Trigonometric Identities and Equations

Students expand on their knowledge of trigonometry, developing a foundation for calculus concepts by expanding their conception of trigonometric functions and looking at connections between functions.

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Unit Summary

Unit 7, Trigonometric Identities and Equations, builds on the previous unit on trigonometric functions to expand students’ knowledge of trigonometry. Students develop a foundation for calculus concepts by expanding their conception of trigonometric functions and looking at connections between trigonometric functions. Reasoning flexibly about trigonometric functions and seeing that expressions that look different on the surface can actually act the same on certain domains sets the stage for a study of differentiation and integration, where periodic functions have many useful properties and act as useful tools to study calculus.

Students also apply algebraic techniques to trigonometry. This part of the unit reinforces algebraic skills while also helping students to better understand trigonometric functions graphically and through the unit circle. As students move more flexibly between representations of trigonometric functions, they develop skills in seeing structure in those functions and practice looking at mathematical objects from multiple perspectives and bringing prior knowledge to bear on a new context. This type of relational thinking helps students to see the power of algebraic manipulation and structure in expressions, allowing them to work more flexibly and to see connections more readily.

Pacing: 18 instructional days (16 lessons, 1 flex day, 1 assessment day)

Assessment

This assessment accompanies Unit 7 and should be given on the suggested assessment day or after completing the unit.

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Unit Prep

Intellectual Prep

Internalization of Standards via the Unit Assessment

Take unit assessment. Annotate for:
- Standards that each question aligns to
- Purpose of each question: spiral, foundational, mastery, developing
- Strategies and representations used in daily lessons
- Relationship to Essential Questions of unit
- Lesson(s) that assessment points to

Internalization of Trajectory of Unit

Read and annotate “Unit Summary.”
Notice the progression of concepts through unit using the “Unit at a Glance.”
Do all target tasks. Annotate the target tasks for:
- Essential questions
- Connection to assessment questions
Answer the essential questions. (In the beginning, submit them to your instructional leader; toward the end, just bring them to the meeting.)

Essential Understandings

Work flexibly with trigonometric identities to rewrite expressions.
Define and use inverse trigonometric functions.
Solve trigonometric equations, including quadratic equations and equations that require u-substitution.
Apply equation-solving knowledge to modeling contexts.
Use the Law of Sines and the Law of Cosines to find missing angles and side lengths in acute triangles.

Vocabulary

Identity	Pythagorean identity
Cofunction identities	Inverse trig functions (arcsin, arccos, arctan, arcsec, arccsc, arccot)
Reciprocal identities	$${\mathrm{sin}^{-1}x}$$ notation for inverse trig functions
Negative angle identities	Linear trigonometric equations
$$u$$-substitution	Quadratic trigonometric equations
General solution	Exact solution
Double angle formula	Sum formula
Difference formula	Law of Cosines
Law of Sines

Materials

Trig Cheat Sheet by Paul Dawkins (Page 2 has all of the relevant identities for this unit. Students should memorize reciprocal identities and be able to use other identities with a reference when necessary.)

Graphing Calculator Skills

Graphing trig functions/system of trig function and linear function
Calculating inverse of trig functions on the calculator
Switching between radian and degree mode on the calculator
Finding the intersection, value, zero, minimum, maximum
Altering the window
Using zoom to graph trigonometric functions from $${-{2π}}$$ to $${2π}$$

Lesson Map

Topic A: Basic Trigonometric Identities and Equivalent Expressions

1

F.TF.C.8

Derive and verify trigonometric identities using transformations and equivalence of functions.

2

F.TF.C.8

Derive and use the Pythagorean identity to write equivalent expressions.

3

F.TF.C.8

Verify trigonometric identities using Pythagorean and reciprocal identities.

Topic B: Solve Trigonometric Equations

4

F.TF.B.6

F.TF.B.7

Find angle measures using inverse trig functions in right triangles.

5

F.IF.C.7.E

F.BF.B.4.D

F.TF.B.6

F.TF.B.7

Analyze inverse trigonometric functions graphically.

6

F.TF.B.7

Solve linear trigonometric equations.

7

F.TF.B.7

Solve linear trigonometric equations using $$u$$-substitution.

8

F.TF.B.7

Use inverse trigonometric functions to solve contextual problems.

9

F.TF.B.6

F.TF.B.7

Solve quadratic trigonometric equations.

