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