Lesson 9 | Trigonometric Identities and Equations | 11th Grade Mathematics

Objective

Solve quadratic trigonometric equations.

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

Foundational Standards

The foundational standards covered in this lesson

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.
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.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Describe how the zero product property is applied to trigonometric functions.
Use GCF factoring, trinomial factoring, and difference of two squares to solve quadratic trigonometric equations.
Determine the general solutions and exact solutions within a specific domain.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

The first three problems are grouped together even though they employ different quadratic strategies. This is because when you solve each of these, the role of the solution is readily apparent AND they all have solutions (which is not always the case with solving in this way, but in these examples, it is).

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Solve for the general solution:

$${3\mathrm{tan}^2x-3=0}$$

$${2\mathrm{cos}^2(x)-\sqrt3 \mathrm{cos}x=0}$$

$${2\mathrm{cos}^2x+\mathrm{cos}x-1=0}$$

Guiding Questions

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Problem 2

Solve for the domain $${0 \leq \theta \leq 2\pi}$$:

$${2\mathrm{sin}^2\theta+\mathrm{cos}\theta\mathrm{sin}^2\theta=0}$$

Guiding Questions

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Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Solve the following equations for all values $${0\leq x\leq2\pi}$$.

$${3\mathrm{sin}\theta+2\mathrm{sin}^2\theta=-1}$$

$${\mathrm{sin}\theta\mathrm{cos}\theta-{1\over2}\mathrm{sin}\theta=0}$$

Lesson 8

Lesson 10

Lesson Map

Topic A: Basic Trigonometric Identities and Equivalent Expressions

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

F.TF.C.8

Derive and use the Pythagorean identity to write equivalent expressions.

F.TF.C.8

Verify trigonometric identities using Pythagorean and reciprocal identities.

F.TF.C.8

Topic B: Solve Trigonometric Equations

Find angle measures using inverse trig functions in right triangles.

F.TF.B.6 F.TF.B.7

Analyze inverse trigonometric functions graphically.

F.BF.B.4.D F.IF.C.7.E F.TF.B.6 F.TF.B.7

Solve linear trigonometric equations.

F.TF.B.7

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

F.TF.B.7

Use inverse trigonometric functions to solve contextual problems.

F.TF.B.7

Solve quadratic trigonometric equations.

F.TF.B.6 F.TF.B.7

Solve trigonometric equations using identities.

F.TF.B.7 F.TF.C.8

Topic C: Advanced Identities and Solving Trigonometric Equations

Evaluate expressions using sum and difference formulas.

F.TF.C.9

Solve equations and prove identities using sum and difference formulas.

F.TF.C.9

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

F.TF.C.9

Use trigonometric identities to analyze graphs of functions.

F.TF.C.8 F.TF.C.9

Topic D: Applications and Extensions of Trigonometric Functions

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

G.SRT.D.10

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

G.SRT.D.10

Trigonometric Identities and Equations

Objective

Common Core Standards

Core Standards

Trigonometric Functions

Trigonometric Functions

Foundational Standards

Interpreting Functions

Trigonometric Functions

Trigonometric Functions

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Target Task

Lesson Map

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