OR a 2 b 2 c 2 c a b leg leg hypotenuse Pythagorean Triples A set of nonzero whole numbers a b and c that satisfy the equation a 2 b 2 c 2 Here are some common ones to remember ID: 904582
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Slide1
Pythagorean Theorem
the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides
. OR a2 + b2 = c2
c
a
b
leg
leg
hypotenuse
Slide2Pythagorean Triples
A set of nonzero whole numbers a, b, and c that satisfy the equation a2 + b2 = c2Here are some common ones to remember:
3, 4, 55, 12, 138, 15, 177, 24, 25
Slide3Using the Pythagorean Theorem to solve for a missing side
Solve for the missing side.
30a24
Slide4Using the Pythagorean Theorem to solve for a missing
side
1218
c
Solve for the missing side.
Slide5Converse of the Pythagorean Theorem
If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.
How will we know which is the hypotenuse?Given the sides of a triangle equal to 16, 30, 34. Does this form a right triangle?
Slide6Which of the following are right triangles?
,
, 5 3, 4, 621, 72, 75
2.5, 6.25, 43.3125
Slide7Showing Triangles are Obtuse or Acute
Obtuse Triangles – A triangle where one angle is greater than 90 degrees.If c2
> a2 + b2, where c is the longest side of the triangle then the triangle is an obtuse triangle.Acute Triangles – A triangle where all angles are less than 90 degrees.If c2 < a2 + b2, where c is the longest side of the triangle then the triangle is an acute triangle.
Slide8Classify the following triangles acute, obtuse, or right.12, 16, 20
11, 12, 1531, 23, 12.3, .4, .5
Slide9End of Day 1P 361 2-38 evens
Slide10Special Right Triangles
Some right triangles are more common and have some short cuts we can use instead of having to use the Pythagorean Theorem.
45 – 45 – 90 Right Triangle9045
45
x
x
x
Slide11Solve for the missing side(s).
10
10y
10
y
z
Slide1230 – 60 – 90 Right Triangle
30
6090
x
2x
x
Slide13Solve for the missing side(s).
30
6090
x
2x
25
Slide1430
60
9018y
x
Solve for the missing side(s).
Slide15End of Day 2P 3692-20 even and 24-29 all
Slide16Math 2
Unit 4: "Right Triangle Trigonometry”Title
: Using Trigonometric Ratios to find Sides.Objective: To use the trigonometric ratios to find the lengths of sides in right triangles.
Slide17Setting Up Triangles
NEVER USE RIGHT ANGLE TO SET UP!
1. Label the side across from 90° angle as Hyp for hypotenuse.2. Circle the angle you need to use.3. Draw a line from the angle to the opposite side. Label that side
Opp for the side opposite the given angle.
4. Label the remaining side
Adj for the side adjacent to the given
angle.Using Angle
A.
A
B
C
Slide18Setting Up Triangles
Using Angle C
.
A
B
C
NEVER USE RIGHT ANGLE
TO SET UP!
1. Label the side across from 90°
angle as
Hyp
for hypotenuse.
2. Circle the angle you need to use.
3. Draw a line from the angle to the
opposite side. Label that side
Opp
for the side opposite the
given angle.
4. Label the remaining side
Adj
for
the side adjacent to the given
angle.
Slide19D
K
L
Setting Up Triangles
Using Angle
D
.
Slide20Trigonometric Ratios
Sine of a Angle = Leg Opposite
Hypotenusesin (θ) =
Opp
Hyp
Cosine of a Angle = Leg Adjacent
Hypotenuse
cos
(
θ
) =
Adj
Hyp
Tangent of a Angle = Leg Opposite
Leg Adjacent
tan (
θ
) =
Opp
Adj
"SOH CAH TOA"
Slide21Find the following trigonometric ratios
. 1. sin A
2. cos A 3. tan A
A
B
C
3
4
5
Slide22A
B
C
3
4
5
Find the following trigonometric ratios
.
1. sin C
2.
cos
C
3. tan C
Slide23T
G
R
Find the following.
1. sin G
2.
cos G3. tan G
4. sin T
5. cos T
6. tan T
24
10
26
Slide24Every angle has it's own sine, cosine, and tangent ratio!
In Geometry Textbook p. 731
Slide25Degrees
STOP
Check your mode
Degrees
Degrees
Officer Be N. Degrees
Degrees
Degrees
Degrees
Degrees
Degrees
Degrees
Degrees
Slide2640°
x
6
How to find Missing Sides
1. Circle given angle and label
sides.2. Decide on set up.
