ch 4 trig functions. 4.1 radian and degree measures converting from radians to degrees converting...

Ch 4 Trig Functions

4.1 Radian and Degree Measures

Converting from Radians to Degrees

Converting from Degrees to Radians

180

radiansdegrees

180

degreesradians

4.1 Angles in Standard Position

vertex at origin initial side

on positive x-axis

terminal sidecounter-clockwise from initial side

40

50complement

140supplement

4.1 Arc Length

240

rs length arcs

radiansin angle

4.1 Arc Length

rs

Find the length of the arc intercepted by a central angle of 45o in a circle with a radius of 10cm.

410

cms

cms2

5

4.1 Linear and Angular Speed

240

time

angle centralspeedAngular

time

length arcspeedLinear

4.1 degrees, minutes, seconds

Converting minutes and seconds to degrees.

Converting decimal degrees to minutes and seconds.

3600

seconds

60

minutes + degrees = degrees

'8.2535

'6043.3543.35

"608.0'2535

"48'2535

4.2 Unit Circle

4.4 All Students Take Calculus

4.4 Evaluating Trig Functions of Any Angle

Given and , find and .4

5tan

4145 22 r

41

41

415sin

0cos secsin

tan (-) and cos (+) = QIV

Draw angle from origin to x-axis.-5

4

4

41

cos

1sec

4.4 Reference Angle

Angle to x-axis.

' '

' 2'

4.5 Graphs of Sine and Cosine Functions

4.5 Graphs of Sine and Cosine Functions

bxaybxay cosor sin

aamplitude

xy 2cos3Find the amplitude and period.

b

π2period

3amplitude period

4.5 Graphs of Sine and Cosine Functions

cbxaky sin

kShift Vertical

a

c Shift Horizontal

Vertical and Horizontal Shifts

4.7 Inverse Trig Functions

h

osin

h

oarcsin

Take the sin of an angle to get a ratio

Take the arcsin of a ratio to get an angle

4.7 Inverse Trig Functions

5

3arcsincos

5

4cos

Find the exact value.

-35

4

5.1 Identities

Pythagorean Identities Quotient Identities

sin2 cos2 1

1 tan2 sec21 cot2 csc2

x

xx

cos

sintan

x

xx

sin

coscot

Be familiar with identities on the inside of front and back cover of book (on blue cheat sheet).

6.1 Law of Sines

The ratios of angles and corresponding sides are equal.

C

c

B

b

A

a

sinsinsin

A

b

12c73B

a

80CFind b.

6.2 Law of Cosines

Abccba cos222

A

b

12c73B

9a

CFind b.

bc

acbA

2cos

222

6.3 Vectors

VectorsA vector whose initial point is at the origin is in standard position.

The magnitude of a vector is its length.

Vectors

The magnitude (length) of v

v x2 x1 2 y2 y1 2

initial point P x1,y1

terminal point Q x2,y2

VectorsThe components (direction) of v

one unit to the lefttwo units up

Scalar Multiplication of VectorsVectors can be multiplied to change its scale.

Vectors AdditionVectors can be added.

=commutative

Addition of Vectors

The Unit VectorTo get a unit vector, divide the vector by its magnitude.

u = unit vector

vv

or 1v

v

i and jhorizontal and vertical components

horizontal component ivertical components j

v =

2,4 2i 4 j

v1,v2 v1 v2v = i j

The Unit VectorWrite a vector as a combination of unit vectors.

i represents a horizontal unit vectorj represents a vertical unit vector

i i ijj

jj

Unit Vectors on Unit Circleu

x,y cos, sin

cosi sinj

Find the magnitude and direction angle of the vector.

v 9 cos30 i sin 30 j

v 9

30