Solve "real world" problems using trig.

• Solve problems involving bearings using trig.

Example 1


Student example 1


Example 2


Example 3


Student example 2


Problem set 5.8:##3, 4, 15, 20, 21, 22, 31

5.7: Inverse Trig Functions

Objectives:

• Define inverse trig functions.
• Evaluate inverse trig functions.
• Evaluate compositions of inverse trig functions.

Example 1 

Student example 1

Example 2

Example 3

Problem set 5.7:

# 3-7, 19-23, 44-47, 49-51

5.6b: Graphs of Other Trig Functions

Objectives:

• Sketch graphs of secant and cosecant functions.

Example 1

Student example 1


Example 2


Student example 2


Problem set 5.6b
#13, 14, 17, 20
38, 40 (include a window that will allow for 2 periods of the function)

5.6a: Graphs of Tangent and Cotangent

Objectives:

• Sketch graphs of Tangent and Cotangent.

Example 1

Student Example 1

Example 2

Student Example 2

Problem set 5.6a:

#7, 10, 20, 24, 30, 35, 36

5.5b: Graphs of Sine and Cosine Functions

Objective:

• Graph translated sine and cosine functions.

Example 1

Example 2

Problem set 5.5b:

15, 16, 21, 22, 33,  46-49

5.5a Graphs of Sine and Cosine Functions

Objectives:

• Sketch graphs of sine and cosine functions.
• Identify period and amplitude of sine and cosine functions.

Example 1

Example 2

Student Example

Problem set 5.5a:

#2-14 even, 35-43 odd

5.4: Trig Functions of any Angle

Objectives:

• Evaluate trig functions of any angle.
• Evaluate trig functions using reference angles.
• Evaluate trig functions of Real numbers.

Example 1

Student Example 1

Example 2

Student Example 2

Example 3 & Student Example 3

Problem set 5.4:

# 4, 6, 12, 14, 19, 20, 24, 29, 30, 32, 34, 48, 56, 60,   68, 70

5.3: Right Triangle Trig

Objectives:

•  Evaluate trig functions of acute angles.
• Use trig functions to model and solve real life functions.

Examples:

Example 1

Student example 1


The Special Triangles Review


Example 2

Student example 2


Example 3

Student example 3



Example 4



Example 5


Student example 5


Problem Set 5.3:
 #2, 10, 18, 27, 28, 34, 36, 38, 53, 54, 60, 63, 64

5.2: The Unit Circle

Objectives:

• Evaluate trig functions using the Unit Circle.

Example 1 & Student Example 1

Example 2

Student Example 2

Problem set 5.2:

# 6, 8, 10, 24, 26, 32, 34, 43-46

5.1b: Radian Measure

Objectives:

üDescribe angles using radian measure.

ü Solve applied problems involving angles.

Example 1 and Student Example 1

Example 2 and Student Example 2

Example 3

Student Example 3

Example 4

Problem set 5.1b

#8, 11, 14, 18, 48, 52, 88, 92, 104, 107

5.1a: Radian and Degree Measure

Objectives:

• Describe angles in standard form using degree measure.

Example 1

Student Example 1

Example 2

Student Example 2

Example 3

Student Example 3

Problem Set 5.1a

# 25-30, 31, 33, 35, 38, 40-44