Aim: What good is the Unit Circle and how does it help us to understand the Trigonometric Functions?. Do Now:. A circle has a radius of 3 cm. Find the length of an arc cut off by a central angle of 270 0. Q II. Quadrant I. terminal side. 90 < < 180. 0 < < 90. terminal side.
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Do Now:
A circle has a radius of 3 cm. Find the length of an arc cut off by a central angle of 2700.
Q II to understand the Trigonometric Functions?
Quadrant I
terminal side
90 < < 180
0 < < 90
terminal side
initial side
t.s.
Q III
Q IV
t.s
180 < < 270
270 < < 360
y
x
Q II to understand the Trigonometric Functions?
Quadrant I
90 < < 180
0 < < 90
initial side

t.s.
Q III
Q IV
180 < < 270
270 < < 360
Coterminal and Negative Anglesy
3000 =
x
600
Q II to understand the Trigonometric Functions?
Quadrant I
90 < < 180
0 < < 90
Q III
Q IV
180 < < 270
270 < < 360
y
4850
1250
x
1250 and 4850 are coterminal
215o
315o
25o
xaxis
QIV
QII
hypotenuse to understand the Trigonometric Functions?
side opp.
cos
side adj.
Unit Circley
1
radius = 1
center at (0,0)
cos , sin
(x,y)
x
1
1
1
Do Now:
Find the measure of an angle between 00 and 3600 coterminal with an angle whose measure is 1250.
3 to understand the Trigonometric Functions?
Hypotenuse = 2 shorter leg
Longer leg = shorter leg
Value of Sine & Cosine: Quadrant Iy
1
radius = 1
center at (0,0)
cos 600, sin 600
(x,y)
600
x
1
1
What is the value of coordinates (x,y)?
300600900 triangle
Sine and Cosine values for angles in
Quadrant I are positive.
( to understand the Trigonometric Functions?x,y)
1200
side adj.
3
Hypotenuse = 2 shorter leg
Longer leg = shorter leg
Sine values for angles in Quadrant II are positive.
Value of Sine & Cosine: Quadrant IIy
1
cos 1200, sin 1200
What is the value
of coordinates (x,y)?
1
60º is the
reference
angle
(180º120º)
600
x
1
1
directed distance
A reference angle for any angle in standard
position is an acute angle formed by the terminal
side of the given angle and the xaxis.
What is the cosine/sine of a 1200 angle?
300600900 triangle
Cosine values for angles in Quadrant II are negative.
side opp. to understand the Trigonometric Functions?
side adj.
Value of Sine & Cosine: Quadrant IIIy
1
What is the value
of coordinates (x,y)?
What is the cosine/sine
of a 2400 angle?
2400
directed distance
60º is the
reference
angle
(240º180º)
x
1
600
1
directed dist.
1
(x,y)
cos 2400, sin 2400
Sine and Cosine values for angles in
Quadrant III are both negative.
side opp. to understand the Trigonometric Functions?
(x,y)
Sine values for angles in Quadrant IV are negative.
Value of Sine & Cosine: Quadrant IVy
1
What is the value
of coordinates (x,y)?
What is the cosine/sine
of a 3000 angle?
60º is the
reference
angle
(360º300º)
3000
x
1
600
1
directed dist.
1
cos 3000, sin 3000
Cosine values for angles in Quadrant IV are positive.
Periodic to understand the Trigonometric Functions?
Unit Circle – 8 Equal ArcsNegative Angles Identities
y to understand the Trigonometric Functions?
Quadrant II
Quadrant I
x
Quadrant III
Quadrant IV
Value of Sine & Cosine in Coordinate Planecos is +
sin is +
cos is –
sin is +
cos is +
sin is –
cos is –
sin is –
for any angle in standard
position is an acute angle formed by the terminal
side of the given angle and the xaxis.
The reference angle:
y to understand the Trigonometric Functions?
1
x
1
1
1
Regents PrepOn the unit circle shown in the diagram below, sketch an angle, in standard position, whose degree measure is 240 and find the exact value of sin 240o.
