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Course Title: Trigonometry/Statistics
Date Adopted: June 27, 2001
Department: Mathematics
Fulfills UC/CSU Requirement: Pending
Pre-Requisite: C-average in Algebra 2
Fulfills CSF Requirement: Yes
Length of Course: 1 Year
Fulfills H/S Graduation
Credit As:
Required x Elective x
Semester Units/Credits: 10 units total
Grade Level: 11 and 12
I.
Course Description
This is a one year course that covers trigonometry and introductory statistics. The
trigonometry component covers the Unit Circle, Right Triangles, Trigonometric
Functions and their graphs, Analytical Trigonometry, Complex Numbers, Exponential
and Logarithmic Functions, Analytical Geometry and Real-Life Applications.
The statistics component of the course covers the basic and essential topics of
understandable statistics. The topics covered are Descriptive Statistics, Probability,
Estimation, Hypothesis Testing, and Linear Regression. This course will use applications
involving the TI-83 calculator, ComputerStats and new coverage of Excel.
II.
Rationale
This course is designed to give students an in-depth study of trigonometry as well as an
introduction to statistics. Students will use technology through the course utilizing
graphing calculators and computers. They will also be taught through a variety of
technology based lectures and examples.
III.
Goals, Objectives, and Performance Indicators
1
Trigonometry
1.1
Radian and Degree Measure
The student will demonstrate an understanding of how to describe an angle and convert
between radian and degree measure.
The student will learn:
1.1.1 How to describe angles.
1.1.2 How to use radian measure.
1.1.3 How to use degree measure
1.1.4 How to use angles to model and solve real-life problems.
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1.2
The Unit Circle
The student will identify a unit circle and its relationship to real numbers.
The student will evaluate trigonometric functions using the unit circle, domain and range,
and technology.
The student will learn:
1.2.1 How to identify a unit circle and its relationship to real numbers.
1.2.2 How to evaluate trigonometric functions using the unit circle.
1.2.3 How to use domain and period to evaluate sine and cosine functions.
1.2.4 How to use a calculator to evaluate trigonometric functions.
1.3
Right Triangle Trigonometry
The student will evaluate trigonometric functions using acute angles, fundamental
identities, technology and real-life applications.
The student will learn:
1.3.1 How to evaluate trigonometric functions of acute angles.
1.3.2 How to use the fundamental trigonometric identities.
1.3.3 How to use a calculator to evaluate trigonometric functions.
1.3.4 How to use trigonometric functions to model and solve real-life problems.
1.4
Trigonometric Functions of Any Angle
The student will evaluate trigonometric functions of any angle.
1.5
The student will learn:
1.4.1 How to evaluate trigonometric functions of any angle.
1.4.2 How to use reference angles to evaluate trigonometric functions.
1.4.3 How to evaluate trigonometric functions of real numbers.
Graphs of Sine and Cosine Functions
The student will sketch the graphs and translations of graphs of sine and cosine functions.
The student will learn:
1.5.1 How to sketch the graphs of basic sine and cosine functions.
1.5.2 How to use amplitude and period to help sketch the graphs of sine and
Cosine functions.
1.5.3 How to sketch translations of the graphs of sine and cosine functions.
1.5.4 How to use sine and cosine functions to model real-life data.
1.6
Graphs of Other Trigonometric Functions
The student will sketch the graphs and translations of other trigonometric functions.
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The student will learn
1.6.1 How to sketch the graph of tangent functions.
1.6.2 How to sketch the graph of cotangent functions.
1.6.3 How to sketch the graphs of secant and cosecant functions.
1.6.4 How to sketch the graphs of damped trigonometric functions.
1.7
Inverse Trigonometric Functions
The student will evaluate the inverse trigonometric functions.
The student will learn:
1.7.1 How to evaluate the inverse sine function.
1.7.2 How to evaluate the other inverse trigonometric functions.
1.7.3 How to evaluate the compositions of trigonometric functions.
1.8
Applications and Models
The student will solve real-life problems.
