Algebra
2012-2013
Pythagorean Theorem &
Trigonometric Ratios
Name:______________________________
Teacher:____________________________
Pd: _______
Table of Contents
DAY 1: SWBAT: Calculate the length of a side a right triangle using the Pythagorean Theorem
Pgs: 1 - 4
HW: 5 - 6
DAY 2: SWBAT: Find the three basic trigonometric ratios in a right triangle
Pgs: 7 - 10
HW: 11 - 12
DAY 3: SWBAT: Use Trigonometric Ratios to find missing lengths of a right triangle
Pgs: 13 - 17
HW: 18 -19
DAY 4: SWBAT: Use Trigonometric Ratios to find a missing angle of a right triangle
Pgs: 20 - 23
HW: 24 - 25
Day 5-6: Review
Pgs: 26 - 32
Day 7: Test
Trig Overall Notes
Pgs: 33 - 34
SWBAT: Calculate the length of a side a right triangle using the Pythagorean Theorem
Pythagorean Theorem – Day 1
Warm – Up
Introduction:
Over 2,500 years ago, a Greek mathematician named Pythagoras popularized the concept that a
relationship exists between the hypotenuse and the legs of right triangles and that this relationship is
true for all right triangles. Thus, it has become known as the Pythagorean Theorem.
Pythagorean Theorem
a 2  b2  c 2
*************************SHOW SKETCHPAD ANIMATION ************************
Identify
1
Example 1: Find the value of x in the following diagrams. Round to the nearest tenth if necessary.
A)
B)
Practice Problems: Find the value of x in the following diagrams. Round to the nearest tenth if necessary.
1)
x
8
2)
x
5
15
3)
12
4)
52
x
8
48
5)
10
x
6)
29
20
x
12
8
x
2
Example 2: Pythagorean Theorem Word Problems
A 15 foot ladder is leaning against a wall. The foot of the ladder is 7 feet from the wall. How high up the wall
is the ladder?
Practice Problems: Pythagorean Theorem Word Problems
7) If the length of a rectangular television screen is 20 inches and its height is 15 inches, what is the length of
its diagonal, in inches?
8) An 18-foot ladder leans against the wall of a building. The base of the ladder is 9 feet from the building on
level ground. How many feet up the wall, to the nearest tenth of a foot, is the top of the ladder?
9) A cable 20 feet long connects the top of a flagpole to a point on the ground that is 16 feet from the base of
the pole. How tall is the flagpole?
3
10) Regents Problem
Challenge Problem
In the accompanying diagram of right triangles ABD and DBC, AB = 5, AD = 4, and CD = 1. Find the length
of BC , to the nearest tenth.
Summary:
Exit Ticket:
4
Homework - Pythagorean Theorem – Day 1
Directions: Find the length of the missing side in the following examples. Round answers to the nearest tenth, if
necessary.
5
6
SWBAT: Find the three basic trigonometric ratios in a right triangle
Trigonometric Ratios – Day 2
Warm – Up
Two joggers run 8 miles north and then 5 miles west. What is the shortest distance, to the nearest tenth of a
mile, they must travel to return to their starting point?
____________________________________________________________________________
O
S
H
A
C
H
O
T
A
7
Example 2:
S
O
H
C
A
H
T
O
A
8
Practice Problems:
S
O
H
C
A
H
T
O
A
7)
8)
Example 3
Practice (for example 3)
9
Challenge Problem:
Summary:
Exit Ticket:
10
Homework - Trigonometric Ratios – Day 2
Write the ratio that represents the trigonometric function in simplest form.
11
12
SWBAT: Use Trigonometric Ratios to find missing lengths of a right triangle
Trigonometry: Solving for a Missing Side - Day 3
Warm Up
Determine the trigonometric ratios for the following triangle:
(a) Sin A =
A
(b) Cos A =
(c) Tan A =
20
12
(d) Sin B =
(e) Cos B =
(f) Tan B =
C
15
B
TRIGONOMETRIC RATIOS
Recall that in a right triangle with acute angle A, the following ratios are defined:
Example 1: Determine the length of side x and y of each right triangle using trigonometric ratios.
y
13
Practice Problems: Determine the length of side x and y of each right triangle using trigonometric ratios.
y
Example 2: Determine the length of side x of each right triangle using trigonometric ratios.
14
Practice
1) A ladder leans against a building as shown in the
picture below. The ladder makes an acute angle
with the ground of 72. If the ladder is 14 feet
long, how high, h, does the ladder reach up the
wall? Round your answer to the nearest tenth of a
foot.
