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

1

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.

*************************SHOW SKETCHPAD ANIMATION ************************

Identify

Pythagorean Theorem

222cba

2

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) 2)

3) 4)

5) 6)

8

15

x

x

48

52

20

x

29

5

12

x

x

8

10

8

x

12

3

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?

4

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:

5

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.

6

7

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?

____________________________________________________________________________

SO

H C

A

H T

O

A

8

Example 2:

SO

H C

A

H T

O

A

9

Practice Problems:

7)

8)

Example 3

Practice (for example 3)

SO

H C

A

H T

O

A

10

Challenge Problem:

Summary:

Exit Ticket:

11

Homework - Trigonometric Ratios – Day 2

Write the ratio that represents the trigonometric function in simplest form.

12

13

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 =

(b) Cos A =

(c) Tan A =

(d) Sin B =

(e) Cos B =

(f) Tan B =

Example 1: Determine the length of side x and y of each right triangle using trigonometric ratios.

TRIGONOMETRIC RATIOS

Recall that in a right triangle with acute angle A, the following ratios are defined:

12

15

20

y

A

B C

14

Practice Problems: Determine the length of side x and y of each right triangle using trigonometric ratios.

Example 2: Determine the length of side x of each right triangle using trigonometric ratios.

y

15

h

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.

2)

14 feet

16

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.

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.

Challenge Problem:

d

72

14 ft h

x

17

17

Summary

Exit Ticket:

18

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 AB 58 8 and . Find the length of each of the following.

Round your answers to the nearest tenth.

(a) AC (b) BC (Hint: Use Pythagorean’s Thm)

2) In right triangle ABC, m B AB 44 15 and . Find the length of each of the following.

Round your answers to the nearest tenth.

(a) AC (b) BC (Hint: Use Pythagorean’s Thm)

3) In right triangle ABC, m C AB 32 24 and . Find the length of each of the following.

Round your answers to the nearest tenth.

(a) AC (b) BC (Hint: Use Pythagorean’s Thm)

A

8

C

B

A

15

C

B

A

24

C

B

19

20

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.

Example 1:

125

C

B A

SO

H C

A

H T

O

A

21

Example 2:

Practice: Solve for the missing angle.

3.

4.

5.

22

Example #3:

7) In right triangle ABC, leg BC = 15 and leg AC =

20. Find angle A to the nearest degree.

8) Triangle ABC has legs BC = 10 and AB = 16.

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.

75 feet

125 feet

E 52 feet

45 feet

A

23

Exit Ticket:

24

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)

(c) (d)

2) Given the following right triangle, which of the following is closest to m A ?

(1) 28 (3) 62

(2) 25 (4) 65

3) In the diagram shown, m N is closest to

(1) 51 (3) 17

(2) 54 (4) 39

39 27

x

19 11

x

36

21

x

51

29

x

28

A

C B 13

17

M

P N 21

y

y

25

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?

26

REVIEW SECTION

Pythagorean’s Theorem Review

2.

3.

4.

27

Applications of the Pythagorean’s Theorem

5.

6.

28

TRIG RATIOS REVIEW

Multiple Choice Practice

7.

8.

9. Use the diagram below to find the ratio of Sin C?

29

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.

30

6.

Applications

7.

8.

31

Using Trigonometry to Solve for Missing Angles

7.

8.

9.

32

Multiple Choice

10.

11.

Applications

12.

13.

33

TRIG NOTES OVERALL

34