Download Module 17 Notes (PPT- pages not included)

Goal: Understand radian measure of an
angle as the length of the arc on the unit
circle subtended by the angle.
• Draw and find angles in standard position.
• Convert between degree measures and
radian measures.
NEW VOCABULARY
A curved
arrow is drawn
to the ending
position of the
ray, or terminal
side of the angle.
Angle of
rotation: is an
angle formed
by the starting
and ending
positions of a
ray.
The angle is in standard
position in a coordinate plane
when the starting position of the
ray, or initial side of the angle,
is on the positive x-axis.
Angle Rotations
Positive Theta
(between 0° − 360°)
the arrow is drawn
Counterclockwise
Negative Theta
(between 0° − 360°)
the arrow is drawn
clockwise
Example: Standard Form
A. Draw an angle with a
measure of 210° in
standard position.
B. Draw an angle with a
measure of –45° in standard
position.
Draw an Angle in Standard Position
C. DIVING In a springboard diving competition, a diver
made a 900-degree rotation before slicing into the
water. Draw an angle in standard position that
measures 900°.
900
= 2.5
360
Co-Terminal Angles: Twin Angles
Co-terminal angles are angles that share the
same terminal side.
 Can be negative or > 360°
A) What is the positive co-terminal
angle for 420° between [0,360]?
420 − 360 = 60°
B) What is the negative
co-terminal angle for 420°?
between [0,-360]? 420 − 360 = 60
60 − 360 = −300°
Your Turn!
SNOWBOARDING While riding down the mountain, a
snowboarder goes off a jump and turns 600° before touching
down onto the snow again. Determine how many degrees past
the positive x-axis the snowboarder lands.
A. 120°
B. 180°
C. 240°
D. 300°
Your Turn!
A. Draw an angle with a measure of 225° in
standard position.
C.
A.
B.
D.
B. Draw an angle with a measure of –60° in standard position.
A.
B.
C.
D.
Your Turn!
A. Find an angle with a positive measure and an angle with
a negative measure that are co-terminal with 330°.
A.
–30°, 690°
C.
–60°, 630°
B.
–30°, 630°
D.
–60°, 720°
B. Find an angle with a positive measure and an angle with a
negative measure that are co-terminal with –80°.
What is a Radian?
• The radian measure of the angle is the
𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ
ratio 𝜃 =
. We use radius=1
𝑟𝑎𝑑𝑖𝑢𝑠
• Recall that there are 360° in a full circle
and the circumference is 2𝜋.
• Therefore 360° =
2𝜋
1
= 2𝜋
Radian to Degrees to Radians
Convert Between Degrees and Radians
A. Rewrite 30° in radians.
B. Rewrite
𝜋
𝜋
30° ∙
=
180° 6
in degrees.
−5𝜋 180°
∙
= −300°
3
𝜋
C. Complete the Table
22.5°
180°
𝜋
240 °
Your Turn!
A. Rewrite 45° in radians.
A.
C.
B.
D.
C. Rewrite -495° in radians.
B. Rewrite
D. Rewrite
13𝜋
12
in degrees.
in degrees.
The Unit Circle
𝑹𝒂𝒅𝒊𝒖𝒔(𝒓) = 𝟏
The unit circle
below shows the
measures of angles
of rotation that are
commonly used in
trigonometry, with
radian measures
outside the circle
and degree
measures inside
the circle. Provide
the missing
measures.
Your Turn!
The Unit Circle
𝑹𝒂𝒅𝒊𝒖𝒔(𝒓) = 𝟏
The Unit Circle
𝑹𝒂𝒅𝒊𝒖𝒔(𝒓) = 𝟏
𝑃𝑟𝑖𝑜𝑟 𝐾𝑛𝑜𝑤𝑙𝑒𝑑𝑔𝑒 𝑁𝑒𝑒𝑑𝑒𝑑:
𝑺𝑶𝑯 − 𝑪𝑨𝑯 − (𝑻𝑶𝑨)
𝑂𝑝𝑝
=
𝐻𝑦𝑝
𝐴𝑑𝑗
=
𝐻𝑦𝑝
𝐼𝑓 𝑡ℎ𝑒 𝑟𝑎𝑑𝑖𝑢𝑠 𝑖𝑠 1:
𝐹𝑜𝑟 𝐴𝑁𝑌 𝑟𝑎𝑑𝑖𝑢𝑠:
( x ,y )
𝑆𝑖𝑛 = 𝑦
𝐶𝑜𝑠 = 𝑥
𝑦
𝑆𝑖𝑛 =
𝑟
x
𝐶𝑜𝑠 =
r
Class Activity:
• Finding the 𝑠𝑖𝑛𝑒 𝜃, 𝑐𝑜𝑠𝑖𝑛𝑒 𝜃 of each angle
in each quadrant for the special triangles
• I will: do the first one.