10

F.TF.B.7

F.TF.C.8

Solve trigonometric equations using identities.

Topic C: Advanced Identities and Solving Trigonometric Equations

11

F.TF.C.9

Evaluate expressions using sum and difference formulas.

12

F.TF.C.9

Solve equations and prove identities using sum and difference formulas.

13

F.TF.C.9

Derive double angle formulas and use them to solve equations and prove identities.

14

F.TF.C.8

F.TF.C.9

Use trigonometric identities to analyze graphs of functions.

Topic D: Applications and Extensions of Trigonometric Functions

15

G.SRT.D.10

Use the Law of Sines to find missing side lengths and angle measures in acute triangles.

16

G.SRT.D.10

Find missing side lengths and angle measures using the Law of Cosines in acute triangles.

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

Building Functions

F.BF.B.4 — Find inverse functions.

Building Functions

F.BF.B.4 — Find inverse functions.
F.BF.B.4.D — Produce an invertible function from a non-invertible function by restricting the domain.

Building Functions

F.BF.B.4.D — Produce an invertible function from a non-invertible function by restricting the domain.

Creating Equations

A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Creating Equations

A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A.CED.A.4 — Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

Creating Equations

A.CED.A.4 — Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

Interpreting Functions

F.IF.C.7 — Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Interpreting Functions

F.IF.C.7 — Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
F.IF.C.7.E — Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Interpreting Functions

F.IF.C.7.E — Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Similarity, Right Triangles, and Trigonometry

G.SRT.D.10 — Prove the Laws of Sines and Cosines and use them to solve problems.

Similarity, Right Triangles, and Trigonometry

G.SRT.D.10 — Prove the Laws of Sines and Cosines and use them to solve problems.

Trigonometric Functions

F.TF.B.6 — Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

Trigonometric Functions

F.TF.B.6 — Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
F.TF.B.7 — Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

Trigonometric Functions

F.TF.B.7 — Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
F.TF.C.8 — Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

Trigonometric Functions

F.TF.C.8 — Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
F.TF.C.9 — Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Trigonometric Functions

F.TF.C.9 — Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Foundational Standards

Arithmetic with Polynomials and Rational Expressions

A.APR.D.6

Arithmetic with Polynomials and Rational Expressions

A.APR.D.6 — Rewrite simple rational expressions in different forms; write ^a(x/_b(x) in the form q(x) + ^r(x)/_b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

Building Functions

F.BF.B.3

Building Functions

F.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Congruence

G.CO.A.2

Congruence

G.CO.A.2 — Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

Creating Equations

A.CED.A.2

Creating Equations

A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A.CED.A.4

Creating Equations

A.CED.A.4 — Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

Geometry

8.G.B.6

Geometry

8.G.B.6 — Explain a proof of the Pythagorean Theorem and its converse.
8.G.B.7

Geometry

8.G.B.7 — Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.B.8

Geometry

8.G.B.8 — Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Interpreting Functions

F.IF.C.8

Interpreting Functions

F.IF.C.8 — Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
F.IF.C.8.A

Interpreting Functions

F.IF.C.8.A — Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Reasoning with Equations and Inequalities

A.REI.A.1

Reasoning with Equations and Inequalities

A.REI.A.1 — Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Seeing Structure in Expressions

A.SSE.A.2

Seeing Structure in Expressions

A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ — y⁴ as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
A.SSE.B.3

Seeing Structure in Expressions

A.SSE.B.3 — Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
A.SSE.B.3.A

Seeing Structure in Expressions

A.SSE.B.3.A — Factor a quadratic expression to reveal the zeros of the function it defines.

Similarity, Right Triangles, and Trigonometry

G.SRT.C.6

Similarity, Right Triangles, and Trigonometry

G.SRT.C.6 — Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
G.SRT.C.7

Similarity, Right Triangles, and Trigonometry

G.SRT.C.7 — Explain and use the relationship between the sine and cosine of complementary angles.
G.SRT.C.8

Similarity, Right Triangles, and Trigonometry

G.SRT.C.8 — Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Trigonometric Functions

F.TF.A.2

Trigonometric Functions

F.TF.A.2 — Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
F.TF.A.3

Trigonometric Functions

F.TF.A.3 — Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π-x, π+x, and 2π-x in terms of their values for x, where x is any real number.
F.TF.A.4

Trigonometric Functions

F.TF.A.4 — Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
F.TF.B.5

Trigonometric Functions

F.TF.B.5 — Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 — Attend to precision.
CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.