Trigonometric Identities

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Trigonometric Identities. Uses and examples.

Text of Trigonometric Identities

  • 1. Trigonometric Identities

2. Trigonometric Identity Equalities that involve trigonometric functions and are true for every single value of the occurring variables. Identities involving certain functions of one or more angles. 3. 3 Groups or Relation Reciprocal Relation Quotient Relation Pythagorean Relation 4. Reciprocal Relation The inverse trigonometric functions are partial inverse functions for the trigonometric functions. 5. tan = y x and cot = x y therefore, tan and cot are reciprocals of each other. The same thing can be said about sin and csc as well as cos and sec. = = = 6. Since the product of a number and its reciprocal equals 1, these relations may also be written as: tancot=1 cossec=1 sincsc=1 7. Quotient Relation 8. Simplifying, sin cos = y x . But = y x . So by transivity; = 9. Since cot is the reciprocal of tan the quotient can be derived to get = 10. Pythagorean Relation The basic relationship between the sine and the cosine is the Pythagorean trigonometric identity: where cos2 means (cos())2 and sin2 means (sin())2. This can be viewed as a version of the Pythagorean theorem, and follows from the equation x2 + y2 = 1for the unit circle. 11. By Pythagorean Theorem, 2 + 2 = 2 . Dividing both members by r results to x2 r2 + y2 2 = 1. Since = x r and = y r , then, cos + sin=1 12. Dividing both members or + = by x you get; 1 + tan = sec 13. dividing by y, you get; cot + 1 = csc 14. Activity 15. A. Fill in the blanks to complete the table. The Fundamental Trigonometric Identities and Their Alternate Forms sincsc = 1 1. = 1 csc 2. = 1 cos = 1 sec tancot = 1 = 1 tan 3. 4. = 1 tan 5. = sin cos 6. 7. sin + cos = 1 8. cos = 1 - sin 9. tan = sec - 1 sec - tan = 1 1 + cot = csc cot = csc - 1 10. 16. B. Use the fundamental identities to find the values of the other trigonometric functions. 1. tancot = ___________ 2. csc = ____________ 3. sin cos = ___________ 4. cos = ____________ 5. sin = ___________ 17. Assignment What are the terminologies used in the graphs of trigonometric function? Define each. Reference: Trigonometry pages 141-142