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Calculus section 1.3 Trigonometry Def 2 1 radians = 360 degrees = 1 revolution 2 1 rad = 360‰ = 1 rev Ex1, 2 31 Ex2, 135‰ rad Arclength and Sector area formulas (1) 6= r) where r radius, ) central angle, 6 arclength, A sector area 1 2 1 (2) A = 2 r ) = 2 r6 Def Let there be a unit circle, x2 + y2 = 1. Draw a ray with initial point (0, 0) and standard angle ). Define (cos ) , sin ) ) be the intersection of the ray and the circle. Def sin ) (1) tan ) ´ cos ) (2) sec ) ´ cos1 ) (3) csc ) ´ sin1 ) ) 1 (4) cot ) ´ cos sin ) =* tan ) (* when ) Á n1 + Basic Identities (1) Square (2) negative angle (3) complementary (4) supplementary (5) half revolution (6) period (6)' period cos2 ) + sin2 ) = 1, cos ( )) = cos ), cos( 12 )) = sin ), cos(1 ))= cos ), cos(1 + ))= cos ), cos(21 + ))= cos ), cos(2n1 + ))= cos ), 1 2 ,n−™) 1 + tan2 ) = sec2 ), sin ( )) = sin ), sin ( 12 )) = cos ), sin (1 )) = sin ), sin (1 + )) = sin ), sin (21 + )) = sin ) sin (2n1 + )) = sin ), 1 + cot2 ) = csc2 ) tan ( )) = tan ) tan ( 12 )) = cot ) tan (1 )) = tan ) tan (1 + )) = tan ) tan (n1 + )) = tan ) even/odd where n − ™ Ex3, Find the (a) arclength (b) sector area of a sector of radius 8 inches and central angle 135‰ . Ex4, Graphs of trig functions Ex5, cos x (2 cos x + 3) + 1 = 0 Ex6, 4 (cos x 1) = sin2 x Def (1) y = cos1 x (2) y = sin1 x Í (3) y = tan1 x Í Ex7, Evaluate sin1 ( Ex8, Evaluate tan1 ( 3) Thm Ex9, Í x = cos y x = sin y x = tan y where 1 Ÿ x Ÿ 1 and 0 Ÿ y Ÿ 1. where 1 Ÿ x Ÿ 1 and 12 Ÿ y Ÿ 12 . where 12 y 12 . 3 2 ) (1A) cos( cos1 x) = x (2A) sin( sin1 x) = x (3A) tan( tan1 x) = x when 1 Ÿ x Ÿ 1 (1B) cos1 (cos x) = x when 0 Ÿ x Ÿ 1 when 1 Ÿ x Ÿ 1 (2B) sin1 (sin x) = x when 12 Ÿ x Ÿ for every x (3B) tan1 (tan x) = x when 12 x 12 . Evaluate cos(cos1 ( 12 )) Ex10, Evaluate cos1 (cos 17 ) Ex11, Evaluate cos1 (cos 91 7 ) Ex12, Evaluate sin1 (sin ( 471 )) 1 2