The student will learn:
1.8.1 How to solve real-life problems involving right triangles.
1.8.2 How to solve real-life problems involving directional bearings.
1.8.3 How to solve real-life problems involving harmonic motion.
2
Analytical trigonometry
2.1
Using Fundamental Identities
The student will use fundamental trigonometric identities to evaluate trigonometric
functions and simplify trigonometric expressions.
The student will learn:
2.1.1 How to recognize and write the fundamental trigonometric identities
2.1.2 How to use the fundamental trigonometric identities to evaluate trigonometric
functions, simplify trigonometric expressions, and rewrite trigonometric
expressions.
2.2
Verifying Trigonometric Identities
The student will verify trigonometric identities.
2.2
The student will learn:
2.2.1 To plan a strategy for verifying trigonometric identities.
2.2.2 How to verify trigonometric identities.
Solving Trigonometric Equations
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The student will use standard algebraic techniques and inverse trigonometric functions to
solve trigonometric equations.
The student will learn:
2.2.1 How to use standard algebraic techniques to solve trigonometric equations.
2.2.2 How to solve trigonometric equations of quadratic type.
2.2.3 How to solve trigonometric equations involving multiple angles.
2.2.4 How to use inverse trigonometric functions to solve trigonometric equations.
2.3
Sum and Difference Formulas.
The student will use sum and difference formulas to rewrite and evaluate trigonometric
functions.
The student will learn:
2.3.1 How to use sum and difference formulas to evaluate trigonometric functions.
2.3.2 How to use sum and difference formulas to verify identities and solve
trigonometric equations.
2.4
Multiple-Angle and Product-to-Sum Formulas
The student will use multiple-angle formulas, power-reducing formulas, half-angle, and
product-to-sum formulas to rewrite and evaluate trigonometric functions.
The student will learn:
2.4.1 How to use multiple-angle formulas to rewrite and evaluate trigonometric
functions.
2.4.2 How to use power-reducing functions to rewrite and evaluate trigonometric
functions.
2.4.3 How to use half-angle functions to rewrite and evaluate trigonometric functions.
2.4.4 How to use product-to-sum formulas to rewrite and evaluate trigonometric
functions.
3
Additional Topics in Trigonometry
3.1
Law of Sines
The student will use Law of Sines to solve and find the areas of oblique triangles
3.2
The student will learn:
3.1.1 How to use the Law of Sines to solve oblique triangles(AAS, ASA, or
SSA).
3.1.2 How to find the areas of oblique triangles.
3.1.3 How to use Law of Sines to model and solve real-life problems.
Law of Cosines
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The student will use Law of Cosines to solve oblique triangles.
The student will learn:
3.2.1 How to use the Law of Cosines to solve oblique triangles (SSS or SAS) .
3.2.2 How to use the Law of Cosines to model and solve real-life problems.
3.2.3 How to use Heron’s formula to find the area of a triangle.
3.3
Vectors in the Plane
The student will write the component forms of vectors and perform basic vector
operations.
The student will learn:
3.3.1 How to represent vectors as directed line segments.
3.3.2 How to write the component forms of vectors.
3.3.3 How to perform basic vector operations and represent them graphically.
3.3.4 How to write vectors as linear combinations of unit vectors.
3.3.5 How to find the direction angles of vectors.
3.3.6 How to use vectors to model and solve real-life problems.
3.4
Vectors and Dot Products
The student will find the direction angles of vectors and the angle between two vectors.
The student will learn:
3.4.1 How to find the dot product of two vectors and use the Properties of the
Dot Product.
3.4.2 How to find the angle between two vectors.
3.4.3 How to determine whether two vectors are orthogonal.
3.4.4 How to write a vector as a sum of two vector components.
3.4.5 How to use vectors to find work done by a force.
4
Complex Numbers
4.1
Complex Numbers
The student will perform operations with complex numbers.
