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11. Basic Trigonometric Identities

11. Basic Trigonometric Identities. TRIGONOMETRIC IDENTITIES. An identity is an equation that is true for all defined values of a variable. We are going to use the identities to "prove" or establish other identities. Trigonometric Identities. Quotient Identities. Reciprocal Identities.

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11. Basic Trigonometric Identities

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  1. 11. Basic Trigonometric Identities

  2. TRIGONOMETRIC IDENTITIES An identity is an equation that is true for all defined values of a variable. We are going to use the identities to "prove" or establish other identities.

  3. Trigonometric Identities Quotient Identities Reciprocal Identities Pythagorean Identities sin2q + cos2q = 1 tan2q + 1 = sec2q cot2q + 1 = csc2q sin2q = 1 - cos2q tan2q = sec2q - 1 cot2q = csc2q - 1 cos2q = 1 - sin2q

  4. Two types of problems Simplifying – there will be no = sign Verifying – there will be an equal sign – you must prove that one side equals the other

  5. Strategies Change everything that is not already sin or cos into sin/cos Use Pythagorean identities to simplify things with squares Factor out common factors or difference of squares Get common denominator to add or subtract fractions Distribute if something outside of parentheses

  6. Simplifying

  7. sin x cos x Example simplify tanxcosx tanx cosx tanxcosx = sin x

  8. sec x sec x sin x sinx 1 1 1 = tan x = = x cos x cos x cos x csc x csc x sin x 1 Example simplify

  9. Example Simplify: Factor out cot x = cot x (csc2 x - 1) Use pythagorean identity = cot x (cot2 x) Simplify = cot3 x

  10. = 1 = sin2 x + cos2x = sin x (sin x) + cos x = sin2 x + (cos x) cos x cos x cos x cos x cos x cos x Example Simplify: Use quotient identity Simplify fraction with LCD Simplify numerator Use pythagorean identity Use reciprocal identity = sec x

  11. Simplify each expression.

  12. Simplify. b) a)

  13. Verifying In establishing an identity you should NOT move things from one side of the equal sign to the other. Instead – start with the more complicated side, substitute using identities you know, and simplifying until both sides match.

  14. tan(x) cos(x) = sin(x)

  15. tan2(x) = sin2(x) cos-2(x)

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