Trigonometry/Advanced Math

PA Common Core Mathematics Standards

• CC.2.1.HS.F.1  Simplify radical expressions.
• CC.2.1.HS.F.2  Simplify and perform operations on rational expressions.
• CC.2.2.HS.C.2 Graph and analyze functions and use their properties to make connections between the different representations. 
• CC.2.2.HS.D.2  Write expressions in equivalent forms to solve problems. 
• CC.2.2.HS.D.6  Extend the knowledge of rational functions to rewrite in equivalent forms. 
• CC.2.2.HS.D.10  Represent, solve and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically. 

• Expressions versus equations.
• Rounding numbers to the correct place.
• Divisibility rules.
• Evaluating functions from an equation and a graph.
• Operations on rational expressions.
• Properties of exponents.
• Simplifying radicals.
• System of equations. 

• Skills and concepts from previous math courses that will be the foundation for the remainder of the course.

• State the difference between an expression and an equation.
• State the difference between evaluate, simplify, and solve.
• Round numbers to the given place.
• Decide if a number is divisible by 2, 3, 4, 5, 6, 9, and 10.
• Evaluate functions given an equation and a graph.
• Add, subtract, multiply, and divide rational expressions.
• Simplify expressions using properties of exponents.
• Simplify radical expressions (including variables).
• Simplify radical expressions with higher indices. 
• Solve a system of equations by graphing, substitution, or elimination.

• How do we use basic prerequisites in high mathematics; i.e. rounding, divisibility rules, function notation, and exponent properties?
• How do foundational mathematical concepts (such as algebra and geometry) help us understand more advanced topics and solve complex problems?
   

• What are ways to use operations on rational expressions?
• How do you simplify radicals (including higher roots)?
• How can solving systems of equations help us solve problems with multiple variables?
• What patterns and properties emerge from divisibility rules, and how can these be used to analyze and predict the divisibility of numbers?

• Desmos: https://www.desmos.com/calculator 
• ixl: https://www.ixl.com/math/ 
• Graphing/Scientific Calculators
   

• Expressions
• Conjugates  
• Equations
• Rationalizing Denominator 
• Rounding  
• Exponents
• Function Notation
• System of equations

• Weekly quizzes (including all material from the course)
• Monthly tests (including all material from the course)
• Informal assessments through warm-ups and/or exit tickets
• Informal check of daily assignments

PA Common Core Mathematics Standards

• CC.2.2.HS.C.7 Evaluate trigonometric and inverse trigonometric expressions, using technology when appropriate.
• CC.2.3.HS.A.7 Apply trigonometric ratios to solve problems involving right triangles. 
• CC.2.2.HS.C.7 Apply radian measure of an angle and the unit circle to analyze the trigonometric functions. 

College Board Standards (AP Precalculus)

• 1.B  Express functions, equations, or expressions in analytically equivalent forms that are useful in a given mathematical or applied context. 
• 3.B Apply numerical results in a given mathematical or applied context. 

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• Rotation of angles in the Cartesian plane (positive/negative/amount of rotation).
• Conversion of angles between degree mode and radian mode.
• The side relationships in special right triangles.
• Angles of elevation and depression in an application problem.
• Evaluating a trigonometric function of any angle.
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• Evaluate any angle in the Cartesian plane using trigonometric functions.

• Find coterminal angles.
• Convert from radians to degrees and vice versa.
• Convert from degrees to degrees, minutes, and seconds and vice versa.
• Use proper vocabulary to describe and apply concepts related to the Cartesian plane.
• Determine the sine, cosine, tangent, cosecant, secant, and cotangent of any angle using the unit circle. 
• Solve a right triangle for missing side lengths and/or angles, including applications of right triangle trigonometry.

• How do we extend the concept of  right triangle trigonometry from geometry to trigonometric functions in circular form?
• What are the similarities and differences between radians and degrees, and why are both necessary in trigonometry?

• How do we convert from degrees to radians?
• What does it mean to solve a triangle?
• How do we evaluate trigonometric functions for all types of angles?
• How do we determine the sign of a trigonometric function based on its quadrant location?
   

