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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 = cos1 x
(2) y = sin1 x Í
(3) y = tan1 x Í
Ex7,
Evaluate sin1 ( 
Ex8,
Evaluate tan1 (  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( cos1 x) = x
(2A) sin( sin1 x) = x
(3A) tan( tan1 x) = x
when  1 Ÿ x Ÿ 1
(1B) cos1 (cos x) = x when 0 Ÿ x Ÿ 1
when  1 Ÿ x Ÿ 1
(2B) sin1 (sin x) = x when  12 Ÿ x Ÿ 
for every x
(3B) tan1 (tan x) = x when  12  x   12 .
Evaluate cos(cos1 (  12 ))
Ex10, Evaluate cos1 (cos 17 )
Ex11, Evaluate cos1 (cos
91
7 )
Ex12, Evaluate sin1 (sin ( 471 ))
1
2