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Trigonometry Lecture Notes Section 1.4 Page 1 of 8 Section 1.4: Using the Definitions of the Trigonometric Functions Big Idea: Using the definitions of the trig functions, we can state relationships between the functions (even though we don’t know how to compute them yet), and also make some basic quantitative predictions about the output of the functions. Big Skill: You should be able to use the identities below to compute quantitative relationships between the trig functions. The Six Trigonometric Functions (Section 1.3) r x2 y 2 y (sine) r y tan (tangent) x r sec (secant) x sin x (cosine) r x cot (cotangent) y r csc (cosecant) y cos The Reciprocal Identities With the above definitions of the trigonometric functions of an angle in standard position, we can begin to see relationships between the functions. One category of these relationships is called the reciprocal identities. 1 csc 1 csc sin sin The Reciprocal Identities (Section 1.4) 1 1 cos tan sec cot 1 1 sec cot cos tan Note: we can re-state these reciprocal identities as: sin csc 1 cos sec 1 tan cot 1 Trigonometry Lecture Notes Section 1.4 Page 2 of 8 Practice: 1. Prove the reciprocal identities using the definitions of the trig functions. 2. Find tan given that cot 4 . 3. Find sec given that cos 4. Find sin given that csc 12 . 5 2 . 20 Trigonometry Lecture Notes Section 1.4 Page 3 of 8 Signs and Ranges of the Trigonometric Functions Combining the signs of x and y in the four quadrants with the definitions of the trig functions allows us to state the sign of the trig functions for any angle in a given quadrant. Practice: 5. Label the four quadrants below and the signs of x and y in each quadrant. Then use that information to fill in the table below showing the sign of each trig function for any given angle in a given quadrant. Signs of the Trigonometric Functions for Angles in the Various Quadrants Quadrant Sign of Sign of Sign of Sign of Sign of Sign of y x y r x r of sin cos tan sec cot csc r r x x y y I II III IV Trigonometry Lecture Notes Section 1.4 Practice: 6. State the signs of the six trig functions if 97 . 7. State the signs of the six trig functions if 260 . 8. State the signs of the six trig functions if 80 . 9. State which quadrant is in if sin 0 and cos 0 . 10. State which quadrant is in if cot 0 and cos 0 . Page 4 of 8 Trigonometry Lecture Notes Section 1.4 Page 5 of 8 Range of the Six Trigonometric Functions Practice: 11. Draw eight angles below (the four quadrantal angles and four angles in between) and note the relationships between the values for r, x, and y. Use that information to fill in the table below summarizing the range of the trig functions. Trig Function y r x cos r y tan x x cot y r sec x r csc y sin Range of the Six Trigonometric Functions Range stated in interval Range stated in set builder notation notation Trigonometry Lecture Notes Section 1.4 Page 6 of 8 Practice: 12. State whether it is possible for cos 28 . 13. State whether it is possible for cot 129 . 14. State whether it is possible for sec 0.5 . 15. Suppose is in quadrant III and tan 5 . Find the value of the remaining five 9 trigonometric functions. 16. Suppose is in quadrant IV and csc trigonometric functions. 2 . Find the value of the remaining five 3 Trigonometry Lecture Notes Section 1.4 Page 7 of 8 The Pythagorean Identities The Pythagorean Theorem can be used to state three more relationships between the trig functions. sin cos 1 2 2 The Pythagorean Identities (Section 1.4) tan 2 1 sec 2 cot 2 1 csc 2 Practice: 17. Derive the first Pythagorean Identity from the Pythagorean Theorem. 18. Derive the second Pythagorean Identity from the Pythagorean Theorem. 19. Derive the third Pythagorean Identity from the Pythagorean Theorem. Trigonometry Lecture Notes Section 1.4 Page 8 of 8 The Quotient Identities The Quotient Identities are two more relationships between the trig functions. sin tan cos Practice: The Quotient Identities (Section 1.4) 20. Prove tan sin using the definitions of the trig functions. cos 21. Prove cot cos using the definitions of the trig functions. sin