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Trigonometry III Fundamental Trigonometric Identities. By Mr Porter Summary of Definitions θ Adjacent Opposite α Opposite Hypotenuse cosecq = Hypotenuse Opposite Adjacent Hypotenuse cosq = secq = Hypotenuse Adjacent Opposite Adjacent tan q = cot q = Adjacent Opposite sin q = Reciprocal Relationships cosecq = Complementary Relationships a = 90° - q sin(90 - q ) = cosq cosec(90 - q ) = secq cos(90 - q ) = sinq sec(90 - q ) = cosecq tan(90 - q ) = cot q cot(90 - q ) = tanq 1 sin q secq = 1 cosq cot q = 1 tan q Negative Angle 0 £ a £ 90 sin(-a ) = -sin a cos(-a ) = cosa tan(-a ) = - tan a Pythagorean Identities of Trigonometry. Using Pythagoras’ Theorem For any angle θ a 2 + b2 = c2 then x 2 + y2 = r 2 cos2 q + sin 2 q = 1 ( r cosq )2 + ( r sinq )2 = r 2 1+ tan q = sec q 2 2 cot q +1 = cosec q 2 2 r 2 cos2 q + r 2 sin 2 q = r 2 ( ) r 2 cos2 q + sin 2 q = r 2 , divide by r 2 cos2 q + sin 2 q = 1 r y θ x Now x = r cosq and y = r sinq cos2 q sin 2 q 1 2 + = , divide by sin q 2 2 2 sin q sin q sin q 1 cot 2 q +1 = 2 sin q cot 2 q +1 = cosec2q Likewise, 1+ tan 2 q = sec2 q Examples: Simplify the following 1- cos a 1- sin 2 a b) 2 a) = 1- (1- sin a ) 1- (1- cos2 a ) 2 1-1+ sin 2 a = 1-1+ cos2 a sin 2 a = cos2 a æ sin a ö =ç è cos a ÷ø = ( tan a ) = tan 2 a 2 Write down the identities cos2 q + sin 2 q = 1 Write down the identities Sometimes, we need to take small steps! cos q + sin q = 1 Use the 3rd identity to replace denominator 2 Options: (1) replace the ‘1’ with a trig expression (2) Rearrange an identity and replace 2 cot a 1 + cot 2 a 2 1+ tan q = sec q 2 2 cot 2 q +1 = cosec2 q In this case, rearrange the 1st identity sin2θ = 1 – cos2θ, and cos2θ = 1 – sin2θ 2 cot a = cosec 2a 1+ tan 2 q = sec2 q cot 2 q +1 = cosec2 q Now, replace cot and cosec with their sin and cos equivalents. cos a = sin a 2 æ 1 ö çè ÷ sin a ø 2 Fraction rearrange cosa sin 2 a =2 ´ sin a 1 1 tan q 1 secq = cosq 1 cosecq = sin q cot q = = 2 cosa sin a Extension student would continue to the next step. 2 = sin 2a Examples: Simplify the following ( ) c) sec 2 a -1 tan ( 90° - a ) Write down the identities cos2 q + sin 2 q = 1 Use the 2 identity, rearranged. nd = tan 2 a tan ( 90° - a ) = tan a cot a 1 = tan 2 a ´ tan a = tan a 2 1+ tan 2 q = sec2 q cot 2 q +1 = cosec2 q Use the complementary trig angles. tan(90 - q ) = cot q Use the reciprocal trig angles. cot q = 1 tan q Write down the identities d) sin3 a + sin a cos2 a cos2 q + sin 2 q = 1 No matches, FACTORISE! 1+ tan 2 q = sec2 q ( = sin a sin 2 a + cos2 a Now use an identity (try number 1). = sin a (1) = sin a ) cot 2 q +1 = cosec2 q Exercise a) Simplify 1+ cos a sin a sin a 1- cos a ans : 0 b) Simplify cosec 2a - cot 2 a cos2 a ans : sec2 a c) Simplify 1 cot a + tan a 1 ans : sin a cosa = sin 2a 2 d) Simplify 1 - 1 sec a -1 sec a +1 ans : 2cot 2 a Trigonometric Identity Proofs. a) Prove that cos a - tan a = sec a 1- sin a cos a Break into terms of sin and cos - tan a 1- sin a cos a sin a = Common denominator. 1- sin a cosa cos2 a - sin a (1- sin a ) Expand numerator = cos a (1- sin a ) cos2 a - sin a + sin 2 a Rearrange numerator = cos a (1- sin a ) cos2 a + sin 2 a - sin a Write down the identities = cos a (1- sin a ) cos2 q + sin 2 q = 1 LHS = 1- sin a = cos a (1- sin a ) = 1 cos a = seca 1+ tan 2 q = sec2 q cot 2 q +1 = cosec2 q Trigonometric Identity Proofs. b) Prove tan a sin a + cosa = sec a LHS = tan a sin a + cosa sin a = sin a + cos a cos a sin 2 a + cos2 a = cos a 1 = cos a = seca Break into terms of sin and cos Common denominator. Write down the identities cos2 q + sin 2 q = 1 1+ tan 2 q = sec2 q cot 2 q +1 = cosec2 q Trigonometric Identity Proofs. d) Prove sin 2 a tan a + cos2 a cot a + 2sin a cosa = tan a + cot a LHS = sin 2 a tan a + cos2 a cot a + 2sin a cosa Break into terms of sin and cos and rearrange sin a cosa = sin 2 a + sin a cosa + cos2 a + sin a cosa Factorise cosa sin a æ sin 2 a ö æ cos2 a ö Express brackets as a common denominator. = sin a ç + cos a ÷ + cos a ç + sin a ÷ è cosa ø è sin a ø æ sin 2 a + cos2 a ö æ cos2 a + sin 2 a ö = sin a ç ÷ø + cos a çè ÷ø cosa sin a è æ 1 ö æ 1 ö = sin a ç + cos a çè ÷ è cosa ÷ø sin a ø sin a cos a Use definitions = + cos a sin a Use identity Expand brackets = tan a + cot a This was NOT an easy question! Exercise 2 1tan a a) Prove = cos2 a - sin 2 a 2 1+ tan a b) Prove 1+ tan 2 a = tan 2 a 2 1+ cot a c) Prove sin a cosa = d) Prove cot a 1+ cot 2 a sin a cosa + = sec a coseca cosa sin a