4.2
The student will learn:
4.1.1 How to use the imaginary i to write complex numbers.
4.1.2 How to add, subtract, and multiply complex numbers.
4.1.3 How to use the conjugates to divide complex numbers.
4.1.4 How to use the Quadratic Formula to find complex solutions of quadratic
equations.
Complex Solutions of Equations
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The student will determine and find the number of zeros of polynomial functions.
The student will learn:
4.2.1 How to determine the numbers of solutions of polynomial equations.
4.2.2 How to find the zeros of a polynomial function.
4.2.3 How to find a polynomial function given the zeros of the function.
4.3
Trigonometric Form of a Complex Number.
The student will multiply and divide complex numbers written in trigonometric form.
The student will learn:
4.3.1 How to plot complex numbers in the complex plane.
4.3.2 How to write the trigonometric forms of complex numbers.
4.3.3 How to multiply and divide complex numbers written in trigonometric
form.
4.4
DeMoivre’s Theorem
The student will find the powers and nth roots of complex numbers.
The student will learn:
4.4.1 How to use DeMoivre’s Theorem to find powers of complex numbers.
4.4.2 How to find nth roots of complex numbers.
5
Exponential and Logarithmic Functions
5.1
Exponential Functions and Their Graphs
The student will recognize, evaluate and graph exponential functions.
The student will learn:
5.1.1 How to recognize and evaluate exponential functions with base a.
5.1.2 How to graph exponential functions.
5.1.3 How to recognize and evaluate exponential functions with base e.
5.1.4 How to use exponential functions to model and solve real-life
applications.
5.2
Logarithmic Functions and Their Graphs.
The student will recognize, evaluate and graph logarithmic functions.
The student will learn:
5.2.1 How to recognize and evaluate logarithmic functions with base a.
5.2.2 How to graph logarithmic functions.
5.2.3 How to recognize and evaluate natural logarithmic functions.
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5.2.4
5.3
How to use logarithmic functions to model and solve real-life
applications.
Properties of Logarithms
The student will rewrite logarithmic functions with different a base and use properties of
logarithms to evaluate, rewrite, expand, or condense logarithmic expressions.
The student will learn:
5.3.1 How to rewrite logarithmic functions with a different base.
5.3.2 How to use properties of logarithms to evaluate or rewrite logarithmic
expressions.
5.3.3 How to use properties of logarithms to expand or condense logarithmic
expressions.
5.3.4 How to use logarithmic functions to model and solve real-life applications.
5.4
Exponential and Logarithmic Equations
The student will solve exponential and logarithmic equations.
The student will learn:
5.4.1 How to solve simple exponential and logarithmic equations.
5.4.2 How to solve more complicated exponential equations.
5.4.3 How to solve more complicated logarithmic equations.
5.4.4 How to use exponential and logarithmic equations to model and solve
real-life applications.
5.5
Exponential and Logarithmic Models
The student will use exponential growth models, exponential decay models, Gaussian
models, logistic models, and logarithmic models to solve real-life problems.
The student will learn:
5.5.1 How to recognize the five most common types of models involving
exponential and logarithmic functions.
5.5.2 How to use exponential growth and decay functions to model and solve
real-life problems.
5.5.3 How to use Gaussian functions to model and solve real-life problems.
5.5.4 How to use logistic growth functions to model and solve real-life
problems.
5.5.5 How to use logarithmic functions to model and solve real-life problems.
6
Topics in Analytic Geometry.
6.1
Lines
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The student will find the inclination of a line, the angle between two lines, and the
distance between a point and a line.
The student will learn:
6.1.1 How to find the inclination of a line.
6.1.2 How to find the angle between two lines.
6.1.3 How to find the distance between a point and a line.
6.2
Introduction to Conics: Parabolas
The student will write the standard forms of the equations of a parabola.
The student will learn:
6.2.1 How to recognize a conic as the intersection of a plane And a double
napped cone.
6.2.2 How to write the standard form of the equation of a parabola.
6.2.3 How to use the reflective property of parabolas to solve real-life problems.
6.3
Ellipses
The student will write the standard forms of the equations of an ellipse.
