B.3 Formulas from Trigonometry

5144_Demana_APPpp839-876
1/13/06
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Page 859
APPENDIX B.3 Formulas from Trigonometry
B.3 Formulas from Trigonometry
Cofunction Identities
Angular Measure
p
cos }} 2 u
2
p
sin }} 2 u
2
p
tan }} 2 u
2
p
cot }} 2 u
2
p
sec }} 2 u
2
p
csc }} 2 u
2
p radians 5 180°
180
So, 1 radian 5 }} degrees,
p
p
and 1 degree 5 }} radians.
180
Reciprocal Identities
1
sin x 5 }}
csc x
1
csc x 5 }}
sin x
1
cos x 5 }}
sec x
1
sec x 5 }}
cos x
1
tan x 5 }}
cot x
1
cot x 5 }}
tan x
Quotient Identities
sin x
tan x 5 }}
cos x
cos x
cot x 5 }}
sin x
11
11
cot2
x5
x
x5
csc2
x
5 sin u
5 cos u
5 cot u
5 tan u
5 csc u
5 sec u
Double-Angle Identities
sin 2u 5 2 sin u cos u
cos 2u 5 cos2 u 2 sin2 u
5 2 cos2 u 2 1
2 tan u
tan 2u 5 }}
1 2 tan2 u
sin2 x 1 cos2 x 5 1
sec2
)
)
)
)
)
)
5 1 2 2 sin2 u
Pythagorean Identities
tan2
(
(
(
(
(
(
Power-Reducing Identities
Odd-Even Identities
sin s2xd 5 2sin x
csc s2xd 5 2csc x
cos s2xd 5 cos x
sec s2xd 5 sec x
tan s2xd 5 2tan x
cot s2xd 5 2cot x
Sum and Difference Identities
sin su 1 vd 5 sin u cos v 1 cos u sin v
sin su 2 vd 5 sin u cos v 2 cos u sin v
cos su 1 vd 5 cos u cos v 2 sin u sin v
cos su 2 vd 5 cos u cos v 1 sin u sin v
tan u 1 tan v
tan su 1 vd 5 }}
1 2 tan u tan v
tan u 2 tan v
tan su 2 vd 5 }}
1 1 tan u tan v
1 2 cos 2u
sin2 u 5 }}
2
1
1
cos 2u
cos2 u 5 }}
2
1
2
c
os 2u
tan2 u 5 }}
1 1 cos 2u
Half-Angle Identities
!§§§§§§
!§§§§§§
!§§§§§§
u
1 2 cos u
sin }} 5 6 }}
2
2
u
1 1 cos u
cos }} 5 6 }}
2
2
u
1 2 cos u
tan }} 5 6 }}
2
1 1 cos u
1 2 cos u
sin u
5 }} 5 }}
sin u
1 1 cos u
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