SOH CAH TOA3. Set up the proportion.
4. Solve by cross multiplication. Round answers to three decimals
Find the missing side.
Slide2750°
x
10
Find the missing side.
Slide2830°
x
12
Find the missing side.
Slide2934°
x
7.2
Find the missing side.
Slide30Find the missing measures.
x
40°
18
y
20°
z
Slide31Find the missing measures.
8
x
68°
21
y
z
Slide32End of Day 3
P 479 1-10, 26, 28, 41-43
Slide33Unit 4: "Right Triangle Trigonometry“
Title: Using Trigonometric Ratios to find Angles.Objective
: To use the trigonometric ratios to find the measures of angles in right triangles.
Degrees
STOP
Check your mode
Degrees
Degrees
Officer Be N. Degrees
Degrees
Degrees
Degrees
Degrees
Degrees
Degrees
Degrees
Slide34How to find Missing Angles
1. Circle missing angle and label sides 2. Decide on set up.
SOH CAH TOA3. Set up proportion.4. Solve by inverse trig ratio. sin-1 cos-1 tan
-1 Round answer to nearest degree.
Find the missing angle
.
11
x°
7
Slide35Find the missing angle
.
8
x°
5
Slide36Find the missing angle.
10.4
x°
12.6
Slide37Find the missing angle.
20
x°
16
Slide38Find the missing angles.
87
x°
63
y°
Slide39Find the missing angles.
7
w°
x°
5
13
y°
z°
Slide40Find the missing angles.
x°
8
17
y°
z°
19
Slide41Find the missing angles
.6
x°8
y°
Slide42Honors Math 2 Assignment:
In the Geometry Textbook,pp. 473-474 #2-16 evens, 27-29
pp. 479-480 #2-16 evens, 22-24
Slide43Find the missing angles.
7
w°
x°
5
13
y°
z°
Find the missing measures.
8
x
68°
21
y
z
Slide44Math 2Unit 4
Lesson 2Unit 4
: "Right Triangle Trigonometry“Title: Application - Angles of Elevation and DepressionObjective: To use angles of elevation and angles of depression to find measures.
Slide45angle of elevation
- is the angle measured from a horizontal line up
to an object.angle of depression is the angle measured from a horizontal line down to an object.
Slide46Angles of Elevation and Depression
Describe each angle as it relates to the situation shown.
Angle 1Angle 2Angle 3Angle 4
Slide47Example 1
Slide48Example 3
"Line-of-Sight Distance"
Slide49Example 4
"Line-of-Sight Distance"
Slide50Example 5A blimp is flying 500 ft above the ground. A person on the ground sees the blimp by looking at a 25° angle. The person’s eye level is 5 ft above the ground. Find the distance from the blimp to the person to the nearest foot.
Slide51Example 6
A surveyor sets up a 5 ft tall theodolite (an instrument for measuring angles) 300 ft from a building to measure its height. The angle measured to the top of the building using the theodolite is 35°. How tall is the building?
Slide52Example 7
You sight a rock climber on a nearly vertical cliff at a 32° angle of elevation. The horizontal ground distance to the base of the cliff is 1,000 ft. Find the line-of-sight distance to the rock climber.
Slide53Example 8
An airplane flying at an altitude of 3,500 ft begins a 2° descent to land at an airport. How many miles from the airport is the airplane when it starts its descent?
Slide54Example 9
Two buildings are 30 ft apart. The angle of depression from the top of higher building to the top of the other building is 19°. What is the difference in the heights of the buildings?
Slide55Honors Math 2 Assignment:
In the Geometry Textbook,pp. 484-486 #9-18, 21, 23, 33, 34
Slide56Example 10
In a galaxy far, far away, a spaceship is orbiting the planet Obar. The ship needs to land in a large, flat crater, but the captain wants to make sure the crater is large enough to hold the ship. When the ship is 4 miles above the planet, the onboard guidance system measures the angles of depression from the ship to both sides of the crater. The angles measure 22° and 37°, respectively. What is the distance across the crater? If the spaceship is 2,500 ft long, will it fit in the crater?
Slide57How tall is the flag?
Slide58End of Day 4
Slide59Example 6
Find the area of the triangle to the nearest hundredth.
Slide60Math 2
Unit 4Lesson 4
Unit 4: "Right Triangle Trigonometry"Title: Application - Finding Area Using Trigonometric RatiosObjective: To find the area of triangles and regular polygons using trigonometry.
