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Mastering Trigonometric Identities: Proofs and Applications

Explore the fundamental and complex aspects of trigonometric identities with geometry and algebraic principles, including sum and difference identities, verifying equations, and finding exact cosine values.

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Mastering Trigonometric Identities: Proofs and 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. Sum and Difference Identities for Cosine 5.3 Difference Identity for Cosine ▪ Sum Identity for Cosine ▪ Cofunction Identities ▪ Applying the Sum and Difference Identities ▪ Verifying an Identity

  4. Cosine of a Sum or Difference

  5. To prove this we need….. • A few ideas from geometry and algebra. • 1. 2.

  6. Difference Identity for Cosine (Take a look at the diagram on the sheet that I gave you.) • Point Q is on the unit circle, so the coordinates of Q are (cos B, sin B). • The coordinates of S are (cos A, sin A). • The coordinates of R are (cos(A – B), sin (A – B)).

  7. Difference Identity for Cosine • Since the central angles SOQ and POR are equal, PR = SQ. • On the sheet that you have, write what PR = SQ in terms of the distance formula.

  8. Sum Identity for Cosine • To find a similar expression for cos(A + B) rewrite A + B as A– (–B) and use the identity for cos(A–B). • Cosine difference identity • Negative-angle identities

  9. FINDING EXACT COSINE FUNCTION VALUES (USING THE COSINE SUM AND DIFFERENCE IDENTITIES) Example 1(a) Find the exact value of cos 15. Think of the angles in degrees in quadrant one that we can find the exact value of. Write them down. See if this helps you.

  10. FINDING EXACT COSINE FUNCTION VALUES Example 1(b) Find the exact value of Think of the angles in radians in quadrant one that we can find the exact value of and write them down. Translate them to a denominator of 12.

  11. FINDING EXACT COSINE FUNCTION VALUES Example 1(c) Find the exact value of cos 87cos 93 – sin 87sin 93.

  12. The same identities can be obtained for a real number domain by replacing 90 with Cofunction Identities

  13. USING COFUNCTION IDENTITIES TO FIND θ Example 2 Find one value of θ or x that satisfies each of the following. (a) cot θ = tan 25° (b) sin θ = cos (–30°)

  14. USING COFUNCTION IDENTITIES TO FIND θ (continued) Example 2 Find one value of θ or x that satisfies the following. (c)

  15. Note Because trigonometric (circular) functions are periodic, the solutions in Example 2 are not unique. We give only one of infinitely many possiblities.

  16. Applying the Sum and Difference Identities • If either angle A or B in the identities for cos(A + B) and cos(A–B) is a quadrantal angle, then the identity allows us to write the expression in terms of a single function of A or B.

  17. REDUCING cos (A – B) TO A FUNCTION OF A SINGLE VARIABLE Example 3 Write cos(180° – θ) as a trigonometric function of θ alone.

  18. FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND t Example 4 • Suppose that and both s and t are in quadrant II. Find cos(s + t).

  19. Example 6 VERIFYING AN IDENTITY • Verify that the following equation is an identity. Work with the more complicated left side.

  20. Example 6 VERIFYING AN IDENTITY (continued) The left side is identical to the right side, so the given equation is an identity.

  21. Assignment • Page 212 #1-6, all, • 7-15 odd

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