1-28-16 to 2-1-16
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T2.1a To define and apply the Six Trigonometric
Functions to Standard Angles
1
Active Learning Assignment Questions?
OPENER: Solve the following equations.
1.
2x +3 = 21
2.
0.8x = 17
3.
45x = x +5
4.
2x – 85 = 13x + 58
2
The Legend of SOH CAH TOA
Trigon on my tree.
Trigonometry
Sine language
OPPOSITE
SINE 
HYPOTENUSE
Hypoten News
Hypotenuse
ADJACENT
COSINE 
HYPOTENUSE
TANGENT 
OPPOSITE
ADJACENT
3
The Six-Trigonometric Functions (Ratios):
Hypotenuse
Opposite

r

y
x
Adjacent
opp y
sineθ = sinθ =
=
hyp r
1
0.4
3
1.2
From geometry, we know
similar triangles have the
same ratios of sides. What is
this length?
This is what the 6 trig ratios
are based on: combinations
of opposite, adjacent, and 4
hypotenuse!
The Six-Trigonometric Functions (Ratios):
Hypotenuse

Opposite
r
y
y
x
r

y
r
y
x
r
Adjacent
opp y
sineθ = sinθ =
=
hyp r
Watch carefully! No
matter which quadrant
the reference angle is in,
the x value remain
adjacent and the y value
remains opposite!
5
The Six-Trigonometric Functions (Ratios):
Hypotenuse

Opposite
r

y
x
Adjacent
*
opp y
sineθ = sinθ =
=
hyp r
adj x
cosineθ = cosθ =
=
hyp r
opp y
tangent θ = tanθ =
=
adj x
hyp r
cosecant θ = cscθ =
=
opp y
hyp r
secant θ = secθ =
=
adj x
adj x
cotangent θ = cot θ =
=
opp y
6
Reciprocal Functions: Definitions are reciprocated
The Six-Trigonometric Functions (Ratios):
You will have a definitions quiz on these. This is what
you will write on your quiz. (In this exact order.)
*
opp y
sineθ = sinθ =
=
hyp r
adj x
cosineθ = cosθ =
=
hyp r
opp y
tangent θ = tanθ =
=
adj x
hyp r
cosecant θ = cscθ =
=
opp y
hyp r
secant θ = secθ =
=
adj x
adj x
cotangent θ = cot θ =
=
opp y
Reciprocal Functions: Definitions are reciprocated
7
QUIZ: ON THE SIX TRIG DEFINITIONS
(FUNCTIONS, RATIOS, IDENTITIES, ETC.)
Ex: If given 1. _______ = _____ = __
You will write: sin  
opp
y

hyp
r
Write functions in a “U”!
8
Quadrangle Angles
y 1
sin 90 
 1
r
1
II
y
0
sin180 
0

r
1
y
x y
tan 90 
(

1,
0)
x
180 
1
  (Undefined)

0
90° III
x y
(0, 1)
90

0
zero in num.
K
(0,0)
N
zero in den.
0

270
(0, 1)
x
y
I
r 1
x y
(1, 0)
 0
360
IV
9
Find all the
EXACT
values for the
6 trig functions
for 0°
0
zero in num.
K
0
y

sin 0 
1
r
0°
r=1
(1, 0)
x, y
0
1
x
 1
cos 0 
1
r
0
y
 0
tan 0 
1
x
N
zero in den.
0
1
r
 
csc 0 
0
y
1
r

sec 0 
1
1
x
x 1
cot 0    
y 0 10
30° Reference Angles: Can we multiply or scale up each side
by 2 and still have a similar triangle?
1 2  3
II
An easier set
of ratios to
remember!
I
1 2  2 60 1
2  1
2
30°
3
2 
2
III
3
IV
11
Find all the
EXACT values
of the 6 trig
functions for 30°:
30*  60  90
2
30
3
yields 1, 2, 3 ratios
60
1
1
sin 30 
2
2
 2
csc 30 
1
3
cos 30 
2
2
3
2 3
sec 30 


3
3
3
1
3
3
tan 30 


3
3
3
3

cot 30 
1
3
12
Let’s look at 150º Find the
EXACT values for:
1 2  3
II
I
150
1
sin150 
2
cos150   3
2
1
3
tan150 

 3
3
2
60
1
30
 3
180
III
IV
 3

3
Why is the sin 150 ° positive and the others are negative?
Could you do the rest of them?
13
60° Reference Angles: Will it have the same ratios as a
30° reference angle?
1 2  3
II
An easier set
of ratios to
remember!
I
2
30°
3
60°
1
III
IV
14
Let’s try 120º (Can we write rules?)
120
30
2
3
180
60
1
3
sin 120 
2
1

cos 120  
2
tan 120  3   3
1
*
30  60  90
yields 1, 2, 3 ratios
Find all the EXACT values
for the 6 trig functions:
2
3
2 3
csc 120  3  3  3
2
 2
sec 120 
1
3
1
3




cot
120

3
3
3
15
45° Reference Angles: Can we “scale up” on a 45°
reference angle and still have a
similar triangle?
11 2
An easier set
of ratios to
remember!
II
I
1 2 
45°
2
2  2 1
2
2 21
2
III
IV
16
Find all the
EXACT values
of the 6 trig
functions for 45°:
*
45  45  90
2
1
yields 1, 1, 2 ratios
45
1
1
2
sin 45    2
2
2
2
csc 45
2
1
2


cos 45 
2
2
2
sec 45 
1
tan
45

 1
1
2

1
2
2
 2
1
1
 117
cot 45 
1
Let’s look at 225º
45  45  90
yields 1, 1, 2 ratio
1
sin 225 

2
1
180
1
225
2
2
45
2
2
 
2
Why is the hypotenuse always positive?
Could you do the rest of them?
18
Let’s write some rules for finding the exact values of
trigonometric functions for standard angles.
*
1. Which quadrant?
2. Reference angle?
3. Draw a triangle.
4. Ratios?
5. Positive or negative?
6. Trig functions?
19
Active Learning Assignment:
Find all 6 trig functions for:
135°, 180°, 240°, AND 330°.
WOW:
Spend less time worrying who’s
right and more time on what’s right.