Download Trig Summary

Summary of Trigonometric Facts
Formulas Involving Radian Angular Measure
180
s
v
1 rad = π deg
=
=
r
r
π
1 deg = 180 rad
Trigonometric Function Definitions
b opp
=
R hyp
R hyp
cosecant x = csc x = b = opp
sine x = sin x =
A
sin A
cos A
tan A
0˚
0
0
30˚
π/6
1/ 2
1
3/2
3/ 3
0
45˚
π/4
a adj
=
R hyp
R hyp
secant x = sec x = = adj
a
b opp
=
a adj
a adj
cotangent x = cot x = = opp
b
tangent x = tan x =
y
Q II
QI
sin x > 0
All
are
positive.
csc x > 0
Others are negative.
90˚
π/2
2/2
3/ 2
1
0
2 / 2 1/2
1
3 undef'd
y 1
.5
 R = a2 + b2


cosine x = cos x =
60˚
π/3
A = 12 r2
Q III
Q IV
tan x > 0
cos x > 0
cot x > 0
sec x > 0
Others are negative. Others are negative.
y
y = sin x
Amplit ude = 1
π/2
π
y = cos x
π
2π
x
3π/2
3π/2
2π
x
π/2
–.5
Amplit ude = 1
–1
Period = 2π
Period = 2π
y
y
y = cot x
y = tan x
–π/2
–π/4
1
π/4
1
x
π/2
3π/4
π/2
π/4
π
x
Period = π
Period = π
y
y
y = csc x
1
π/2
π
Period = 2π
3π/2
y = sec x
2π
x
1
π/2
π
Period = 2π
3π/2
2π
x
csc x =
Reciprocal Identities
1
sec x = cos
cot x = tan1 x
x
1
sin x
Tangent and Cotangent Identities
sin x = tan x
cos x = cot x
cos x
sin x
Sum and Difference Formulas
sin(x ± y) = sin x cos y ± cos x sin y
cos(x ± y) = cos x cos y m sin x sin y
tan( x ± y) =
Pythagorean Identities
sin2 x + cos2 x = 1
1 + tan2 x = sec 2 x
1 + cot2 x = csc 2 x
tan x ± tan y
1 m tan x tan y
Double Angle Fomulas
sin 2x = 2 sin x cos x
cos 2x = cos2x – sin2x
= 2 cos2x–1
=1–2 sin2x
Cofunction Identities
sin x = cos(π/2 – x)
cos x = sin(π/2 – x)
tan x = cot(π/2 – x)
csc x = sec(π/2 – x)
sec x = csc(π/2 – x)
cot x = tan(π/2 – x)
Even-Odd Identities
sin(− x) = − sin x
csc(− x) = − csc x
cos(− x) = cos x
sec(− x) = sec x
tan(− x) = − tan x
cot(− x) = − cot x
tan 2x =
2tan x
1–tan 2 x
Half Angle Formulas
sin x2
sin
sin
=±
1–cos x
2
cos = ±
x
2
x cos y = 12 [(sin(x + y)
x sin y = 12 [cos(x – y)
1+cos x
2
x
1−cos x
sin x
tan 2x = ± 1−cos
1+cos x = sin x = 1+ cos x
Product to Sum or Difference Formulas
+ sin( x – y)]
cos x sin y = 12 [sin(x + y) – sin( x – y)]
– cos( x + y)]
cos x cos y = 12 [cos(x + y) + cos( x – y)]
Inverse Trigonometric Functions
arcsin x or sin-1x ∈
,2
arccsc x or csc–1x ∈ −2 ,0 U 0, 2
arcsec x or sec–1x ∈ 0, 2 U 2 ,
arccos x or cos–1x ∈ [0,π]
[
−
2
(
arctan x or tan-1x ∈ −2 , 2
]
)
[ )( ]
[ )( ]
arccot x or cot–1x ∈ ( −2 ,0) U(0, 2 ]
Law of Sines
a
b
c
=
=
sin A sinB sinC
Law of Cosines
= b 2 + c 2 – 2bc cos A
= a 2 + c 2 – 2ac cos B
c = a 2 + b 2 – 2ab cos C
a2
b2
2
Area of a Triangle
area = 12
area = 12
area = 12
bc sin A
ac sin B
ab sin C
Heron's Formula
area =
s(s – a)(s – b)(s – c) ,
a+b +c
where s =
2
Sum of Sine and Cosine with Same Period
a sin cx + b cos cx = A sin(cx + ),
b
a
where A = a 2 + b 2 , sin =
, and cos =
a2 + b2
a2 + b2