• Desmos: https://www.desmos.com/calculator 
• IXL: https://www.ixl.com/math/ 
• Graphing/Scientific Calculators
• Teacher created notes and assignments
   

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• Sine
• Cosine
• Tangent
• Cosecant
• Secant
• Cotangent
• Terminal side
• Radian
• Reference angle
• Solving a right triangle
• Standard position
• Degrees
• Minutes
• Seconds (DMS)
• Angle of elevation
• Angle of depression
• Coterminal angles
• Quadrantal angle
• Initial side
• Unit circle
• Inverse trig functions
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• Weekly quizzes (including all material from the course)
• Monthly tests (including all material from the course)
• Informal assessments through warm-ups and/or exit tickets
• Informal check of daily assignments

PA Common Core Mathematics Standards

• CC.2.2.HS.C.8 Choose trigonometric functions to model periodic phenomena and describe the properties of the graphs. 
• CC.2.2.HS.C.2  Graph and analyze functions and use their properties to make connections between the different representations. 

College Board Standards (AP Precalculus)

• 2.A Identify information from graphical, numerical, analytical, and verbal representations to answer a question or construct a model, with and without technology.
• 2.B Construct equivalent graphical, numerical, analytical, and verbal representations of functions that are useful in a given mathematical or applied context, with and without technology. 

• Properties of sine and cosine equations from a graph.
• Properties of sine and cosine equations from an equation.
• Applications of sine and cosine graphs.

• Use the properties of sine and cosine to graph and write an equation that models a real world situation.

• Identify amplitude, period, phase shift and vertical shift of a trigonometric function.
• Graph the six trigonometric functions.
• Model periodic behavior with sinusoids.

• Using the basic concepts of sine and cosine graphs, how can we find an equation that will allow us to determine the output of the function at any time?

• How and why do we graph trigonometric functions?
• How does the amplitude and period affect the graph of a trigonometric function?
• In which ways does phase shift and vertical shift translate the graph of a trigonometric function?
• How can we use the graphs of trigonometric functions to model periodic phenomena in the real world?
   

• Desmos: https://www.desmos.com/calculator 
• IXL: https://www.ixl.com/math/ 
• Graphing/Scientific Calculators
• Teacher created notes and assignments
   

• Domain
• Phase Shift
• Cosine
• Range
• Vertical Shift
• Asymptotes
• Amplitude
• Cycle
• Frequency
• Period
• Sine

• Weekly quizzes (including all material from the course)
• Monthly tests (including all material from the course)
• Informal assessments through warm-ups and/or exit tickets
• Informal check of daily assignments

PA Common Core Mathematics Standards

• CC.2.1.HS.F.2  Apply properties of rational and irrational numbers to solve real world or mathematical problems. 
• CC.2.2.HS.D.4 Understand the relationship between zeros and factors of polynomials to make generalizations about functions and their graphs. 
• CC.2.2.HS.D.6 Extend the knowledge of rational functions to rewrite in equivalent forms. 

• Factor binomials and trinomials.
• Simplify rational expressions.
• Solve quadratic equations by factoring and using square roots.

• Perform operations to simplify expressions and solve quadratic equations.

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• Solve quadratic equations by factoring.
• Solve quadratic equations using square roots.
• Simplify rational expressions by factoring first.
• Perform operations on rational expressions by factoring first.
• Simplify complex fractions.
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• How can we use algebraic concepts and extend them to higher mathematical concepts?

• What are the various ways to solve a quadratic function?
• When is the appropriate time to simplify a rational expression?

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• Desmos: https://www.desmos.com/calculator 
• IXL: https://www.ixl.com/math/ 
• Graphing/Scientific Calculators
• Teacher created notes and assignments
•  

• Factoring
• Monomial
• Rational expressions Complex fractions
• Trinomial
• Greatest Common Factor
• Binomial

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• Weekly quizzes (including all material from the course)
• Monthly tests (including all material from the course)
• Informal assessments through warm-ups and/or exit tickets
• Informal check of daily assignments
   
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PA Common Core Mathematics Standards

• CC.2.2.HS.C.9  Prove the Pythagorean identity and use it to calculate trigonometric ratios. 
• CC.2.2.HS.D.2  Write expressions in equivalent forms to solve problems. 
• CC.2.2.HS.D.9  Use reasoning to solve equations and justify the solution method. 

