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Comprehensive Guide to Sum and Difference Identities for Tangent in Trigonometry

This resource delves into the sum and difference identities for tangent, providing essential techniques for verifying identities and simplifying trigonometric expressions. We will explore the derivation of tangent identities using cosine sum and difference identities, apply fundamental identities, and demonstrate how to combine fractions and manipulate terms. Examples are included for clarity, showcasing how to find exact values of sine and tangent functions as well as the verification process for identities. Ideal for students looking to deepen their understanding of trigonometric identities.

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Comprehensive Guide to Sum and Difference Identities for Tangent in Trigonometry

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Presentation Transcript

  1. Warm Up Verify that the equation is an identity.

  2. Trig Game Plan Date: 12/09/13

  3. Sum and Difference Identities for Tangent We can use the cosine sum and difference identities to derive similar identities for sine and tangent. Fundamental identity Sum identities Multiply numerator and denominator by 1.

  4. Sum and Difference Identities for Tangent Multiply. Simplify. Fundamental identity

  5. Sum and Difference Identities for Tangent Replace B with –B and use the fact that tan(–B) to obtain the identity for the tangent of the difference of two angles.

  6. Tangent of a Sum or Difference tan ( A ± B ) = tan A ± tan B 1 tan A tan B ± -

  7. WRITING FUNCTIONS AS EXPRESSIONS INVOLVING FUNCTIONS OF θ Example Write each function as an expression involving functions of θ. (a) (b)

  8. FINDING EXACT SINE AND TANGENT FUNCTION VALUES Example 1(d) Find the exact value of

  9. FINDING EXACT SINE AND TANGENT FUNCTION VALUES Example 1(e) 5.4 Example 1 Finding Exact Sine and Tangent Function Values (cont.) (b)

  10. FINDING EXACT SINE AND TANGENT FUNCTION VALUES Example 1(g) 5.4 Example 1 Finding Exact Sine and Tangent Function Values (cont.) (c)

  11. VERIFYING AN IDENTITY USING SUM AND DIFFERENCE IDENTITIES Example 4a Combine the fractions. Verify that the equation is an identity.

  12. VERIFYING AN IDENTITY USING SUM AND DIFFERENCE IDENTITIES Example 4a 5.4 Example 4 Verifying an Identity Using Sum and Difference Identities (cont.) Expand the terms. Combine terms. Factor. So, the identity is verified.

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