The student will learn:
6.3.1 How to write the standard form of the equation of an ellipse.
6.3.2 How to use properties of ellipses to model and solve real-life problems.
6.3.3 How to find the eccentricity of an ellipse.
6.4
Hyperbolas
The student will write the standard forms of the equations of a hyperbola.
The student will learn:
6.4.1 How to write the standard form of the equation of a hyperbola.
6.4.2 How to find the asymptotes of a hyperbola.
6.4.3 How to use properties of hyperbolas to solve real-life problems.
6.4.4 How to classify a conic from its general equation.
6.5
Rotation of Conics
The student will eliminate the xy-term in the equation of a conic and use the discriminant
to identify a conic.
The student will learn:
6.5.1 How to rotate the coordinate axes to eliminate the xy-term in the equation
of a conic.
6.5.2 How to use the disciminant to classify a conic.
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6.6
Parametric Equations
The student will rewrite a set of parametric equations as a rectangular equation and find a
set of parametric equations for a graph.
The student will learn:
6.6.1 How to evaluate a set of parametric equations for a given value of the
parameter.
6.6.2 How to sketch the curve that is represented by a s et of parametric
equations.
6.6.3 How to rewrite a set of parametric equations as a single rectangular
equation.
6.6.4 How to find a set of parametric equations for a graph.
6.7
Polar Coordinates
The student will write equations in polar form.
The student will learn:
6.7.1 How to plot points in the polar coordinate system.
6.7.2 How to convert points from rectangular to polar form and vice versa.
6.7.3 How to convert equations from rectangular to polar form and vice versa.
6.8
Graphs of Polar Equations
The student will graph polar equations.
The student will learn:
6.8.1 How to graph a polar equation by point plotting.
6.8.2 How to use symmetry, zeros, and maximum r-values as graphing aids.
6.8.3 How to recognize special polar graphs.
6.9
Polar Equations of Conics
The student will write equations of conics in polar form.
The student will learn:
6.9.1 How to define a conic in terms of eccentricity.
6.9.2 How to write equations of conics in polar form.
6.9.3 How to use equations of conics in polar form to model real-life problems.
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1.
Organizing Data
1.1
Populations, Samples and Data
The student will understand the meaning of populations, samples and data
The student will learn:
1.1.1 How to identify the population in a study.
1.1.2 How to learn ways to generate data.
1.1.3 How to become aware of some of the pitfalls of data collection.
1.2
Random Samples
The student will generate random samples from specific populations.
The student will learn:
1.2.1 How to generate a random sample.
1.2.2 How to identify the characteristics of a random sample.
1.2.3 How to recognize other sampling techniques.
1.3
Graphs
The student will construct and interpret various statistical graphs..
The student will learn:
1.3.1 How to interpret as well as create a bar graph.
1.3.2 How to interpret as well as create a Pareto chart.
1.3.3 How to interpret as well as create a circle graph or pie chart.
1.3.4 How to interpret as well as create a time plot.
1.3.5 How to interpret as well as create a time series
1.4
Histograms and Frequency Distribution
The student will organize data in the form of a frequency distribution or histogram as
appropriate.
The student will learn:
1.4.1 How to produce a histogram.
1.4.2 How to create relative-frequency tables
1.4.3 How to evaluate distribution shapes.
2.
Averages and Variation
2.1
Measures of Central Tendency
The student will compute the mean, mode and median.
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The student will learn:
2.1.1 How to compute the mean, sample mean, population mean and trimmed
mean of a set of data.
2.1.2 How to compute the mode of a set of data.
2.1.3 How to compute the median of a set of data.
2.2
Measures of Variation
The student will compute range, standard deviation, and coefficient of variation for a data
set.
The student will learn:
2.2.1 How to compute the range for a set of data.
2.2.2 How to compute the sample standard deviation.
2.2.3 How to compute the sample variance.
2.2.4 How to compute the population mean and standard deviation.
2.2.5 How to compute the coefficient of variation.
2.2.6 How to use and interpret Chebyshev’s theorem.
3.