College Board Standards (AP Precalculus)

• 1.A  Solve equations and inequalities represented analytically, with and without technology. 
• 1.B  Express functions, equations, or expressions in analytically equivalent forms that are useful in a given mathematical or applied context. 
• 3.B  Apply numerical results in a given mathematical or applied context. 

• Pythagorean Identities.
• Basic Identities (Reciprocal, Even/Odd, Cofunction, Tangent/Cotangent).
• Trigonometric Proofs.
• Double Angle Identities.
• Sum and Difference Identities.
• Trigonometric solutions to equations.

• Manipulate trigonometric expressions, prove identities, solve equations, and apply trigonometry to various problems in mathematics.

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• Use Pythagorean Identities to evaluate trigonometric functions of any angle.
• Simplify trigonometric expressions using the basic trigonometric identities, using algebra as needed.
• Solve basic trigonometric equations.
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• How can we use trigonometric identities to simplify using algebraic techniques and solve trigonometric expressions and equations?
• How can understanding and applying trigonometric identities help to simplify complex expressions and solve trigonometric equations?
   
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• How do we use the basic identities to simplify and prove trigonometric identities?
• What is the importance of simplifying trigonometric expressions to a given solution?
• How do we use our previous experience to solve multi-step trigonometric equations?
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• Desmos: https://www.desmos.com/calculator 
• IXL: https://www.ixl.com/math/ 
• Graphing/Scientific Calculators
• Teacher created notes and assignments

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• Tangent/Cotangent Identities
• Cofunction Identities
• Reciprocal Identities
• Double Angle Identities
• Pythagorean Identities
• Sum and Difference Identities
• Even/Odd Identities
• Half Angle Identities
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• Weekly quizzes (including all material from the course)
• Monthly tests (including all material from the course)
• Informal assessments through warm-ups and/or exit tickets
• Informal check of daily assignments

PA Common Core Mathematics Standards

• CC.2.1.HS.F.6.B  Add, subtract, multiply, and divide complex numbers in standard form.
• CC.2.1.HS.F.6.A  Represent complex numbers in polar form.
• CC.2.1.HS.F.6.C  Multiply, divide, and find powers and roots of complex numbers in polar form.

College Board Standards (AP Precalculus)

• 2.B  Construct equivalent graphical, numerical, analytical, and verbal representations of functions that are useful in a given mathematical or applied context, with and without technology. 
• 3.A  Describe the characteristics of a function with varying levels of precision, depending on the function representation and available mathematical tools. 

• Complex numbers in standard form.
• Trigonometric form of a complex number.
• Polar coordinates.
• Rectangular coordinates.
• Operations with complex numbers, including DeMoivre’s Theorem.
• Characteristics of polar graphs, specifically rose and limaçon curves.

• Understand the mathematical mechanics of polar coordinates and graphs.
• Manipulate complex numbers in trigonometric (polar) form, solve equations involving complex numbers, and understand their geometric interpretations. The ability to switch between rectangular and polar forms is crucial for solving real-world problems in various fields such as engineering, physics, and applied mathematics.

• Perform operations on complex numbers, using complex conjugates as needed.
• Represent complex numbers in trigonometric (or polar) form.
• Use trigonometric form to perform operations on complex numbers, including DeMoivre’s Theorem.
• Convert between rectangular and polar coordinates.
• Convert between rectangular and polar equations.
• Explain characteristics (ie petal length and type of symmetry) of rose and limaçon curves.
• Graph rose and limaçon curves, both of hand and using technology.

• How can polar coordinates and polar graphs be used to represent and analyze shapes and relationships that are difficult to express in rectangular coordinates?
• How can complex numbers be used to extend our understanding of numbers?

