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Basic Trigonometric Identities

Basic Trigonometric Identities. T, 3.1: Students prove that this identity is equivalent to the Pythagorean theorem (i.e., students can prove this identity by using the Pythagorean theorem and, conversely, they can prove the Pythagorean theorem as a consequence of this identity).

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Basic Trigonometric Identities

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  1. Basic Trigonometric Identities T, 3.1: Students prove that this identity is equivalent to the Pythagorean theorem (i.e., students can prove this identity by using the Pythagorean theorem and, conversely, they can prove the Pythagorean theorem as a consequence of this identity). T,3.2: Students prove other trigonometric identities and simplify others by using the identity cos2 (x) + sin2 (x) = 1. For example, students use this identity to prove that sec2 (x) = tan2 (x) + 1.

  2. Basic Trigonometric Identities Objectives Key Words Identity Trigonometric Identity Reciprocal Identities Quotient Identity Pythagorean Identities Symmetry Identities Opposite-Angle Identities • Derive the Pythagorean Trigonometry Identity • Identify and use reciprocal identities, quotient identities, Pythagorean identities, symmetry identities, and opposite-angle identities.

  3. Deriving the Pythagorean Trigonometric Identity Pick a coordinate point on the unit circle. Draw the appropriate triangle ABC. Write the Pythagorean theorem for this triangle with sides a, b, and c. Divide by c-squared. Replace the appropriate values with respect to the trigonometric function sine, and cosine. Finish with the last touch ups. Study this exact proof. You will be tested on it. The exact same question and solution.

  4. Example 1 Prove that sec x cot x = sin x is not a trigonometric identity by producing a counterexample.

  5. Reciprocal Identities

  6. Quotient Identities

  7. Pythagorean Identities

  8. Example 2 Use the given information to find the trigonometric value. If cot =6/5 , find tan . If sec =5/4 , find cot .

  9. 02-15-2012Derive the Pythagorean Trigonometric Identity (10 Minutes) • Draw the appropriate triangle ABC. • Write the Pythagorean theorem for this triangle with sides a, b, and c. • Divide by c-squared. • Apply appropriate exponential property • Replace the appropriate values with respect to the trigonometric function sine, and cosine. • Finish with the last touch ups.

  10. Symmetry Diagram

  11. Symmetry Identities

  12. Example 3 Express each value as a trigonometric function of an angle in Quadrant I. a. sin 765° b. sin (-19π/3) cos935° cot 11π/4

  13. Example 3 Express each value as a trigonometric function of an angle in Quadrant I. a. sin 765° b. sin (-19π/3) cos935° cot 11π/4

  14. Opposite-Angle Identities

  15. Example 4 Simplify cosx cot x + sin x.

  16. Example 5

  17. Conclusions Summary Assignment Pg427 #(18-43 ODD, 44-53 ALL) • Write an expression that contains all six trigonometric functions and is equal to 3. sin2x+cos2x+sec2x-tan2x+csc2x-cot2x

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