Download Fundamental Identities Cofunction Identities Double

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Fundamental Identities
▼
Reciprocal Identities
Ratio Identities
1
sec ⫽
cos sin tan ⫽
cos csc ⫽
1
sin cot ⫽
Pythagorean Identities
2
▼
cos sin Cofunction Identities
▼
cosa
⫺ b ⫽ sin 2
cos1 ⫾ 2 ⫽ cos cos ⫿ sin sin tan a
⫺ b ⫽ cot 2
cota
⫺ b ⫽ tan 2
sin1 ⫾ 2 ⫽ sin cos ⫾ cos sin seca
⫺ b ⫽ csc
2
csca
⫺ b ⫽ sec
2
tan1 ⫾ 2 ⫽
Half-Angle Identities
▼
1 ⫺ cos2
2
cos122 ⫽ cos2 ⫺ sin2
1 ⫹ cos cos a b ⫽ ⫾
2
A
2
cos2 ⫽
1 ⫹ cos2
2
1 ⫺ cos tan a b ⫽
2
sin tan2 ⫽
1 ⫺ cos2
1 ⫹ cos2
sin 1 ⫹ cos ▼
sin ⫹ sin ⫽ 2sin a
cos sin ⫽
1
冤sin1 ⫹ 2 ⫺ sin1 ⫺ 2冥
2
sin ⫺ sin ⫽ 2cos a
Area of a Triangle
1
Area ⫽ bc sin A
2
ISBN: 0-07-351948-0
Author: John W. Coburn & J.D.Herdlick
Title: Trigonometry, 2e
c
B
Front endsheets
Color: 5
Pages: 2, 3
0
1
0
—
1
—
6
4
3
2
1
2
13
2
1
13
2
2
13
13
12
2
12
2
1
12
12
1
13
2
1
2
13
2
13
2
1
13
1
0
—
1
—
0
B
45
√2x
A 45
1x
1x
C
⫺
⫹
b sina
b
2
2
(degrees cancel)
180°
30
A
180°
(radians cancel)
By Side Length
Right
x
r
r
sec , x 0
x
Obtuse
Equilateral
Isoceles
Scalene
y
(x, y)
y
r
y
tan , x 0
x
r
csc , y 0
y
x
cot , y 0
y
sin a2 ⫽ b2 ⫹ c2 ⫺ 2bc cos A
2
▼
adj
hyp
sin opp
hyp
hyp
csc opp
tan c2 ⫽ a2 ⫹ b2 ⫺ 2ab cos C
␪r ␪
x
x
sin t y
1
sec t ; x 0
x
1
csc t ; y 0
y
adj
(x, y)
For any real number t and point P1x, y2 on the unit circle associated with t:
y
tan t ; x 0
x
x
cot t ; y 0
y
opp
␪
adj
cot opp
Trigonometric Functions of a Real Number
cos t x
hyp
opp
adj
2
b ⫽ a ⫹ c ⫺ 2ac cos B
r
y
Right Triangle Trigonometry
hyp
sec adj
Law of Cosines
C
√3x
radians to degrees: multiply by
Trigonometry and the Coordinate Plane
cos ▼
1x
Triangle Classifications
Acute
⫺
⫹
bcos a
b
2
2
60
2x
Degree and Radian Conversions
By Angle Measure
▼
B
cos 2
a
cot␪
⫹
⫺
cos ⫺ cos ⫽ ⫺2sin a
b sina
b
2
2
C
A
sec ␪
For right ^ ABC with indicated sides adjacent and opposite to acute angle :
▼
b
csc␪
⫺
⫹
b cos a
b
cos ⫹ cos ⫽ 2cos a
2
2
Law of Sines
sin B
sin A
sin C
⫽
⫽
a
c
b
tan␪
degrees to radians: multiply by
