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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. Basic Trigonometric Identities Objectives • Derive the Pythagorean Trigonometry Identity • Identify and use reciprocal identities, quotient identities, Pythagorean identities, symmetry identities, and oppositeangle identities. Key Words • Identity • Trigonometric Identity • Reciprocal Identities • Quotient Identity • Pythagorean Identities • Symmetry Identities • Opposite-Angle Identities 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. Example 1 Suppose x = . 3 Prove that sec x cot x = sin x is not a trigonometric identity by producing a counterexample. ? sec x cot x sin x ? sec cot sin 3 3 3 ? 3 2 3 2 3 ? (2) 3 3 2 1 1 2 1 2 2 3 3 ≠ 3 2 Replace x with . 3 Reciprocal Identities Quotient Identities Pythagorean Identities 6 a. If cot = , find tan . 5 1 Example 2 tan = Choose an identity that involves tan and cot . cot 1 5 6 = or Substitute for cot and evaluate. Use the given information to find the 6 6 5 trigonometric value. 5 a. b. If cot =6/5 , find tan . If sec =5/4 , find cot . 5 b. If sec = , find cot . 4 Since there are no identities relating sec and cot , we must use two identities, one relating sec and tan and another relating cot and tan . sec2 = 1 + tan2 5 4 2 = 1 + tan2 Pythagorean identity 5 Substitute for sec . 4 25 =1 + tan2 16 9 = tan2 16 3 = tan 4 Now find cot . 1 tan 4 3 cot = Reciprocal identity 02-15-2012 Derive 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. Symmetry Diagram Symmetry Identities Example 3 Express each value as a trigonometric function of an angle in Quadrant I. a. sin 765° a. sin 765° b. sin (-19π/3) 765° and 45° differ by a multiple c. cos 935° 765° = 45° + 2(360°) d. cot 11π/4 sin 765° = sin (45° + 2(360°)) = sin 45° of 360°. Case 1, with A = 45° and k = 2 19 b. sin () 3 19 The sum of and is a multiple of 2. 3 3 19 - 3 = -6 - 3 19 sin () = sin (-6 - ) Case 3, with A = and k = -3 3 3 3 = - sin 3 Example 3 Express each value as a trigonometric function of an angle in Quadrant I. a. sin 765° c. cos 935° Relate 935° b. sin (-19π/3) 935° = 35° + 5(180°) c. cos 935° cos 935° = cos (35° + 5(180°)) d. cot 11π/4 = - cos 35° d. cot 935° and 35° differ by an odd multiple of 180°. Case 2, with A = 35° and k = 3 11 4 11 12 and , which is or 3, is an odd multiple of . 4 4 4 = 3 - 4 Case 4 with A = 4 and k = 2 11 cos 4 = Rewrite using the quotient identity. 11 sin 4 cos 3 - 4 = sin 3 - 4 - cos 4 = or - cot 4 Quotient identity sin 4 The sum of 11 4 11 cot 4 Opposite-Angle Identities Example 4 Simplify cos x cot x + sin x. cos x cos x cot x - sin x = cos x sin x + sin x cos2 x = + sin x sin x 1 - sin2 x = + sin x sin x 1 = sin x - sin x + sin x 1 = or csc x sin x Definition of cot x Pythagorean identity: sin2 x + cos2 x = 1 Reciprocal identity Example 5 PROBLEM SOLVING In the Cartesian coordinate system, points are expressed as ordered pairs of their x- and y-coordinates, (x, y), on a coordinate plane where x is the signed horizontal distance from the origin and y is the signed vertical distance from the origin. Another way of expressing a point P is in polar coordinates, (r, ), where r is the signed distance of a point from a fixed origin O, or pole, and is the signed angle from the initial ray to the ray OP. Using polar coordinates, the equation ke r= , where 0 < e < 1, represents an ellipse 1 + e cos F fM A X that has a focus at the origin and directrix x = k to a. μS = F N the right of the origin. Show that the equation mg sin μS = ke sin csc2 mg cos r= represents the same ellipse. csc + e cot μS = tan b. μS = tan 60° μS = 3 μS 1.732050808 The coefficient of static friction is about 1.73. Conclusions Summary • Write an expression that contains all six trigonometric functions and is equal to 3. sin2x+cos2x+sec2x-tan2x+csc2x-cot2x Assignment • Pg427 #(18-43 ODD, 44-53 ALL)