Regression and Correlation
3.1
Introduction to Paired Data and Scatter Diagrams
The student will make a scatter diagram from data pairs.
The student will learn:
3.1.1 How to construct a scatter plot
3.1.2 How to model a linear correlation against the data
3.2
Linear Regression
The student will find the equation of the least squares line and make forecasts based on
the least squares line.
The student will learn:
3.2.1 How to differentiate between an explanatory variable and a response variable.
3.2.2 How to determine the criteria for a least-squares line.
3.2.3 How to draw a line of best fit.
3.2.4 How to analyze data as being interpolation or extrapolation.
3.2.5 How predict the effects of extreme data points.
3.3
The Linear Correlation Coefficient
The student will use the correlation coefficient to assess the strength of the linear forecast
model.
The student will learn:
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3.3.1
3.3.2
3.3.3
4.
How to compute the correlation coefficient.
How to match a graph against the correlation coefficient.
How to compute the coefficient of determination.
Elementary Probability Theory
The student will learn some of the basic properties of probability necessary for working
with inferential statistics.
4.1
What is Probability?
The student will estimate probabilities.
The student will learn:
4.1.1 How to use relative frequency to estimate the probability of an event.
4.1.2 How compute the probability of outcomes that are equally likely.
4.1.3 How to determine a sample space.
4.1.4 How to find the complement of an event.
4.1.5 How to compare probability related to statistics.
4.2
Some Probability Rules-Compound Events
The student will apply basic rules of probability to find the probability of mutually
exclusive events and independent events.
The student will learn:
4.2.1 How to find the probability for independent and dependent events.
4.2.2 How to find the probability for compound events.
4.2.3 How to find the probability of mutually exclusive events.
4.2.4 How to find the probability of a combination of several events.
5.
Introduction to Probability Distributions and The Binomial Probability Distribution
5.1
Introduction to Random Variables and Probability Distributions
The student will find the expected value and standard deviation of a discrete probability
distribution.
The student will learn:
5.1.1 How to determine if a random variable is discrete.
5.1.2 How to determine if a random variable is continuous.
5.1.3 How to construct a probability distribution.
5.2
Binomial Probabilities
The student will find probabilities associated with specific success/failure relationships.
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The student will learn:
5.2.1 How to analyze a binomial experiment.
5.2.2 How to compute the probabilities for a binomial experiment.
5.2.3 How to use the formula for binomial probability distribution.
5.2.4 How to use the binomial distribution table to find the probability of an event.
5.2.5 How to compute the probability of a range of successes.
5.3
The Mean and Standard Deviation of the Binomial Distribution
The student will find the expected value and standard deviation of binomial distributions.
The student will learn:
5.3.1 How to analyze the graph of a binomial distribution.
5.3.2 How to compute the mean of binomial probability distributions.
5.3.3 How to compute the standard deviation of binomial probability distributions.
6.
Normal Distributions
6.1
Graphs of Normal Probability Distributions
The student will analyze and construct graphs of normal probability distributions.
The student will learn:
6.1.1 How to use the properties of a normal distribution graph.
6.1.2 How to compute the area under a normal curve.
6.1.3 How to apply the empirical rule and compute definite percentages.
6.1.4 How to construct control charts.
6.1.5 How to develop out-of-control warning signals.
6.2
Normal Units and Areas Under the Standard Normal Distribution
The student will convert raw scores to standard scores and standard scores to raw scores
and find the areas under the standard normal curve.
The student will learn:
6.2.1 How to compute standard scores.
6.2.2 How to compute a z-score.
6.2.3 How to compute a raw score.
6.2.4 How to compute the areas under the standard normal curve.
6.3
Areas Under any Normal Curve
The student will combine the skill of using the table of areas for a standard normal
distribution with the skill of converting a raw score to a z score.
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The student will learn:
6.3.1 How to convert normal distributions to standard normal.
6.3.2 How to find x or z values that correspond to a given area under the normal curve.
6.3.3 How to find the area under any normal curve.
7.