Slide61Example 1
Find the area of the triangle to the nearest hundredth.
Slide62Area of a Triangle with an Angle
Slide63Example 2
Find the area of the triangle to the nearest hundredth.
Slide64Example 3F
ind the area of the triangle to the nearest hundredth.
15 m
24 m
120
°
Slide65Example 4F
ind the area of the triangle to the nearest hundredth.
14 yd
6 yd
87
°
49
°
Slide66Example 5F
ind the area of the triangle to the nearest hundredth.
Slide67Example 6
A triangular plot of land has two sides that measure 300 ft and 200 ft and form a 65 angle. Find the area of the plot to the nearest square foot.
Slide68radius
- is a line segment from center to the vertex of a polygon.
apothem - is a line segment from the center of a regular polygon perpendicular to any of its sides.Area of Regular Polygons
Area = 1/2 * side * apothem * number of sides
Slide69Example 7F
ind the area of a regular pentagon with 8 cm sides
to the nearest hundredth.
Slide70Example 8F
ind the area of a regular decagon with a perimeter
of 120 ft to the nearest hundredth.
Slide71Example 9F
ind the area of a regular hexagon with a apothem
of 8 cm to the nearest hundredth.
Slide72Example 10F
ind the area of a regular nonagon with a apothem
of 14 in to the nearest hundredth.
Slide73Example 11F
ind the area of a regular octagon with a radius
of 16 m to the nearest hundredth.
Slide74Example 12F
ind the area of a regular dodecagon with a radius
of 26 in to the nearest hundredth.
Slide75Honors Math 2 Assignment:
In the Geometry Textbook,
pp. 500-501 #1-4, 6-9, 11-18
Slide76End of Day 5
Slide77Math 2Unit 4
Lesson 4Title: Law of
SinesObjectives: To learn to use the law of Sines to find missing sides and angles in triangles.
Slide78Slide79Example: Finding a side of a triangle
Slide80Example: Finding a side of a triangle
Slide81Example: Finding an angle of a triangle
Slide82Example: Finding an angle of a triangle
Slide83Slide84Assignment:
Law of Sines Handout
Slide85Law of Sines: "
The Ambiguous Case
"
Slide86Law of Sines: "The Ambiguous Case
"
Slide87Law of Sines: "The Ambiguous Case
"
Slide88Questions?
Slide89Math 2Unit 4
Lesson 5Title: Law of Cosines
Objectives: To learn to use the law of cosines to find missing sides and angles in triangles.
Slide90Find the missing sides and angles of the triangles below
.
Slide91Slide92Example: Finding a side of a triangle
Slide93Example: Finding a side of a triangle
Slide94Example: Finding an angle of a triangle
Slide95Example: Finding an angle of a triangle
Slide96Assignment:
Law of Cosines Handout
Slide97End of Day 6
Slide98Math 2 Warm Up
1. Find the area of each triangle to the nearest tenth. a. 11 yd, 14
yd, 16 yd b. 100 ft, 80 ft, 140 ft c. 6 m, 12.8 m, 13 m2. In the Geometry Practice Workbook, Reteaching 11-5 (p. 139) #1, 3, 5-10, 12Practice 11-5 (p. 140) #13, 14
Slide99Unit 4 Test Review: Trigonometry
trigonometric ratios sin (angle) = cos
(angle) =
tan (angle) =
F
ind the length of
sides
of a
right triangle
.
F
ind the measure of
angles
of a
right triangle
.
Unit 4 Test Review: Trigonometry
angles of elevation
measured from horizon up to objectangles of depression measured from horizon down
to object
Unit 4 Test Review: Trigonometry
Area of Regular Polygons
Use ½ measure of the central angle for the right triangle to find measures. A = ½ ∙ side ∙ apothem ∙ # of trianglesArea of Oblique Triangles
A = ½ ∙ side ∙ side ∙ sin(Included Angle)
A =
s = ½(a + b + c)
Equations of Circles
Math 2 Assignment
In the Geometry Textbook,pp. 505-507 #8-25, 40-46
Slide103Math 2 Test Day Warm Up
In the Geometry Textbook, p. 508 #
9-12, 13-16 , 26, 28
Slide104TESTING!
No talking during test!When finished, turn in test and test review that you finished for homework to appropriate folders on the front table!
Slide105Test Review: Trigonometry
Law
of Sines =
=
Law of Cosines
b
c