Sum-to-Product Identities
1
冤sin1 ⫹ 2 ⫹ sin1 ⫺ 2冥
2
1
冤cos1 ⫺ 2 ⫺ cos1 ⫹ 2冥
2
cos␪
For P1x, y2 a point on the terminal side of an angle in standard position:
sin cos ⫽
sin sin ⫽
▼
Power Reduction Identities
sin2 ⫽
Product-to-Sum Identities
90° ▼
1 ⫺ cos sin a b ⫽ ⫾
2
A
2
1
cos cos ⫽ 冤cos1 ⫹ 2 ⫹ cos1 ⫺ 2冥
2
▼
tan ⫾ tan 1 ⫿ tan tan sin122 ⫽ 2sin cos ⫽
60° Sum and Difference Identities
⫺ b ⫽ cos 2
⫽ 1 ⫺ 2sin2
▼
tan1⫺2 ⫽ ⫺tan sin a
▼
30° 45° 2
sin␪
0° 0
cos1⫺2 ⫽ cos 1 ⫹ cot ⫽ csc Double-Angle Identities
␪
sin1⫺2 ⫽ ⫺sin tan2 ⫹ 1 ⫽ sec2
2
⫽ 2cos2 ⫺ 1
▼
2
sin ⫹ cos ⫽ 1
1
cot ⫽
tan ▼
Identities due to Symmetry
Special Triangles and Special Angles
t
r1
1
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/Volumes/MHDQ-New/MHDQ146/MHDQ146-ES
Special Constants
3.1416
1.5708
2
1.0472
3
e 2.7183
12 1.4142
13 1.7321
0.7854
4
12
0.7071
2
0.5236
6
13
0.8660
2
0.2618
12
13
0.5774
3
s
Arcs and Sectors
For a circle of radius r and angle in radians:
arc length: s r
␪
r
1
area of sector: A r 2
2
Graphs of the Trigonometric Functions
y csc t
y
y sec t
y
␲
␲
2
␲4 , 1
y cos t
2␲
3␲
2
t
␲
␲
2
t
2␲
3␲
2
␲
␲
␲
␲
2
2
1
1
4
Domain: t 僆 1q, q2
Range: sin t 僆 31, 1 4
Domain: t 僆 1q, q2
Range: cos t 僆 3 1, 14
y cot t
Domain: t 12k 12; k 僆 Z
2
Range: tan t 僆 R
Transformations of Basic Trig Graphs
Transformation of y ⴝ f 1x2
Given Function
y f 1x2
For y A sin c B ax C
bd D
B
S
S
y Af c B ax horizontal shift, opposite
direction of sign
north/south reflections;
vertical stretches and compressions
▼
y tan t
4
y sin t
▼
y
1
1
S
▼
vertical shift, same
direction as sign
C
C
2
b d D we have: amplitude: ƒ A ƒ , period:
, horizontal shift: , vertical shift: D
B
B
B
The Inverse Trigonometric Functions
For y sin t with t 僆 c , d and y 僆 31, 1 4 , the inverse function is y sin1t, where t 僆 31, 1 4 and y 僆 c , d .
2 2
2 2
For y cos t with t 僆 30, 4 and y 僆 31, 1 4 , the inverse function is y cos1t, where t 僆 31, 1 4 and y 僆 30, 4 .
For y tan t with t 僆 a , b and y 僆 R, the inverse function is y tan1t, where t 僆 R and y 僆 a , b.
