Lesson 12 | Trigonometric Identities and Equations | 11th Grade Mathematics

Objective

Solve equations and prove identities using sum and difference formulas.

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.

Criteria for Success

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

Use the sum and difference formulas to verify cofunction identities.
Use sum and difference formulas to rewrite equations.
Solve trigonometric equations using sum and difference formulas.

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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

Prove the cofunction identity of:

$${\mathrm{cos}\left({\pi\over2}-x\right)=\mathrm{sin}x}$$

Using the sum or difference formula.

Guiding Questions

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Problem 2

Solve:

$${\mathrm{sin}\left(x+{\pi\over4}\right)+1=\mathrm{sin}\left({\pi\over4}-x\right)}$$ for $${0 \leq x \leq 2\pi}$$.

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

Find the general solution for:

$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$

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.

Prove other cofunction identities using the sum and difference formulas.
Prove periodic identities using the sum and difference formulas.
Include error analysis problems.
Ensure that students check solutions with a calculator.

CK-12 Solving Trigonometric Equations Using Sum and Difference Formulas — (This is a great source of problems and an answer key.)

Lesson 11

Lesson 13

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

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Target Task

Additional Practice

Lesson Map

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