Lesson 6 | Trigonometric Identities and Equations | 11th Grade Mathematics

Objective

Solve linear trigonometric equations.

Common Core Standards

Core Standards

The core standards covered in this lesson

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.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 solving a linear trigonometric equation is like solving a linear equation.
Describe how solving trig equations is similar to finding the solution(s) to a system of equations, graphically and algebraically.
Describe the number of solutions in trigonometric equations within a specified domain. Explain the difference between solutions with a function and its inverse with restricted domain.
Identify, based on the sign of the solution, the number and value of the solutions within the specified domain of the equation using the unit circle.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

This YouTube video, LT 8 - Solving a Basic Trig Equation, is pretty concise, informative, and clear.

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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

We did these problems a few lessons ago when we were solving for angles in right triangles.

$${\mathrm{sin}x={1\over2}}$$

$${\mathrm{cos}^{-1}\left(\sqrt2\over2\right)=x}$$

Which has infinite solutions and which has only one solution? Describe your reasoning.

Guiding Questions

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Problem 2

Find all the solutions within the domain of $${0 \leq x \leq 2\pi}$$ for the following equation.

$${2\mathrm{cos}(x)-1=0}$$

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

Problem 1

Solve each trigonometric equation such that $${0\leq x\leq2\pi}$$. Give answers in exact form.

$${2\mathrm{sin}(x)+\sqrt3=0}$$

References

EngageNY Mathematics Precalculus and Advanced Topics > Module 4 > Topic C > Lesson 12 — Exit Ticket, Question #2a

Precalculus and Advanced Topics > Module 4 > Topic C > Lesson 12 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Problem 2

Solve the trigonometric equation such that $${0\leq x\leq2\pi}$$. Round to three decimal places.

$${\sqrt5\mathrm{cos}(x)-2=0}$$

References

EngageNY Mathematics Precalculus and Advanced Topics > Module 4 > Topic C > Lesson 12 — Exit Ticket, Question #3

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Explain what the solution$${{\pi\over3}+2\pi k}$$ means. Can you determine the domain that this solution is under? Why or why not?
Include problems that ask students to solve within a specified domain and to find all real solutions.
Include problems that can be solved with the unit circle and problems requiring a calculator.

The Secondary Classroom Can Be Fun Too Solving Basic Trigonometric Equations Task Cards/Speed Dating
Math Mistakes Inverse Trig Functions — (Error analysis)
EngageNY Mathematics Precalculus and Advanced Topics > Module 4 > Topic C > Lesson 12 — Problem Set

Lesson 5

Lesson 7

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

Foundational Standards

Trigonometric Functions

Trigonometric Functions

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Target Task

Problem 1

References

Problem 2

References

Additional Practice

Lesson Map

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