• As a class: we do the second
• In pairs: do the third
𝟐
(−
𝟐,
𝟐
𝟐)
(
𝟐
𝟐,
𝟐
45-45-90 Triangle
𝟐)
1) Inscribe the 45-45-90 Triangle in
all four quadrants.
2) Find the 𝑠𝑖𝑛 𝜃, 𝑐𝑜𝑠 𝜃
sin 𝜃 =
(−
𝟐
𝟐,
−
𝟐
𝟐)
(
𝟐
𝟐,
−
𝟐
𝑂𝑝𝑝 𝑦
=
𝐻𝑦𝑝 1
𝐴𝑑𝑗 𝑥
𝑐𝑜𝑠𝜃 =
=
𝐻𝑦𝑝 1
𝟐)
135
225
2
− 22
− 22
2
− 22
315
− 22
2
2
(− 𝟏 𝟐,
𝟑
𝟐)
(𝟏 𝟐,
𝟑
60-30-90 Triangle
𝟐)
1) Inscribe the 60-30-90 Triangle in
all four quadrants.
2) Find the 𝑠𝑖𝑛 𝜃, 𝑐𝑜𝑠 𝜃
sin 𝜃 =
(− 𝟏
𝟐,
−
𝟑
𝟐)
(𝟏 𝟐, −
𝟑
𝑂𝑝𝑝 𝑦
=
𝐻𝑦𝑝 1
𝐴𝑑𝑗 𝑥
𝑐𝑜𝑠𝜃 =
=
𝐻𝑦𝑝 1
𝟐)
120
240
300
3
− 32
−1 2
− 32
1
2
2
−1 2
30-60-90 Triangle
1) Inscribe the 30-60-90 Triangle in
all four quadrants.
2) Find the 𝑠𝑖𝑛 𝜃, 𝑐𝑜𝑠 𝜃 .
sin 𝜃 =
𝑂𝑝𝑝 𝑦
=
𝐻𝑦𝑝 1
150
1
2
− 32
𝐴𝑑𝑗 𝑥
𝑐𝑜𝑠𝜃 =
=
𝐻𝑦𝑝 1
210
−1 2
− 3
2
330
−1 2
3
2
You can now be able to tell me the sine
and cosine of any angle OR Point in the
circle or beyond the circle!
MEMORIZE OR KNOW HOW TO SOLVE
Watch:
https://learnzillion.com/lesson_plans/8031-understand-the-wrapping-functionusing-the-unit-circle
0.29 s - end
Other things you can do:
• What if I give you a Triangle?
• What if I give you a point?
Find sin , cos 
𝑜𝑝𝑝
4
12
𝑠𝑖𝑛𝜃 =
ℎ𝑦𝑝 = 15 = 5
9
3
𝑎𝑑𝑗
=
=
𝑐𝑜𝑠𝜃 =
15
5
ℎ𝑦𝑝
15
The terminal side of angle  in standard position intersects
the unit circle at
3
𝑠𝑖𝑛𝜃 = 𝑦 =
4
7
𝑐𝑜𝑠𝜃 = 𝑥 =
4
.Find cos  and sin .
Your Turn!
The terminal side of angle  in standard position intersects
the unit circle at
Find cos  and sin .
The reference angle of a rotation angle is
the acute angle from the terminal side of the
rotation angle to the x-axis.
(+, +)
(−, +)
(𝑪𝒐𝒔, 𝑺𝒊𝒏)
(−, −)
RECALL:
(+, −)
𝐶𝑜𝑠 = 𝑥
𝑆𝑖𝑛 = 𝑦
1. Find the Co-terminal Angle
2. Use table to find Reference Angle
3. Choose the sign (positive or negative) based on
Quadrant (or table)
Sin 7𝜋
1. Divide the number to find how many half rotations
7
1
3 full rotations (2𝜋)
+1 half rotation (𝜋)
Where is 7𝜋 on the circle?
