MARIAN UNIVERSITY
Indianapolis
®
Values of Special Angles from 0° to 90°
cos
π/2
π/3
60°
90°
THE “HAND” METHOD
Instructions
π/4
1. Fold down the finger of the desired angle.
2. Place in the numerator the number of fingers from
the folded finger in the direction of the desired function.
3. Place
over the numerator.
4. Place 2 in the denominator.
45°
sin
MATHEMATICS
Y
π/6
30°
cos θ =
0° 0
top fingers
2
sin θ =
bottom fingers
2
tan θ =
bottom fingers
top fingers
X
UNIT CIRCLE
Y
VALUES CHART
Angle
cos
sin
tan
0
1
0
0
π/6
3/2
1/2
3/3
π/4
2/2
2/2
1
π/3
1/2
3/2
3
π/2
0
1
undefined
www.marian.edu
(cos θ, sin θ)
(0,1)
(_ 1 , 3 )
( 1 ,
2 2
2
π
__
2
2
90°
,
(_
)
2
π
2π
__
__
2 2
3 __
π
3
3π
__
(_ 3 , 1 )
4
120°
60°
4
2 2
5π
__
45°
135°
6
150°
(-1,0)
180°
π
30°
3)
2
( 2 , 2)
2 2
π
__
6
0° 0
360° 2π
( 3,1)
2 2
X
(1,0)
210°
330° 11π
7π
___
__
315° 6
6 5π 225°
300° 7π
__
__
240°
(_ 3 , _ 1 )
( 3 ,_ 1 )
4
4 4π
2
2
2
2
5π
__
__
3
2
2
2
2
3π
3
_
_
_
__
,
,
(
(
)
)
270° 2
2
2
2
2
(_ 1 , _ 3 )
( 1 ,_ 3 )
(0,-1)
2
2
2
2
ANGLE MEASUREMENT AND ALGEBRAIC FORMULAS
r
Slope:
s
θ
r
π radians = 180°
1°=
(θ in radians)
π
rad
180
1 rad =
m=
y2 - y1
x2 - x1
Line:
Quadratic Formula:
y = mx + b
y - y1 = m(x - x1)
x=
-b ± b2 - 4ac
2a
where ax2 + bx + c = 0
Distance:
180°
π
Special Polynomials:
x3 + y3 = (x + y) (x2 - xy + y2)
x3 - y3 = (x - y) (x2 + xy + y2)
(x + y)3 = x3 + 3x2y + 3xy2 + y3
(x - y)3 = x3 - 3x2y + 3xy2 - y3
d = (x2 - x1)2 + (y2 - y1)2
s = rθ
Circular Function Definitions, where θ is any angle:
Y
r
y
r=
x2 + y2
θ
X
x
Right Triangle Definitions, where 0 < θ < π/2:
y
sin θ = r
x
cos θ = r
r
csc θ = y
r
sec θ = x
y
tan θ = x
x
cot θ = y
)
e
us
n
ote
p
(hy
p
hy
θ
opp
sin θ = hyp
adj
cos θ = hyp
opp
tan θ = adj
opposite (opp)
(x, y)
MATHEMATICS
IDENTITIES
adjacent (adj)
hyp
csc θ = opp
hyp
sec θ = adj
adj
cot θ = opp
FORMULAS
Tangent and Cotangent Identities:
tan θ =
sin θ
cos θ
cot θ =
cos θ
sin θ
1
sin θ
1
sec θ =
cos θ
1
cot θ =
tan θ
1
csc θ
1
cos θ =
sec θ
1
tan θ =
cot θ
sin θ =
Double Angle Formulas:
sin(2θ) = 2sin θ cos θ
cos(2θ) = cos2 θ — sin2 θ
= 2cos2 θ — 1
= 1 —2sin2 θ
2tanθ
tan(2θ) =
1 — tan2 θ
Periodic Formulas - If n is an integer:
sin θ + cos θ = 1
sin(θ + 2πn) = sin θ
csc(θ + 2πn) = csc θ
tan2 θ + 1 = sec2 θ
cos(θ + 2πn) = cos θ
sec(θ + 2πn) = sec θ
1 + cot2 θ = csc2 θ
tan(θ + πn) = tan θ
cot(θ + πn) = cot θ
2
Reciprocal Identities:
csc θ =
Pythagorean Identities:
2
Even/Odd Formulas:
Cofunction Formulas:
sin(­— θ) = — sin θ
csc(­— θ) = — csc θ
cos(­— θ) = cos θ
sec(­— θ) = sec θ
tan(­— θ) = — tan θ
cot(­— θ) = — cot θ
CONICS
Circle:
x2 + y2 = a2
Ellipse:
x
y
+ 2 =1
a2
b
2
2
1
Y
y = sin x
0
1
π
2π
X
Y
0
y = cos x
π
2π
X
Parabola: y = 4ax
2
Hyperbola:
π
­— θ) = sin θ
2
π
sec(
­— θ) = csc θ
2
π
cot(
­ θ) = tan θ
—
2
cos(
GRAPHS
x2
y2
— 2 =1
2
a
b
2
1
0
Y
y = tan x
π
2π
X
-1
-1
2
1
0
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π
—
­ θ) = cos θ
2
π
csc(
­— θ) = sec θ
2
π
tan(
—
­ θ) = cot θ
2
sin(
-1
Y
y = csc x
π
2π
X
2
1
0
-2
Y
y = sec x
π
2π
X
2
1
0
-1
-1
-1
-2
-2
-2
Marian University is sponsored by the Sisters of St. Francis, Oldenburg, Indiana.
MAY 2016
Y
y = cot x
π
2π
X