Download 1. RECIPROCAL IDENTITIES secx = 1 cosx cscx = 1 sinx cotx = 1

REFRESHER 1: Trigonometric Identities
MA26200, Gabriel Sosa
June 23, 2014 - Monday
1. RECIPROCAL IDENTITIES
sec x =
1
cos x
csc x =
1
sin x
cot x =
1
tan x
Notice the reciprocal of Sine is COsecant.
The reciprocal of COsine is Secant.
2. PYTHAGOREAN IDENTITIES
sin2 x + cos2 x = 1
Dividing by
cos2 x
we obtain
tan2 x + 1 = sec2 x
Dividing by
sin2 x
we obtain
1 + cot2 x = csc2 x
3. DOUBLE ANGLE IDENTITIES
sin 2x = 2 sin x cos x
cos 2x = cos2 x − sin2 x
= 2 cos2 x − 1
= 1 − 2 sin2 x
These are useful if one wants to
rewrite cos 2x in terms of just
cos x or just sin x.
Also bear in mind that because of the last identity one can rewrite
sin2 x =
1 − cos 2x
1 + cos 2x
and cos2 x =
2
R2
which
is the trick necessary for integrating
R
2
(cos x)dx.
Refresher 1
MA26200 (Gabriel Sosa), Summer ’14
(sin2 x)dx and
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4. ADDITION AND SUBTRACTION OF ANGLES
sin (x±y) = sin x cos y± sin y cos x
cos (x±y) = cos x cos y∓ sin x sin y
tan x± tan y
tan (x±y) =
1∓ tan x tan y
5. DERIVATIVES OF TRIGONOMETRIC FUNCTIONS
(sin x)0 = cos x
(cos x)0 = − sin x
(tan x)0 = sec2 x
(cot x)0 = − csc2 x
(sec x)0 = sec x tan x
(csc x)0 = − csc x cot x
Notice that the derivatives of all the CO functions are negative.
6. INTEGRALS OF TRIGONOMETRIC FUNCTIONS
R
(sin x)dx = − cos x + C
R
(cos x)dx = sin x + C
R
(tan x)dx = − ln | cos x| + C = ln | sec x| + C
R
(cot x)dx = ln | sin x| + C
R
(sec x)dx = ln | sec x + tan x| + C
R
(csc x)dx = ln | csc x − cot x| + C
Refresher 1
MA26200 (Gabriel Sosa), Summer ’14
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