Edge of the circle
𝑵𝑶𝑻𝑬:
𝝅 𝒊𝒔 𝒂 𝒉𝒂𝒍𝒇 𝒓𝒐𝒕𝒂𝒕𝒊𝒐𝒏
(−1,0) 5𝜋
𝜋
7𝜋
3𝜋
What is this
coordinate
point?
Therefore,
sin(7𝜋) = 0
6𝜋
0
4𝜋
2𝜋
Sin 7𝜋
1. Divide the number to find how many half rotations
7
1
3 full rotations (2𝜋)
+1 half rotation (𝜋)
Edge of the circle
2. Rewrite the question without the full rotations
1 half rotation = 𝜋
si𝑛 𝜋 = 0
1. Divide the number to find how many rotations
2 full rotations (2𝜋)
16
1
=5
+1 half rotation (𝜋)
3
3
+ 1/3
2. Rewrite the question without the full rotations
4𝜋
𝜋 + 1/3 𝜋 =
3
4𝜋
cos
3
Third
quadrant
𝑵𝑶𝑻𝑬:
𝝅 𝒊𝒔 𝒂 𝒉𝒂𝒍𝒇 𝒓𝒐𝒕𝒂𝒕𝒊𝒐𝒏
𝜋
5𝜋
3𝜋
What is this
coordinate
point?
1
3
− ,− ,
2
2
60°
4𝜋
3
Therefore,
cos(16𝜋/3) = −1/2
0
4𝜋
2𝜋
4𝜋
cos
3
3. Use Reference table (if necessary)
𝜋 180°
4𝜋
𝜃′ =
−𝜋 = ×
= 60°
3
3
𝜋
cos 𝜃′ = 1/2
**Cos is negative in the
third quadrant**
16𝜋
1
cos
=−
3
2
Extra Practice with Co-terminal and Evaluating Sine
and Cosine
Calculator: Sine and Cosine
1)Press the number
2) Click sin or cos
To find cos 3
3
To find sin
3
∗𝜋 =
2
= −.98999
3
𝜋
2
= −1
Introducing your new FAVORITE
Pythagorean Theorem!
Favorite Identity: The Proof
The terminal side of an angle θ intersects
the unit circle at the point (a, b) as shown.
Write a and b in terms of trigonometric
functions involving θ.
cos 𝜃
a= ________
sin 𝜃
b=________
cos 𝜃
cos 2 𝜃
sin 𝜃
sin2 𝜃
1
1
Favorite Identity: Confirming it works
2
2
sin 𝜃 + cos 𝜃 = 1
𝜋
sin
3
3
2
2
2
𝜋
+ cos
3
2
2
=1
1
+
=1
2
3 1
+ =1
4 4
1. Plug in given value of sin or cosine
2. Solve for the missing trig function
3. Find the quadrant of the given Trig Function.
Use this to decide if and should be positive or
negative
sin2 𝜃 + cos2 𝜃 = 1
0.766
2
(+, +)
+ cos2 𝜃 = 1
0.586 + cos2 𝜃 = 1
cos2 𝜃 = 1 − 0.586
cos2 𝜃 = 0.414
cos2 𝜃 = 0.414
cos 𝜃 = ± 0.643
Cos is positive in
first quadrant
𝑊𝐴𝐼𝑇! 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑜𝑟 𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒?
sin2 𝜃 + cos2 𝜃 = 1
sin2 𝜃 + −0.906
2
=1
sin2 𝜃 + 0.821 = 1
(−, +)
sin2 𝜃 = 1 − 0.821
sin2 𝜃 = 0.179
sin2 𝜃 = 0.179
sin 𝜃 = ±0.423
Sin is positive in
first quadrant
𝑊𝐴𝐼𝑇! 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑜𝑟 𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒?
Your Turn!
A. Find the exact value of
cos 660°.
B. Find the exact
7𝜋
value of sin(− )
4
1. Find the exact value
of sin 225°.
A.
B.
C.
D.