Transcript Unit Circle - Ms. Huls` Math

Welcome Pre-Calc!
o Take Out: Notes from last week and Unit Circle on yellow paper.
o HW:
Period 3: Pg. 296 #15-29 odd, #31, 33, 39
Period 4: Pg. 296 #15-29 odd, #31, 33, 39
Notecards: 45-45-90 and 30-60-90 triangle.
o Updates: Unit 5 Quiz 1 (5.1-5.3)
Tuesday/Wednesday
AGENDA
① Review HW
② Finish U5L2
③ Unit Circle Fun!
④ Cool-Down…
Review HW
Pg. 282 #53-57 odd p 287 # 1-8
2. sine= cosecant
cos= secant
tan= tangent
4. sinA=cosB; cscA=secB; tanA=cotB
6. csc= 2/5
8. sinP= 3/ 10 secP= 10/3
cosP= sqrt(91)/ 10 secP: 10(sqrt(91))/ 91
tanP= 3(sqrt(91))/91 cot P= sqrt(91)/ 3
Learning Objectives
By the end of this period you will be able to:
o Use special right triangles to derive the order pairs of the unit
circle.
Trigonometric Functions on the Unit Circle (5.3)
Lets put everything we have learned today together!!
TIPS:
① Sketch the point and connect the point to the origin.
② Identify the reference angle and name it theta. Create a
right triangle.
③ Identify the 6 trig. ratios.
Trigonometric Functions on the Unit Circle (5.3)
Trigonometric Functions on the Unit Circle (5.3)
Whiteboards!
Suppose θ is an angle in standard position whose terminal
side lies in quadrant IV.
If
, find the values of the remaining five
trigonometric functions of θ.
Special Right Triangles
In order for us to be successful with the Unit Circle we need
to review Special Right Triangles.
Take out a piece of binder paper
Label it: Special Right Triangles: U5L2 Continued.
Trigonometric Ratios in Right Triangles (5.2)
From Geometry and Algebra 2 you studied special right
triangles; 30-60-90 and 45-45-90.
When working with trigonometric ratios you will commonly
refer to the special right triangles to identify the ratio of the
side lengths.
Deriving the 45-45-90 Triangle
Draw a square on your paper with a side length of “s”.
Draw a diagonal through your square to form two triangles.
Solve for the hypotenuse of your triangle using the
Pythagorean Theorem.
What did you get?
Deriving the 45-45-90 Triangle
Deriving the 30-60-90 Triangle
We can use an equilateral triangle to help us derive the
relationship of the sides of a 30-60-90 triangle.
Draw a triangle with each side as “2s”
Cut the triangle in half.
What is the base? What is the other leg? What is the
hypotenuse?
Deriving the 30-60-90 Triangle
Example 1
(3) Practice: Find the value of the variable(s). Give your answer in simplest radical form.
(a)
(b)
(c)
Whiteboards!
Find the value of x.
(a)
(b)
Unit Circle
Unit Circle
o A circle with a radius of 1 unit. For every point P(x, y) on the unit
circle, the value of r is 1.
o The coordinates of P can be written as (cosθ, sinθ) since the radius of
a unit circle is 1.
Unit Circle
o On your yellow Unit Circle, trace
over the y and x axis with a black
pen. Label the degrees. ( this line
divides the circle into fourths)
o Using a different color, trace the
line that divides the circle into
eighths. Label the degrees on
the dotted line.
o Using a different color, trace the
line that is closest to the x-axis.
Label the degrees on the dotted
line.
o Using a different color, trace the
line that is closest to the y-axis.
Label the degrees on the dotted
line.
Unit Circle
Now, lets write the ordered
pairs for each terminal side.
o If a unit circle has a 1 unit
radius what is an order pair
for 0, 90, 180, 270?
Unit Circle
You ONLY need to know the
ordered pairs for the first
quadrant, and then you can
easily fill in the rest of the Unit
Circle!
o Let’s look at the 30 degree
ordered pair. If the radius ( or
the hypotenuse was 1) what
are the lengths of the other
legs? Let’s use special right
triangles
Unit Circle
You ONLY need to know the
ordered pairs for the first
quadrant, and then you can
easily fill in the rest of the Unit
Circle!
o Let’s look at the 45 degree
ordered pair. If the radius ( or
the hypotenuse was 1) what
are the lengths of the other
legs? Let’s use special right
triangles
Unit Circle
You ONLY need to know the
ordered pairs for the first
quadrant, and then you can
easily fill in the rest of the Unit
Circle!
o Let’s look at the 60 degree
ordered pair. If the radius ( or
the hypotenuse was 1) what
are the lengths of the other
legs? Let’s use special right
triangles
Unit Circle
Now, with your table fill out the
rest of the unit circle.
Hint: All the green lines ( for
example) should be the same
ordered pair, but watch out for
the negatives!
Do this in pencil first. When you
are confident, then raise your
hand and write down the
ordered pairs using the same
color you drew the line.
Unit Circle
A way to help you remember
the ordered pairs is remember
the lines closest to the x-axis (
same color) are the same
ordered pairs and so on
OR
3-2-1 1-2- 3
Look at the numerator of the xcoordinate.
Unit Circle
On the Unit Circle, the following are true:
Sinθ= y
cosθ= x
What about the other four trig ratios?
Tanθ=
Secθ=
Cscθ=
cotθ=
Example 2
Write this on your piece of binder paper.
Use the Unit Circle to find each value.
a) cos ( -180°)
b) tan ( 270°)
Whiteboards!
Use the Unit Circle to find each value.
a) Sec ( 90°)
b) cot ( 270°)
Example 3
Use the unit circle to find the values of the six trig functions for a
135° angle.
Whiteboards
Use the unit circle to find the values of the six trig functions for a
210° angle.
Cool-Down…
Name three things you have learned, are still
confused about, or are wondering.
Take 3 minutes to think to yourself.
We will popcorn to share our thoughts.