14 feet
h
2)
15
3) A 14 foot ladder is leaning against a house. The
angle formed by the ladder and the ground is 72 .
(a) Determine the distance, d, from the base of the
ladder to the house. Round to the nearest foot.
(b) Determine the height, h, the ladder reaches up
the side of the house. Round to the nearest
foot.
14 ft
4) In the accompanying diagram, x represents the
length of a ladder that is leaning against a wall of a
building, and y represents the distance from the
foot of the ladder to the base of the wall. The
ladder makes a 60° angle with the ground and
reaches a point on the wall 17 feet above the
ground. Find the number of feet in x and y.
h
72
d
x
17
Challenge Problem:
16
Summary
Exit Ticket:
17
Homework - Trigonometry: Solving for a Missing Side - Day 3
Directions: In problems 1 through 3, determine the trigonometric ratio needed to solve for the missing side and
then use this ratio to find the missing side.
1) In right triangle ABC, mA  58 and AB  8 . Find the length of each of the following.
Round your answers to the nearest tenth.
C
(a) AC
(b) BC
(Hint: Use Pythagorean’s Thm)
A
B
8
2) In right triangle ABC, mB  44 and AB  15 . Find the length of each of the following.
Round your answers to the nearest tenth.
B
(a) AC
(b) BC
(Hint: Use Pythagorean’s Thm)
15
C
3) In right triangle ABC, mC  32 and AB  24 . Find the length of each of the following.
Round your answers to the nearest tenth.
(a) AC
A
B
(b) BC
(Hint: Use Pythagorean’s Thm)
24
C
A
18
19
SWBAT: Use Trigonometric Ratios to find a missing angle of a right triangle
Trigonometry: Solving for a Missing Angle – Day 4
Warm Up
Find the length of AB to the nearest tenth.
C
S
O
H
C
A
H
T
O
A
125
B
A
Example 1:
20
Example 2:
Practice: Solve for the missing angle.
3.
4.
5.
21
Example #3:
7) In right triangle ABC, leg BC = 15 and leg AC = 8) Triangle ABC has legs BC = 10 and AB = 16.
20. Find angle A to the nearest degree.
To the nearest tenth of a degree, what is the
measure of the largest acute angle in the
triangle?
9) A flagpole that is 45-feet high casts a shadow
along the ground that is 52-feet long. What is
the angle of elevation, A, of the sun? Round
your answer to the nearest degree.
10) A hot air balloon hovers 75 feet above the
ground. The balloon is tethered to the ground
with a rope that is 125 feet long. At what angle
of elevation, E, is the rope attached to the
ground? Round your answer to the nearest
degree.
45 feet
A
52 feet
125 feet
75 feet
E
22
Exit Ticket:
23
Homework - Trigonometry: Solving for a Missing Angle – Day 4
1) For the following right triangles, find the measure of each angle, x, and y, to the nearest degree:
(a)
(b)
19
39
11
27
x
x
(c)
(d)
51
21
y
x
29
x
36
y
2) Given the following right triangle, which of the following is closest to mA ?
A
(1) 28
(3) 62
28
(2) 25
(4) 65
C
3) In the diagram shown, mN is closest to
(1) 51
(3) 17
(2) 54
(4) 39
13
B
21
N
M
17
P
24
4) A skier is going down a slope that measures 7,500 feet
long. By the end of the slope, the skier has dropped
2,200 vertical feet. To the nearest degree, what is the
angle, A, of the slope?
5) A person standing 60 inches tall casts a shadow 87
inches long. What is the angle the measure of angle
x to the nearest degree?
6) From the top of an 86 foot lighthouse, the angle to a
ship in the ocean is x. If the ship is 203 feet from the
light house, determine the angle x? Round your
answer to the nearest degree.
7) An airplane takes off 200 yards in front of a 60 foot
building. At what angle must the plane take off in
order to avoid crashing into the building?
25
REVIEW SECTION
Pythagorean’s Theorem Review
2.
3.
4.
26
Applications of the Pythagorean’s Theorem
5.
6.
27
TRIG RATIOS REVIEW
Multiple Choice Practice
7.
8.
9. Use the diagram below to find the ratio of Sin C?
28
Using Trigonometry to Solve for Missing Sides
1.
2.
3.
Multiple Choice
4. Which expression would you use to calculate the value of x?
5.
29
6.
Applications
7.
8.
30
Using Trigonometry to Solve for Missing Angles
7.
8.
9.
31
Multiple Choice
10.
11.
Applications
12.
13.
32
TRIG NOTES OVERALL
33
34