• What are complex numbers and how do they coexist with our number system?
• How do you convert from rectangular coordinates to polar coordinates?
• What is the relationship between the rectangular coordinate system and the polar coordinate system?
• Can you classify common shapes that arise from plotting certain types of polar equations?
• What is the procedure for raising a complex number to a power use DeMoivre's Theorem?
• How can we represent complex numbers in trigonometric form, and how does this form simplify operations like multiplication, division, and finding powers or roots?
   

• Desmos: https://www.desmos.com/calculator 
• IXL: https://www.ixl.com/math/ 
• Graphing/Scientific Calculators
• Teacher created notes and assignments
   

• Polar coordinate
• DeMoivre's Theorem
• Modulus
• Rectangular coordinate
• Complex Number
• Argument 
• Rose curve
• Complex Conjugate
• Limaçon curve
• The Number i

• Weekly quizzes (including all material from the course)
• Monthly tests (including all material from the course)
• Informal assessments through warm-ups and/or exit tickets
• Informal check of daily assignments
   

PA Common Core Mathematics Standards

• CC.2.4.HS.A.14B  Use concepts of trigonometry to find the area of oblique triangles.
• CC.2.4.HS.A.14A  Use the law of sines and the law of cosines to solve oblique triangles.

• Law of sines, including the ambiguous case.
• Law of cosines.
• Area of oblique triangles, including Heron’s Formula.

• Using tools to solve non-right triangles, find missing sides and angles, and calculate areas using the Law of Sines, the Law of Cosines, and appropriate geometric formulas. 
• The ability to apply these methods effectively is essential for tackling complex problems in various applied fields.

• Classify an oblique triangle as one of the four cases.
• Solve AAS, ASA, SAS, and SSS triangle cases.
• Solve ambiguous SSA triangle cases.
• Calculate the area of an oblique triangle, using Heron’s Formula as needed.

• How can the Law of Sines and the Law of Cosines be used to solve real-world problems involving non-right triangles?
• How can we calculate the area of oblique triangles, and in what real-world situations is this useful?
   

• Can you classify an oblique triangle as one of the four cases?
• Why do we need a different method to find the area of oblique triangles?

Desmos: https://www.desmos.com/calculator 
IXL: https://www.ixl.com/math/ 
Graphing/Scientific Calculators
Teacher created notes and assignments
 

• Law of Sines
• Heron’s Formula
• Included angle
• Law of Cosines
• Semiperimeter
• Area 
• Ambiguous case
• Oblique Triangle
• AAS, ASA, SSA, SAS, SSS

• Weekly quizzes (including all material from the course)
• Monthly tests (including all material from the course)
• Informal assessments through warm-ups and/or exit tickets
• Informal check of daily assignments

PA Common Core Mathematics Standards

• CC.2.3.HS.A.10A  Identify properties of the conic sections, including circle, parabolas, ellipses, and hyperbolas.
• CC.2.3.HS.A.10B  Graph circles, parabolas, ellipses, and hyperbolas, using technology when appropriate.

College Board Standards (AP Precalculus)

• 1.B  Express functions, equations, or expressions in analytically equivalent forms that are useful in a given mathematical or applied context. 
• 2.A  Identify information from graphical, numerical, analytical, and verbal representations to answer a question or construct a model, with and without technology. 
• 2.B  Construct equivalent graphical, numerical, analytical, and verbal representations of functions that are useful in a given mathematical or applied context, with and without technology. 

•  Completing the square.
• Center and radius of a circle from an equation.
• Characteristics of an ellipse, hyperbola, and parabola.

• A comprehensive understanding of the properties, equations, and applications of conic sections.

• Graph a circle, parabola, ellipse, and hyperbola using technology when needed.
• Complete the square to put an equation in standard form.
• Find the important characteristics (see the vocabulary) of each conic section.
• Identify the type of conic section from the equation whether or not it is in standard form.
• Change an equation of a conic section into standard form by completing the square.
• Write the equation of a conic section given information regarding its characteristics.