Do Now:
Use the unit circle to find:
a. sin 1800 () b. cos 1800
to understand the Trigonometric Functions?
(1,0)
Model ProblemsUse the unit circle to find:
a. sin 1800 () b. cos 1800
(x, y) = (1, 0)
sin 1800 = y
= 0
cos 1800 = x
= 1
180º  quadrantal angle
sin to understand the Trigonometric Functions?
sin
cos
cos = 1
1
( 1, tan)
( , )?
Tan radius = 1
center at (0,0)
cos , sin
y
1
(x,y)
tan
1
x
1
1
Quadrant I to understand the Trigonometric Functions?
Q II
90 < < 180
0 < < 90
Q III
Q IV
180 < < 270
270 < < 360
Trigonometric Values  A C T Sy
S Sine is +
A
All are +
x
T
Tangent is +
C
Cosine is +
1 to understand the Trigonometric Functions?
y
r
y
x
1
x
1
1
Reciprocal Functions
Negative Angles Identities
csc = 1/y
sec = 1/x
cot = x/y
denominators 0
Need to KnowsWhen r = 1
sin = y
cos = x
tan = y/x
y to understand the Trigonometric Functions?
(x,y)
1
1
sin = y
2
2
450
x
1
1
cos = x
In a 450450900 triangle, the length of the hypotenuse is times the length of a leg.
1
(x)
length of hypo. =
Model ProblemsUsing the unit circle, find
cos 450 (/4)
sin 450
tan 450
450450900 triangle
A 450450900 triangle is an isosceles right triangle.
therefore x = y
cos = sin
y to understand the Trigonometric Functions?
1
1
sin = y
45º
x
1
1
cos = x
=
=
1
Model ProblemsUsing the unit
circle, find
cos 45º(/4)
sin 45º
tan 45º
(x,y)
cos 45º = x
tan 45 = 1
sin 45º = y
to understand the Trigonometric Functions?
0º
0
30º
/6
45º
/4
60º
/3
90º
/2
sin
0
1
cos
1
0
tan
0
1
UND.
Trigonometric Values for Special AnglesWhy is tan 90º undefined?
What is the slope of a line perpendicular to the xaxis?
= slope
What is the cos 510º (17 to understand the Trigonometric Functions?/6)?
What is the tan 135º (3/4)?
≈ .866…
Given:
sin 68o = 0.9272
cos 68o = 0.3746
Find cot 112o
reference angle for 112o is 68o; 112o is in QII; tan and cot are negative in QII
WHAT ELSE DO WE KNOW?
Express sin 285º as the function of an angle
whose measure is less than 45º.
What do we know?
285º in IV quadrant
the sine of a IV quadrant
angle is negative
sin 75º
reference angle for 285º is
(360 – 285) = 75º
> 45º
sine and cosine are cofunctions
complement of 75º is 15º
< 45º
sin 285º =
= cos 15º
sin 75º
Algebraically:
Find:
sin (π/3)
remember:
π/3 radians
π/3
60º
sin 60º =
≈ .866…
Using the calculator:
Use the mode key:
change setting from degrees to radians
then hit:
sin
2nd
π
÷
ENTER
3
Display: .8660254083
y to understand the Trigonometric Functions?
1
1
1
unit circle
1
x
1
Ununit circle is any angle in standard position with (x, y) any point on the terminal side of and
r 1
4 to understand the Trigonometric Functions?
r
= 5
3
Model Problem(3, 4) is a point on the terminal side of . Find the sine, cosine, and tangent of .
Q II
1 to understand the Trigonometric Functions?
r
= 2
Model Problemis a point on the terminal side of . Find , the sine, cosine, and tangent of .
Q III
Tan = 5/4 and cos > 0, find sin and sec
When tangent is negative and cosine is positive angle is found in Q IV.
The terminal side of is in quadrant I and lies on the line y = 6x. Find tan ; find .
y = mx + b  slope intercept form of equation
m = slope of line
y = 6x
m = 6 = tan
Q I
The terminal side of is in quadrant IV and lies on the line 2x + 5y = 0. Find cos .
y = mx + bslope intercept
form of equation
tan = m = 2/5