Introduction to Sampling Distributions
7.1
Sampling Distributions
The student will define terms such as parameter, statistic and sampling distribution.
7.2
The student will learn:
7.1.1 How to apply the term parameter.
7.1.2 How to apply the term statistic.
7.1.3 How to apply the term sampling distribution.
The Central Limit Theorem
The student will use the Central Limit Theorem to analyze the probability distribution of
sample means.
The student will learn:
7.2.1 How to apply the properties of a sample mean distribution.
7.2.2 How to compute the standard error of the sample mean.
7.2.3 How to use the central limit theorem.
8.
Estimation
8.1
Estimating  with Large Samples
The student will use sample information to draw conclusions about a population.
The student will learn:
8.1.1 How to find the estimate of a population parameter.
8.1.2 How to compute the critical value.
8.1.3 How to compute the error of estimate.
8.1.4 How to compute a confidence interval for a large sample.
8.2
Estimating  with Small Samples
The student will estimate  with sample samples.
The student will learn:
8.2.1 How to compute a t distribution.
8.2.2 How to compute the degrees of freedom.
8.2.3 How to use tables to find critical values for confidence intervals.
8.2.4 How to find the maximum error of estimate.
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8.2.5
8.3
How to find the confidence interval for  in a small sample.
Estimating p in the Binomial Distribution
The student will estimate a population proportion.
The student will learn:
8.3.1 How to employ large sample methods.
8.3.2 How to compute the error of estimate.
8.3.3 How to compute the confidence interval for a population.
8.3.4 How to interpret the margin of error.
8.4
Choosing the Sample Size
The student will estimate a mean or as proportion for a sample size.
The student will learn:
8.4.1 How to determine a sample size for estimating .
8.4.2 How to determine the sample size for a proportion.
9
Hypothesis Testing
9.1
Introduction to Hypothesis Testing
The student will develop a hypothesis involving one population and will conclude
whether to accept or reject that hypothesis through a variety of evaluations.
The student will learn:
9.1.1 How to establish a null hypothesis.
9.1.2 How to establish an alternate hypothesis.
9.1.3 How to test for type I and Type II errors.
9.1.4 How to assess the level of significance.
9.1.5 How to evaluate the power of a statistical test.
9.1.6 How to accept or reject the null hypothesis.
9.1.7 How to evaluate the critical regions of the null hypothesis.
9.2
Tests Involving the Mean  (Large Samples)
The student will use the critical value method for concluding hypothesis tests.
The student will learn:
9.2.1 How to find the critical values that form the boundaries of the critical
region(s).
9.2.2 How to use the sample evidence to draw a conclusion regarding whether or not to
reject the null hypothesis.
9.2.3 How to compute the sample test statistic.
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9.3
The P Value in Hypothesis Testing
The student will determine the conclusion of a test using the P value.
The student will learn:
9.3.1 How to find the P value for the tests of the mean.
9.3.2 How to determine the statistical significance based on the P value.
9.3.3 How to interpret results from P values generated on a computer.
9.4
Tests Involving the Mean  (Small Samples)
The student will develop tests involving the mean on small samples.
The student will learn:
9.4.1 How to construct a small sample.
9.4.2 How to obtain critical values from a t distribution.
9.4.3 How to calculate a sample test statistic.
9.4.4 How to use P values for tests of the mean on small samples.
9.5
Tests Involving a Proportion
The student will find confidence intervals for a population proportion.
The student will learn:
9.5.1 How to test for a single proportion.
9.5.2 How to calculate critical values for testing p.
9.5.3 How to compute a sample test statistic.
9.5.4 How to find P values for tests of proportions.
9.6
Testing the Correlation Coefficient
The student will test whether the population correlation coefficient is positive, negative
or just not zero.
The student will learn:
9.6.1 How to use the population correlation coefficient.
9.6.2 How to test the population correlation coefficient.
9.6.3 How to establish the importance of the term significant.
9.6.4 How to get P values from a table.
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