Lesson 7 | Trigonometric Identities and Equations | 11th Grade Mathematics

Objective

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

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

Substitute a variable for a more complicated expression to streamline solving equations.
Describe how the number of solutions to a trigonometric equation is influenced by the coefficient of $${\theta}$$.
Verify solutions to trigonometric equations graphically.
Determine the solutions to a trigonometric equation over a specified domain.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

These are two useful YouTube videos on u-substitution with trig.
- L 10 - Solving a Trig Equation Using U-Substitution
- L 10 - Solving a Trig Equation Using U-Substitution 2
This lesson uses u-substitution to deal with the solutions to a double and half angle identity problem. This is purposeful—you can bring these problems back later on in the unit.
It will be really valuable for you to have students practice identifying the period of the function, as they will need this to be able to find all of the solutions within a specified domain using $$u$$-substitution.

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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

Below are two trigonometric equations and the graphs of the left-hand side of each equation.

$${2\mathrm{cos}x+1=3}$$	$${2\mathrm{cos}\left({x\over2}\right)+1=3}$$

How are the equations and the graphs similar? Different?

Guiding Questions

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Problem 2

Find all of the solutions, algebraically, within the domain $${0\leq x \leq 2\pi}$$.

$${2\mathrm{sin}(2x)+2=3}$$

Guiding Questions

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Problem 3

Solve:

$${-3\mathrm{tan}\left({\pi\over2}x\right)=\sqrt3}$$

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

Write a general solution for:

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

b. $${-\mathrm{tan}(3x)=1}$$

Lesson 6

Lesson 8

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

Problem 3

Guiding Questions

Target Task

Lesson Map

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