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PRE-CALCULUS Semester 2 Final Exam Review Packet
Final Exam for ALL PRE-CALCULUS STUDENTS will be during classes on June 6/7.
Non-Seniors will complete GIFT #9 (small group assessment) on the following days:
Period 2/A:
Period 1:
Period 3/C:
FRIDAY, June 14th
8:15am – 9:55
10:10 – 11:50
12:05 – 1:45
MONDAY, June 17th
Period 7/D: 8:15am – 9:55
Period 5:
10:10 – 11:50
Period 4: 12:05 – 1:45
TUESDAY, June 18th
Period 8:
8:15am – 9:55
Period 6/B:
10:10 – 11:50
Make-up:
12:05 – 1:45
During your final exam you will be expected to display a cumulative and in-depth understanding of important mathematical
concepts discussed in class. Your final exam will measure your ability to:
 recognize major mathematical concepts,
 organize and express mathematical ideas clearly,
 develop and support the main idea,
 use appropriate math vocabulary and ability to define math terms in own words,
 use appropriate instruments and formulas to explain the concept and to solve problems
 ability to think critically using reasoning and evidence
 ability to apply the concepts and skills to solve application problems
Your test will consist of two portions: [I] Short Response and [II] Standardized Multiple Choice.
[ I ] SHORT RESPONSE (Test Grade): This portion of the test will consist of 3 short response questions. Your short response is
not graded on length, it will be graded on quality and clear demonstration of understanding of the concept presented. Your
response on this portion of the test will be scored with 0-5 points per question.
[ II ] STANDARDIZED MULTIPLE CHOICE TEST (Final Exam Grade): This portion of the final exam will consist of multiple
choice questions involving concepts stated in the focus questions as well as Advanced Algebra pre-requisite skills. This test will be
graded on point system (1 point per correct response).
FINAL EXAM FOCUS QUESTIONS:
CHAPTER 4: TRIGONOMETRIC FUNCTIONS.
QUESTION 1. Make a clear connection between the right triangle trigonometry and the unit circle to explain why the following are true:
explain why
cos 30 
3

2

; sin 
; tan is undefined .
2
4
2
2
To fully address this prompt, your response should include the following:
a) Sketch a 30-60-90 special right triangle inside of the unit circle with a reference angle of 30° (reference angle has a vertex at the
origin).
b) Using ratios of sides for special right triangles, find the lengths of all sides of this triangle if the hypotenuse is = 1 unit since it
represents the radius of a unit circle.
c) Show that the adjacent side of the triangle represents the length along the x-axis  value cosine
d) Show that the opposite side of the triangle represents the length along the y-axis  value of sine
e) Sketch a 45-45-90 special right triangle inside of the unit circle with a reference angle of 45° and repeat steps b) through d)
f) For tangent show that vertical length does not represent a function  tangent is undefined at 90 and 270. Also explain that since
tan x = y/x or sin /cos at 90 degrees sine = 1 and cosine = 0; therefore,
tan 90 = 1/0 is undefined.
QUESTION 2. Explain the relationship between the six trigonometric ratios and their Trigonometric Identities.
To fully address this prompt, your response should include the following:
a) List the six trigonometric ratios [ Page 267 blue box] and list the Reciprocal Trigonometric Identities [Page 270 in the blue box].
b) Use a right triangle with specific numerical values and identify all six ratios. [Page 268 Example 1]
c) Use Algebra to verify the reciprocal identities.
d) Create one example problem that demonstrates finding all six trigonometric ratios given one real value. [Page 261 Example 1]
Pre-Calculus
1
2011/2012
QUESTION 3. Using equations and graphs, demonstrate the following shifts of circular functions:
a) vertical shift of a sine function
b) period change of a cosine function
c) phase shift of a tangent function
To fully address this prompt, your response should include the following:
a) Write the standard form of trigonometric equation y = a sin b (x – c) + d
b) For each of the given transformations, identify the variable in the equation that represents the given transformation.
c) Graph one example of each function. You can assume that the remaining coefficients remain unchanged from the common
trigonometric function.
d) Clearly explain the steps to finding the new KEY points of a function after the transformation was performed.
QUESTION 4. Compare and contrast graphs of common trigonometric functions and their reciprocal functions.
To fully address this prompt, your response should include the following:
a) Graphs of common trigonometric functions and their reciprocal functions with an explanation of their commonalities and
differences. [Page 301 lesson intro, Page 302 Example 4, 5 Page 304 Figure 4.67]
CHAPTER 5. TRIGONOMETRIC IDENTITIES and SOLVING TRIGONOMETRIC EQUATIONS.
QUESTION 5 : Fundamental Trigonometric Identities. [5.1]
a) List the fundamental trigonometric identities: reciprocal identities, quotient identities, and Pythagorean identities.
b) Demonstrate using trigonometric identities to find the value of all six trigonometric functions.
c) Demonstrate simplifying trigonometric expressions using identities.
QUESTION 6: Trigonometric Equations. [5.3]
Demonstrate solving trigonometric equations involving the following algebraic operations
a) combining like terms
b) factoring
c) For each equation, check the solution and explain the difference between solutions in the interval (0, 2π) and all general solutions.
QUESTION 7: Trigonometric Equations. [5.3]
Demonstrate solving trigonometric equations involving the following algebraic operations
a) extracting square roots
b) substitution of variables using trig identities
c) For each equation, check the solution and explain the difference between solutions in the interval (0, 2π) and all general solutions.
CHAPTER 8. SEQUENCES AND SERIES.
QUESTION 8: Sequences and Series. [8.1]
a) Define sequences and series and provide examples.
b) Demonstrate writing specific terms of a sequence and writing the n-th term of a sequence.
c) Explain how to use sequence notation to describe terms of a sequence.
d) Use summation notation to write partial sums.
QUESTION 9: Arithmetic sequences and series [8.2, 8.3]
a) Define arithmetic sequences and provide examples.
b) Show examples of finding the nth term of an arithmetic sequence.
c) Provide formulas for finding the sum of finite and infinite arithmetic series.
QUESTION 10: Geometric sequences and series [8.2, 8.3]
a) Define geometric sequences and provide examples.
B) Show examples of finding the nth term of a geometric sequence.
c)Provide formulas for finding the sum of finite and infinite geometric series.
QUESTION 11: Factorials [8.1]
a) Define factorial.
b)Demonstrate evaluating factorial expressions.
c)Provide examples of writing terms of a sequence involving factorials.
QUESTION 12: Binomial Theorem [8.5]
Cr
a)
Show examples of how you can calculate binomial coefficients using
b)
c)
Explain how Pascal’s triangle can be used to calculate binomial coefficients.
Demonstrate expanding a binomial.
D) Demonstrate expanding a binomial expression.
n
notation.
IMPORTANT NOTE: Chapter 9 Conic Sections and Chapter 11 Introduction to Limits are NOT included in the
short response of the Final Exam, but will be included in the standardized (multiple choice) portion of the
test!
Pre-Calculus
2
2011/2012