• How can we identify and graph conic sections, and what do their equations reveal about their geometric properties?
• How do the properties of conic sections (circles, ellipses, parabolas, and hyperbolas) help us model and understand real-world phenomena?
   

• How can you identify which conic section is represented by looking at an equation?
• What is the difference between the focus of a parabola, ellipse, and hyperbola?
• How can you find the asymptotes for a hyperbola?
• What are some applications of conic sections?
   

• Desmos: https://www.desmos.com/calculator 
• IXL: https://www.ixl.com/math/ 
• Graphing/Scientific Calculators
• Teacher created notes and assignments
   

• Parabola
• Directrix
• Hyperbola
• Semiconjugate axis
• Focus (foci)
• Ellipse
• Transverse axis
• Asymptote
• Focal length
• Major/minor axis
• Conjugate axis
• Focal width
• Semimajor/semiminor axis
• Semitransverse axis

• Weekly quizzes (including all material from the course)
• Monthly tests (including all material from the course)
• Informal assessments through warm-ups and/or exit tickets
• Informal check of daily assignments

PA Common Core Mathematics Standards

• CC.2.2.HS.D.8A  Convert from exponential to logarithmic form and vice versa.
• CC.2.2.HS.D.8B  Use properties of logarithms to rewrite expressions and to solve exponential and logarithmic equations.
• CC.2.2.HS.D.8C  Evaluate exponential and logarithmic expressions, using technology when appropriate.
• CC.2.2.HS.C.2B  Perform transformations by composition, including polynomial, exponential, logarithmic, and trigonometric functions.
• CC.2.2.HS.C.4B  Perform transformations by finding inverses, including polynomial, exponential, logarithmic, and trigonometric functions.
• CC.2.2.HS.C.2C  Graph logarithmic and exponential functions using technology when appropriate. 

College Board Standards (AP Precalculus)

• 1.B  Express functions, equations, or expressions in analytically equivalent forms that are useful in a given mathematical or applied context. 
• 2.B  Construct equivalent graphical, numerical, analytical, and verbal representations of functions that are useful in a given mathematical or applied context, with and without technology. 
• 3.A  Describe the characteristics of a function with varying levels of precision, depending on the function representation and available mathematical tools. 
• 3.B  Apply numerical results in a given mathematical or applied context. 

• Graphs of exponential and logarithmic functions, and how they relate to one another.
• Properties of logarithms.
• Methods to solve exponential and logarithmic equations.
• Mathematical finance and applications.

• Exponential and logarithmic functions are essential for solving real-world problems involving growth, decay, and various natural processes.
• Understanding the inverse relationship and key properties of these functions is crucial for algebra, calculus, and applications in science and engineering.

• Convert from exponential to logarithmic form.
• Graph exponential and logarithmic functions.
• Solve exponential and logarithmic equations.
• Solve application problems using exponents and logarithms.
• Differentiate between present value and future value.
   

• How can we solve real-world problems involving exponential functions, and what techniques are useful in those cases?
• What are the key properties of logarithms (such as the product, quotient, and power rules), and how are they used to simplify expressions and solve equations?
• What role do exponential and logarithmic functions play in modeling real-world phenomena like population growth, radioactive decay, or financial investments?
   

• How and why do we use exponential and logarithmic functions?
• What is the difference between exponential and logistic functions?
• How do we use logarithms to solve exponential equations?
• What are some applications of exponential and logarithmic functions?
• What is the difference between future value and present value?
   

• Desmos: https://www.desmos.com/calculator 
• IXL: https://www.ixl.com/math/ 
• Graphing/Scientific Calculators
• Teacher created notes and assignments
   

• Exponential function
• The number “e”
• Common logarithm
• Present Value
• Asymptote
• Compound continuously
• Natural logarithm
• Exponential growth 
• Exponential decay
• Depreciation
• Properties of logarithms
• Compound interest
• Logarithm
• Future Value

• Weekly quizzes (including all material from the course)
• Monthly tests (including all material from the course)
• Informal assessments through warm-ups and/or exit tickets
• Informal check of daily assignments