Lesson 13 | Trigonometric Identities and Equations | 11th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

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

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

Derive the double angle formulas from the sum formulas.
Explain that doubling an angle does not double the value of the sine, cosine, or tangent.
Prove identities using double angle formulas.
Solve equations using double angle formulas.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

The Georgia Standards of Excellence Curriculum Frameworks by the Georgia Department of Education that is referenced in Anchor Problem #1 and the Problem Set Guidance for this lesson is great for a lot of explanations, including the tangent guiding questions of Anchor Problem #1.

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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

Use the sum formula to determine a formula for finding the sine, cosine, and tangent of a double angle.

$${\mathrm{sin}(a+a)=}$$

$${\mathrm{cos}(a+a)=}$$

$${\mathrm{tan}(a+a)=}$$

Guiding Questions

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Problem 2

Is the following statement true or false?

“If an angle is doubled, then the sine value of the angle is doubled.”

Write an equation that describes the statement.
Explain your reasoning algebraically.
Explain your reasoning using the graph shown.

Guiding Questions

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References

Georgia Department of Education Georgia Standards of Excellence Curriculum Frameworks, GSE PreCalculus, Unit 4 — Page 39

Problem 3

Prove the following identity:

$${{{\mathrm{cos}2x}\over{\mathrm{sin}x+\mathrm{cos}x}}=\mathrm{cos}x-\mathrm{sin}x}$$

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 equation for $${0 \leq x \leq 2\pi}$$.

$${\mathrm{sin}2x\mathrm{cos}x=\mathrm{sin}x}$$

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.

Michael Corral Trigonometry — Page 81
Georgia Department of Education Georgia Standards of Excellence Curriculum Frameworks, GSE PreCalculus, Unit 4 — Page 41
Classzone.com Using Double- and Half-Angle Formulas — (Use the double-angle formulas)

Lesson 12

Lesson 14

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

References

Problem 3

Guiding Questions

Target Task

Additional Practice

Lesson Map

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