2 2
2 2
y sin1 t
1,
1
␲
2
␲
1
2
1,
␲
2
3
2
1
␲
2
1
␲
2
ISBN: 0-07-351948-0
Author: John W. Coburn & J.D.Herdlick
Title: Trigonometry, 2e
y
(1, ␲) y
y
␲
2
t
␲2
1
y
(1, 0)
Front endsheets
Color: 5
Pages: 4, 5
2
1
y cos1 t
␲
2
␲
2
3
␲
t
y tan1 t
␲
1, 4 ␲
1
2
3
y ␲
2
t
3␲
2
t
cob19480_es.indd Page 4 11/3/09 3:44:05 PM user-s180
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Commonly used, small case Greek letters
␨
▼
/Volumes/MHDQ-New/MHDQ146/MHDQ146-ES
alpha
zeta
rho
▼
beta
theta
sigma
delta
mu
psi
epsilon
pi
omega
Trigonometric Form
Products and Quotients
ƒ z ƒ ⫽ 2a ⫹ b
z ⫽ r 1cos ⫹ i sin 2
distance from (0, 0) to (a, b)
where r ⫽ ƒ z ƒ
z1z2 ⫽ r1r2 3cos1
1 ⫹ 2 2 ⫹ i sin1
1 ⫹ 2 2 4
z1
r1
⫽ 3cos1
1 ⫺ 2 2 ⫹ i sin1
1 ⫺ 2 2 4
z2
r2
2
2
▼
n
z n ⫽ r n 1cos n
⫹ i sin n
2 for positive integers n
n
1 z ⫽ 1 r acos
a2 ⫺ b2 ⫽ 1a ⫹ b21a ⫺ b2
a2 ⫹ b2 is prime over the real numbers
a3 ⫺ b3 ⫽ 1a ⫺ b21a2 ⫹ ab ⫹ b2 2
a3 ⫹ b3 ⫽ 1a ⫹ b21a2 ⫺ ab ⫹ b2 2
8a, b9
ⱍvⱍ
r ⫽ tan⫺1 `
a
x
y ⫽ logb x 3 b ⫽ x
logb b ⫽ x
x
r
logb MN ⫽ logb M ⫹ logb N
logb
▼
y
␪
x
b
logb 1 ⫽ 0
logb x
logc x ⫽
logb c
⫽x
M
⫽ logb M ⫺ logb N
N
a
2
2
A ⫽ r
r
2
C ⫽ 2r ⫽ d
b
a
2
b
A⫽
2
ab
3
a
b
P(x, y)
x
logb x
Circle
a ⫹b ⫽c
2
b
Right Parabolic Segment
C ⬇ 221a ⫹ b 2
y
logb b ⫽ 1
b
Pythagorean Theorem
C
A
A ⫽ ab
y
`, x Z 0
x
1
A ⫽ bh
2
c
Ellipse
2
h
h
Right Triangle
B
A ⫹ B ⫹ C ⫽ 180°
␪
Logarithms and Logarithmic Properties
y
b
a
Triangle
a
h
A ⫽ 1a ⫹ b2
2
Sum of angles
b
P(x, y) in rectangular coordinates can be represented as P(r, ) in polar coordinates:
r ⫽ 2x 2 ⫹ y 2
Trapezoid
s
P ⫽ ns
a
A⫽ P
2
s
A ⫽ s2
h
Triangle
Regular Polygon
P ⫽ 4s
l
y
Polar Coordinates
y ⫽ r sin Square
w
A ⫽ bh
u
v
• Given the nonzero vectors u and v and angle between them, cos ⫽
•
.
ƒuƒ ƒvƒ
x ⫽ r cos Formulas from Plane Geometry: P S perimeter, C S circumference, A S area
Parallelogram
defined as: u • v ⫽ Ha, bI • Hc, dI ⫽ ac ⫹ bd.
