Transcript 6_2 find exact values of trigonometric ratios _5-2_5-3

5-Minute Check Lesson 5-2A
5-Minute Check Lesson 5-2B
6.2: find exact values of
trigonometric ratios [5-2/5-3]





In this standard we will…
Review the side relationships of 30°-60°-90°
and 45°-45°-90° triangles.
Build trig ratios based 30°-60°-90° and 45°45°-90° triangles.
What exactly is a radian and why the pi?
Can I switch between radians and
degrees?
If they both measure angles why do I need
to learn radians at all?
The 6 Trig Ratios:
B
c
A
b
a
C
Set some up…
S
t
R
s
r
T
2
Right Triangles in Trig:
1


1 
-1
-1
Angles are in
Standard Position
in the Unit Circle.
They work off their
Reference Angle.
The ratios are set
up in terms of x, y
and r.
Try Some…

Example 1
The terminal side of angle θ in standard
position contains (8,-15), find the 6 trig ratios.

Now find the angle.

Try Some…

Example 2
The terminal side of angle θ in standard
position contains (-5,-7), find the 6 trig ratios.

Now find the angle.

Example 3
If the csc θ = - 13/5 and θ lies
in QIII, find all 6 trig values.

Now find the angle.
Example 4
If the cot θ = -1 and θ lies in
QII, find all 6 trig values.

Now find the angle.
Special Triangles from
Geometry:

Who are the 2 special triangles from
geometry?
Special Triangles in Trig

Used as reference angles.
45º- 45º- 90º
30º- 60º- 90º
The Unit Circle:
How to find ratios with the
special triangles
1.
2.
3.
4.
Set up the angle in standard position.
Set up the right triangle and determine the
reference angle.
Fill in sides of the triangle.
*Be Careful!* Place the proper signs for the
quadrant.
Build the required trig ratio.
1  sin
2  cos
3  tan
4  csc
5  sec
1  0
11  240
21  150
2  30
12  270
22  180
3  45
13  300
23  210
4  60
14  330
24  270
5  90
15  360
25  300
6  120
16  30
26  330
7  135
17  45
27  360
8  150
18  60
28  225
9  180
19  90
29  315
10  210
20  120
30  225
6  cot
Radians
What exactly is a radian and
why the pi?

What is a degree?

Radians are based on the circumference of the
unit circle.

Radian measurements are usually shown in
terms of π.

Radians are unitless. No unit or symbol is used.
Degrees and Radians

How do they relate?
Degree/Radian Conversions
180°
1 radian =
p
or approximately 57.3°
p
1 degree =
180°
or approximately 0.017 radians
Converting back and forth…

Example 1: Change 115º to a radian
measure in terms of pi.

Example 2: Change 210º to a radian
measure in terms of pi.
Converting back and forth…

3p
Example 3: Change
radians to degree
4
measure.

7p
Example 4: Change
radians to degree
8
measure.
Learning the standard angles
in radians:
  
,
, Circle
4 4 2
  
,
, Circle
6 3 2
The Unit Circle
Finding Trig Ratios with
Radian Measures:

Memorize the radian measures.

Force yourself to think in and recognize
radian measure without having to convert to
degrees.
Evaluate each expression:
æ 3p ö
÷
ç
÷
Example 1: cos ç÷
çè 4 ø
÷
Example 2: tan5p
7p
Example 3: sin
3
1 = sin
1= 0
2=
3=
4=
5=
6=
7=
8=
p
6
p
4
p
3
p
2
2p
3
3p
4
5p
6
2 = cos
3 = tan
9= p
10 =
11 =
12 =
13 =
14 =
15 =
16 =
7p
6
5p
4
4p
3
3p
2
5p
3
7p
4
11p
6
4 = csc
5 = sec
6 = cot
11p
2
2p
26 = 3
17 = 2p
25 = -
p
6
p
19 = 4
3p
20 = 4
27 = - 6p
18 = -
21 = - p
11p
22 = 6
23 = - 5p
24 =
17 p
6
13p
6
22p
29 =
6
13p
30 = 4
28 = -
Homework:
• P 288 #11 – 21 odd
• P 296 #15 – 43 odd