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Understanding Double-Angle Identities & Their Applications

Learn about the fundamental trigonometric identities, verifying trigonometric identities, sum and difference identities, double-angle identities, and their practical applications. Additionally, explore how to find function values and simplify expressions using these identities.

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Understanding Double-Angle Identities & Their Applications

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  1. 5 Trigonometric Identities

  2. 5 Trigonometric Identities • 5.1 Fundamental Identities • 5.2 Verifying Trigonometric Identities • 5.3 Sum and Difference Identities for Cosine • 5.4 Sum and Difference Identities for Sine and Tangent • 5.5 Double-Angle Identities • 5.6 Half-Angle Identities

  3. Double-Angle Identities 5.5 Double-Angle Identities ▪ An Application ▪ Product-to-Sum and Sum-to-Product Identities

  4. Double-Angle Identities • We can use the cosine sum identity to derive double-angle identities for cosine. • Cosine sum identity

  5. Double-Angle Identities • There are two alternate forms of this identity.

  6. Double-Angle Identities • We can use the sine sum identity to derive a double-angle identity for sine. • Sine sum identity

  7. Double-Angle Identities • We can use the tangent sum identity to derive a double-angle identity for tangent. • Tangent sum identity

  8. Double-Angle Identities

  9. FINDING FUNCTION VALUES OF 2θ GIVEN INFORMATION ABOUT θ Example 1 Given and sin θ < 0, find sin 2θ, cos 2θ, and tan 2θ. Now use the double-angle identity for sine.

  10. FINDING FUNCTION VALUES OF θ GIVEN INFORMATION ABOUT 2θ Example 2 Find the values of the six trigonometric functions of θif

  11. Example 3 VERIFYING A DOUBLE-ANGLE IDENTITY Verify that is an identity. Quotient identity Double-angle identity

  12. Example 4 SIMPLIFYING EXPRESSIONS USING DOUBLE-ANGLE IDENTITIES Simplify each expression. cos2A = cos2A – sin2A Multiply by 1.

  13. Example 5 DERIVING A MULTIPLE-ANGLE IDENTITY Write sin 3x in terms of sin x. Sine sum identity Double-angle identities

  14. Assignment • Page 231 #3-51 M3

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