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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