logb MP ⫽ P # logb M
Applications of Exponentials and Logarithms
▼
A S amount accumulated
P S initial deposit, P S periodic payment
n S compounding periods/year
r S interest rate per year
r
R S interest rate per time period a b
n
t S time in years
Formulas from Solid Geometry: S S surface area, V S volume
Rectangular Solid
Cube
Right Circular Cylinder
V ⫽ lwh
V ⫽ s3
V ⫽ r 2h
S ⫽ lw ⫹ lh ⫹ wh
S ⫽ 6s2
S ⫽ 2r1r ⫹ h2
Right Circular Cone
Right Square Pyramid
Sphere
1
V ⫽ r2h
3
1
V ⫽ s2h
3
S ⫽ r 1r ⫹ 2r 2 ⫹ h2 2
S ⫽ s 2 ⫹ s2s 2 ⫹ 4h2
V⫽
4 3
r
3
S ⫽ 4r 2
Formulas from Analytical Geometry: P1 S (x1, y1), P2 S (x2, y2)
Distance between P1 and P2
d ⫽ 21x2 ⫺ x1 2 2 ⫹ 1y2 ⫺ y1 2 2
Slope of Line Containing P1 and P2
m⫽
¢y
y2 ⫺ y1
⫽
x2 ⫺ x1
¢x
Interest Compounded n Times per Year
Interest Compounded Continuously
r nt
A ⫽ P a1 ⫹ b
n
Equation of Line Containing P1 and P2
Equation of Line Containing P1 and P2
A ⫽ Pert
Point-Slope Form
Slope-Intercept Form (slope m, y-intercept b)
Payments Required to Accumulate Amount A
y ⫺ y1 ⫽ m1x ⫺ x1 2
y ⫽ mx ⫹ b, where b ⫽ y1 ⫺ mx1
AR
P⫽
11 ⫹ R2 nt ⫺ 1
Parallel Lines
Perpendicular Lines
Accumulated Value of an Annuity
P
A ⫽ 冤11 ⫹ R2 nt ⫺ 1冥
R
Topics from Algebra
▼
abx 2 ⫹ 1ad ⫹ bc2 x ⫹ cd ⫽ 1ax ⫹ c21bx ⫹ d2
A ⫽ lw
Vectors and the Dot Product
• Given the vectors u ⫽ Ha, bI and v ⫽ Hc, dI, their dot product is denoted u • v and is
▼
x 2 ⫹ 1c ⫹ d2 x ⫹ cd ⫽ 1x ⫹ c21x ⫹ d2
P ⫽ 2l ⫹ 2w
⫹ 2k
⫹ 2k
⫹ i sin
b for k ⫽ 0, 1, 2, p , n ⫺ 1
n
n
• For a position vector, v ⫽ Ha, bI and angle as shown, a ⫽ ƒ v ƒ cos and b ⫽ ƒ v ƒ sin ,
b
where r ⫽ tan⫺1 ` ` and ƒ v ƒ ⫽ 2a2 ⫹ b2.
a
v
• For any nonzero vector v ⫽ Ha, bI ⫽ ai ⫹ bj, the vector u ⫽
is a unit vector in the
ƒvƒ
same direction as v.
▼
a2 ⫺ 2ab ⫹ b2 ⫽ 1a ⫺ b2 2
Rectangle
Roots and the nth Roots Theorem
Powers and DeMoivres Theorem
▼
a2 ⫹ 2ab ⫹ b2 ⫽ 1a ⫹ b2 2
Complex Numbers z ⴝ a ⴙ bi
Absolute Value
▼
gamma
lambda
phi
Special Factorizations
Slopes Are Equal: m1 ⫽ m2
Special Products
1a ⫹ b2 2 ⫽ a2 ⫹ 2ab ⫹ b2
1a ⫺ b2 2 ⫽ a2 ⫺ 2ab ⫹ b2
1a ⫹ b2 3 ⫽ a3 ⫹ 3a2b ⫹ 3ab2 ⫹ b3
1a ⫺ b2 3 ⫽ a3 ⫺ 3a2b ⫹ 3ab2 ⫺ b3
1x ⫹ c21x ⫹ d2 ⫽ x 2 ⫹ 1c ⫹ d2 x ⫹ cd
1ax ⫹ c21bx ⫹ d2 ⫽ abx 2 ⫹ 1ad ⫹ bc2 x ⫹ cd
ISBN: 0-07-351948-0
Author: John W. Coburn & J.D.Herdlick
Title: Trigonometry, 2e
Back endsheets
Color: 5
Pages: 6, 7
Slopes Have a Product of ⫺1: m1 ⫽ ⫺
1
m2
or
m1m2 ⫽ ⫺1
Intersecting Lines
Dependent (Coincident) Lines
Slopes Are Unequal: m1 ⫽ m2
Slopes and y-Intercepts Are Equal: m1 ⫽ m2, b1 ⫽ b2