Mth 162 Pre Cal 2 Unit 3

Front 1

Law of Sines

Back 1

a/sinA = b/sinB = c/sinC or sinA/a = sinB/b = sinC/c

Front 2

Law of Sines Ambiguous case - no triangle

Back 2

a<h, side a is too short to meet side c

Front 3

Law of Sines Ambiguous case - one right triangle

Back 3

a=h, side a equals the altitude

Front 4

Law of Sines Ambiguous case - one oblique triangle

Back 4

a>b, one triangle is possible

Front 5

Law of Sines Ambiguous case - two triangles one acute and one obtuse

Back 5

h<a<b, side a intersects side c at an obtuse angle or acute angle

Front 6

Area of a triangle given 2 sides and the angle between them

Back 6

(1/2)bcsinA or (1/2)absinC or (1/2)acsinB

Front 7

Law of cosines a

Back 7

a squared = b squared + c squared -2bccosA or cosA = (b squared + c squared - a squared)/2bc

Front 8

Law of cosines b

Back 8

b squared = a squared + c squared - 2accosB or cosB = (a squared + c squared - b squared)/2ac

Front 9

Law of cosines c

Back 9

c squared = a squared + b squared - 2abcosC or cos C = (a squared + b squared - c squared)/2ab

Front 10

Guidelines for solving an iblique triangle given SAS 1

Back 10

Find the length of the side opposite the known angle by using law of cosines

Front 11

Guidelines for solving an iblique triangle given SAS 2

Back 11

Use the law of sines to find the measure of the angle opposite the shorter of the two given sides. This angle will always be acute. Alternatively, use the law of cosines (alternative form) to find either remaining angle

Front 12

Guidelines for solving an iblique triangle given SAS 3

Back 12

Find the measure of the third angle by subtracting the sum of the measures of the other two angles from 180

Front 13

Guidelines for solving an oblique triangle given SSS 1

Back 13

Use the alternative formulas for the law of cosines to find the largest angle of the triangle ( angle opposite the longest side)

Front 14

Guidelines for solving an oblique triangle given SSS 2

Back 14

Find either the remaining two angles using either the law of sines or the law of cosines. Since the largest angle was found in step 1, the two remaining angles are guaranteed to be acute.

Front 15

Guidelines for solving an oblique triangle given SSS 3

Back 15

Find the measure of the third angle by subtracting the sum of the measures of the other two angles from 180

Front 16

Heron's formula for area of a triangle

Back 16

Area = square root of (s(s-a)(s-b)(s-c) where s is the semi perimeter: s=1/2(a+b+c)

Front 17

Polar coordinate system definition

Back 17

Consists of a fixes point O called the pole (or origin), and a ray called the polar axis, with endpoint at the pole. Each point P in the plane is defined by an ordered pair of the form (r, theta) where r is the directed distance from the pole to P

Front 18

Polar coordinate system 2

Back 18

If r > 0, point P is located r units from the pole in the direction of theta

If r < 0, point P is located |r| units from the pole in the direction of theta + pi (the direction opposite theta)

If r = 0, point P is located at the pole

Front 19

Polar coordinate system 3

Back 19

Theta is a directed angle from the polar axis to line OP

Theta > 0 is measured counterclockwise from the polar axis

Theta < 0 is measured clockwise from the polar axis

Front 20

Convert between rectangular and polar coordinates

Back 20

To convert polar coordinates (r, theta) to rectangular coordinates (x,y), use x=rcos theta and y=rsin theta

Front 21

Convert between rectangular and polar coordinates

Back 21

To convert the rectangular coordinates (x,y) to polar coordinates (r,theta), use r squared = x squared + y squared and tan theta = y/x for x does not = 0

Front 22

Scalars

Back 22

Quantities that are described by a single value called the magnitude

Front 23

Magnitude

Back 23

Size or measure of the quantity

Front 24

Vector quantities

Back 24

Quantities that are described by both magnitude and direction

Front 25

Speed

Back 25

Scalar quantity that tells us how fast an object travels

Front 26

Velocity

Back 26

Vector quantity that tells us how fast the object travels and in which direction

Front 27

Vector

Back 27

In a plane it is a line segment with a specified direction

Front 28

Component form of a vector

Back 28

V={x2-x1,y2-y1}

Front 29

Equality of vectors

Back 29

If v=《a1,b1》and w=《a2,b2》, then v=w if and only if a1=a2 and b1=b2

Front 30

Vector addition

Back 30

Let v =《a1,b1》, w=《a2,b2》and c be a real number

V+w=《a1+a2,b1+b2》

Front 31

Vector subtraction

Back 31

Let v =《a1,b1》, w=《a2,b2》and c be a real number

V-w=《a1-a2,b1-b2》

Front 32

Multiplication of a vector by a scalar

Back 32

Let v =《a1,b1》, w=《a2,b2》and c be a real number

Cv=《ca1,cb1》

Front 33

Unit vector

Back 33

Vector that has a magnitude of 1

Front 34

Find a unit vector in the direction of a given vector

Back 34

If v = {a,b}, then a unit vector Uv in the direction of v is given by

Uv = (1/||v||)v=(1/||v||){a,b}={a/||v||,b/||v||}

If v = {a,b}, then a unit vector Uv in the direction of v is given byUv = (1/||v||)v=(1/||v||){a,b}={a/||v||,b/||v||}

Front 35

Represent vectors in terms of I and J

Back 35

The representation of v={a,b} in terms of i and j is v=ai+bj

The values of a and b are called scalar horizontal and vertical components of v, respectively

Front 36

Magnitude direction and components of a vector 1

Back 36

Let v={a,b} be a vector in standard position and let 0 be less than or equal to theta which is less thenan 360° be the direction of v measured counterclockwise from the positive x axis

Front 37

Magnitude direction and components of a vector 2

Back 37

||v||=square root of a squared + b squared and tan theta = b/a (where a is not = to 0) (magnitude and direction of v)

Front 38

Magnitude direction and components of a vector 3

Back 38

a=||v||cos theta and b=||v||sin theta

v={a,b}={||v||cos theta, ||v||sin theta} or v=ai+bj=||v|| (cos theta)i + ||v||(sin theta)j

Front 39

Dot product

Back 39

If v={a1,b1} and w={a2,b2}, the dot product v•w is defined as

v•w=a1a2+b1b2

Front 40

Angle between 2 vectors

Back 40

If theta is the angle between 2 nonzero vectors v and w, then

Cos theta = (v•w)/||v|| ||w|| and theta = cos-1(v•w/||v|| ||w||)

Front 41

Orthogonal vectors

Back 41

Two vectors v and w are orthogonal if and only if v•w=0

Front 42

Decomposition of a vector into orthogonal vectors 1

Back 42

If v and w are 2 nonzero vectors then1. The vector projections of v onto w is given by projwv=((v•w)/||w||squared)w

If v and w are 2 nonzero vectors then1. The vector projections of v onto w is given by projwv=((v•w)/||w||squared)w

Front 43

Decomposition of a vector into orthogonal vectors 2

Back 43

2. To decompose v into vectors v1 and v2 parallel to w and orthogonal to w, respectively, we have

v1=projwv=((v•w)/||w||squared)w and v2=v-v1

Front 44

Definition of work 1

Back 44

If D is the displacement vector of an object in moving the object in a straight line from points A to B under a constant force F, then the work W done is computed by

1. W=F•D

Front 45

Definition of work 2

Back 45

W=||projdF|| ||D|| or W=||F|| ||D||cos theta, where theta is the angle between F and D