Download Summary of Trig Facts

Summary of Trigonometric Facts
π
1 deg = 180 rad
Formulas Involving Radian Angular Measure
180
s
v
1 rad = π deg
θ=
ω=
r
r
Trigonometric Function Definitions
y opp
=
r hyp
r hyp
cosecant θ = csc θ = =
y opp
x adj
=
r hyp
r hyp
secant θ = sec θ = €=
x adj
sine θ = sin θ =
€
cosine θ = cos θ =
Trig Function Values€at Special Angles
°
0
0
A
€
°
°
30
π/6
45
€
π/4
°
60
π/3
°
90
π/2
€
€
€
y 1
.5
€
€
€
π/2
)
y opp
=
x adj
x adj
cotangent θ = cot θ = =
y opp
tangent θ = tan θ =
Q II
y
€
sin A and csc A are
QI
positive; others
All are positive
are negative.
x
Q III
Q IV
tan A and cot A are cos A and sec A are
positive; others are positive; others are
negative.
negative.
y
y = sin x
Amplit ude = 1
x2 + y2
Signs €
of the Trig Functions in the Quadrants
1/ 2 2 / 2 3 / 2
1
sin A
0
€cos A € 1 €
€
€
0
3 / 2 2 / 2 1/ 2
€
€
€
€
€
undef’d
tan A 0
1
3/3
3
€
€
€
€
€
€ €
€
€
€
€
(r =
A = 12 θ r2
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
3π/2
π/2
π
Period = 2π
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  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  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)
tan 2x =
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
2tan x
1 – tan2 x
Half Angle Formulas
sin x2
=±
1– cos x
2
x
2
cos = ±
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)]
sin x cos y = 12 [(sin(x + y) +
sin x sin y = 12 [cos(x – y) –
cos(x + y)]
cos x cos y = 12 [cos(x + y) + cos(x – y)]
Inverse Trigonometric Functions
[
arcsin x or sin-1x ∈ −2π , π2
]
arccos x or cos–1x ∈ [0,π]
arctan x or tan-1x ∈ −2π , π2
(
)
[ )( ]
x ∈ [0, π2 ) ( π2 ,π ]
x ∈ ( π ,0) (0, π ]
arccsc x or csc–1x ∈ −2π ,0  0, π2
arcsec x or sec–1
arccot x or cot–1
€
Law of Sines
a
b
c
=
=
sin A sinB sinC
−
2
2
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
€
1
area = 2
1
area = 2
1
area = 2
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 a Sine and a Cosine with the Same Period
b
2
2
, cosφ =
a sin cx + b cos cx = A sin(cx + φ ) , where A = a + b , sinφ =
a2 + b2
a
a2 + b2