Reciprocal Identities MHF4U: Advanced Functions Recall that the - PDF document

t r i g o n o m e t r i c i d e n t i t i e s t r i g o n o m e t r i c i d e n t i t i e s Reciprocal Identities MHF4U: Advanced Functions Recall that the three primary trigonometric ratios are sin θ , cos θ and tan θ . The three secondary ratios are csc θ , sec θ and cot θ . Since the secondary ratios are reciprocals of the primary ratios, they are often called the reciprocal ratios. Review of Basic Trigonometric Identities This relationship gives us three trigonometric identities . J. Garvin Reciprocal Identities 1 For any value of θ , the reciprocal identities are csc θ = sin θ , 1 1 sec θ = cos θ and cot θ = tan θ . These identities can be used to prove that one trigonometric expression is equivalent to another. J. Garvin — Review of Basic Trigonometric Identities Slide 1/17 Slide 2/17 t r i g o n o m e t r i c i d e n t i t i e s t r i g o n o m e t r i c i d e n t i t i e s Proofs Using Reciprocal Identities Pythagorean Identity Example Let P be any point on a unit circle. Prove that sin θ · csc θ = 1. LHS = sin θ · csc θ 1 = sin θ · sin θ = sin θ sin θ By the Pythagorean Theorem, x 2 + y 2 = 1. = 1 Since x = cos θ and y = sin θ , x 2 + y 2 = cos 2 θ + sin 2 θ = 1. = RHS Pythagorean Identity For any value of θ , sin 2 θ + cos 2 θ = 1. J. Garvin — Review of Basic Trigonometric Identities J. Garvin — Review of Basic Trigonometric Identities Slide 3/17 Slide 4/17 t r i g o n o m e t r i c i d e n t i t i e s t r i g o n o m e t r i c i d e n t i t i e s Proofs Using the Pythagorean Identity Proofs Using the Pythagorean Identity Example Example Prove that sin 2 θ + 4 cos 2 θ = − 3 sin 2 θ + 4. Prove that 1 + sin 2 θ − cos 2 θ = sin θ . 2 sin θ LHS = sin 2 θ + 4 cos 2 θ LHS = 1 + sin 2 θ − cos 2 θ = sin 2 θ + 4(1 − sin 2 θ ) 2 sin θ = sin 2 θ + 4 − 4 sin 2 θ = sin 2 θ + sin 2 θ = − 3 sin 2 θ + 4 2 sin θ = 2 sin 2 θ = RHS 2 sin θ = sin θ = RHS J. Garvin — Review of Basic Trigonometric Identities J. Garvin — Review of Basic Trigonometric Identities Slide 5/17 Slide 6/17

t r i g o n o m e t r i c i d e n t i t i e s t r i g o n o m e t r i c i d e n t i t i e s Tangent and Cotangent Identities Proofs Using Tangent and Cotangent Identities Let P be any point on a unit circle. Example Prove that sin θ · cot θ = cos θ . LHS = sin θ · cot θ = sin θ · cos θ sin θ = cos θ x , tan θ = sin θ Since tan θ = y cos θ . = RHS tan θ , cot θ = cos θ 1 Since cot θ = sin θ . Tangent and Cotangent Identities For any value of θ , tan θ = sin θ cos θ and cot θ = cos θ sin θ . J. Garvin — Review of Basic Trigonometric Identities J. Garvin — Review of Basic Trigonometric Identities Slide 7/17 Slide 8/17 t r i g o n o m e t r i c i d e n t i t i e s t r i g o n o m e t r i c i d e n t i t i e s General Rules For Proving Identities Proofs Using the Basic Identities While there is no specific procedure for proving a Example trigonometric identity, the following general rules may help. Prove that cos θ · cot θ + csc θ = cos 2 θ + 1 . sin θ • Try to simplify, rather than expand, when possible. • Replace all instances of tan θ , cot θ , sec θ and csc θ with sin θ and cos θ . • DO NOT “move” terms or factors across an = sign, LHS = cos θ · cot θ + csc θ since this presupposes that the identity is true. = cos θ · cos θ 1 sin θ + • If necessary, work on both sides of an identity sin θ = cos 2 θ 1 simultaneously and meet somewhere in the middle. sin θ + sin θ = cos 2 θ + 1 sin θ = RHS J. Garvin — Review of Basic Trigonometric Identities J. Garvin — Review of Basic Trigonometric Identities Slide 9/17 Slide 10/17 t r i g o n o m e t r i c i d e n t i t i e s t r i g o n o m e t r i c i d e n t i t i e s Proofs Using the Basic Identities Using the Conjugate Recall that ( x − y )( x + y ) = x 2 − y 2 , a difference of squares. Example Prove that sec 2 θ − 1 = tan 2 θ . We say that x − y is the conjugate of x + y , and vice versa. Similarly, (1 − sin θ )(1 + sin θ ) = 1 − sin 2 θ (also a difference of squares), which simplifies to cos 2 θ . LHS = sec 2 θ − 1 When trying to simplify rational expressions involving cos 2 θ − cos 2 θ 1 1 ± sin θ or 1 ± cos θ , it is often useful to use the conjugate = cos 2 θ and apply the Pythagorean Identity. = 1 − cos 2 θ cos 2 θ = sin 2 θ cos 2 θ = tan 2 θ = RHS J. Garvin — Review of Basic Trigonometric Identities J. Garvin — Review of Basic Trigonometric Identities Slide 11/17 Slide 12/17

t r i g o n o m e t r i c i d e n t i t i e s t r i g o n o m e t r i c i d e n t i t i e s Using the Conjugate Factoring Trigonometric Expressions Example Another useful technique is to factor trigonometric identities, similar to how other binomials and trinomials are factored. 1 + sin θ = 1 − sin θ cos θ Prove that . cos θ If a rational expression contains the same factor in both its numerator and its denominator, then that factor can be cancelled out, leaving an equivalent expression. cos θ LHS = All of the standard factoring techniques (common, simple, 1 + sin θ complex, perfect squares, differences of squares) may apply, 1 + sin θ · 1 − sin θ cos θ = and more than one method of factoring may be required. 1 − sin θ = (cos θ )(1 − sin θ ) 1 − sin 2 θ = (cos θ )(1 − sin θ ) cos 2 θ = 1 − sin θ cos θ = RHS J. Garvin — Review of Basic Trigonometric Identities J. Garvin — Review of Basic Trigonometric Identities Slide 13/17 Slide 14/17 t r i g o n o m e t r i c i d e n t i t i e s t r i g o n o m e t r i c i d e n t i t i e s Factoring Trigonometric Expressions Factoring Trigonometric Expressions Example Example Prove that sin 2 θ − cos 2 θ Prove that cos 3 θ + cos 2 θ = cos θ + 1. = − sin θ − cos θ . 1 − sin 2 θ cos θ − sin θ LHS = sin 2 θ − cos 2 θ LHS = cos 3 θ + cos 2 θ cos θ − sin θ 1 − sin 2 θ = (sin θ + cos θ )(sin θ − cos θ ) = (cos 2 θ )(cos θ + 1) cos θ − sin θ 1 − sin 2 θ = − (sin θ + cos θ )(cos θ − sin θ ) = (cos 2 θ )(cos θ + 1) cos θ − sin θ cos 2 θ = − (sin θ + cos θ ) = cos θ + 1 = − sin θ − cos θ = RHS = RHS J. Garvin — Review of Basic Trigonometric Identities J. Garvin — Review of Basic Trigonometric Identities Slide 15/17 Slide 16/17 t r i g o n o m e t r i c i d e n t i t i e s Questions? J. Garvin — Review of Basic Trigonometric Identities Slide 17/17