MATH 2 - UNIT 5 – NOTE PACKET

MATH 2 Unit 5 Notes: DAY 1 – Pythagorean Theorem

Warm-up

1. Solve for x.

2. Solve for x and give similarity statement.

1

Day 1 Notes: Pythagorean Theorem Pythagorean Theorem

a. Pythagorean Theorem is used to find missing _________ in a triangle.

b. “a” and “b” represent the _________________________________ c. “c” represents the ___________________________

d. Examples: Find the missing sides using Pythagorean Theorem

1. 2.

3. 4.

2

8

Pythagorean Triple: a set of numbers which __________ satisfy the

__________________ __________________.

Ex) 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25

Ex) The hypotenuse of a right triangle has a length 12. One leg has length 6. Find the length of the other leg.

Ex) The Parks Dept. rents paddle boats at a dock near the entrance of the park. About how far is it to paddle from one dock to another?

Ex) A baseball diamond is a square with 90 ft. sides. Home plate and 2nd base are at opposite vertices of the square. About how far is home plate from second base?

3

Given 3 sides of a triangle......

c2 = a2 + b2 ____________________________

c2 ____ a2 + b2 ____________________________

c2 ____ a2 + b2 ___________________________

Classify the following triangles.

a) 13, 85, 84 b) 12, 13, 15

c) 15, 20, 25 d) 6, 11, 14

Day 1 Classwork: Pythagorean Theorem Find the length of the missing side in the following examples. Round answers to the nearest hundredth (2 decimal places), if necessary. Show all work

4

22cm

14cm

X

4cm

7cmX

1. 2. 3.

4. 5. 6.

7. 8. 9.

What is the length of the missing side of the following right angle triangles? Show all work!

10. a = 3 ; b = 4 ; c = ? 11. a = 6 ; b = 8 ; c = ?

5

3cm

6cm X

22cm4cm

3.7cm 4.2cm

X5.3m6.9m

X

2.7cm

8.6cmX

22m17m

X

X

3m

7m6.4cm6.9cm

X

12. a = 12 ; b =? ; c = 22 13. a = 9 ; b = ? ; c = 13

14. To get from point A to point B you must avoid walking through a pond.  To avoid the pond, you must walk 34 meters south and 41 meters east.  To the nearest meter, how many meters would be saved if it were possible to walk through the pond? 

6

MATH 2 Unit 5 Notes: DAY 2 – Special Right Triangles

Warm-up

7

Day 2 Notes: Special Right TrianglesPart 1: 45-45-90 Triangles

In a 45-45-90 triangle, hypotenuse=leg √2

Ex) Solve for h.

Ex) Solve for x.

Ex) Find the length of the hypotenuse of a 45-45-90 triangle with legs of

8

length 5√3.

Ex) Solve for x.

Part 2: 30-60-90 triangles

9

Ex) Solve for x and y.

Ex) Solve for d and f.

Ex) The shorter leg of a 30-60-90 triangle has length √6. What are the lengths of the other 2 sides?

10

Ex) The longer leg of a 30-60-90 triangle has length 18. Find the other 2 lengths.

Ex) Find the other 2 lengths of a 30-60-90 triangle if the hypotenuse is 4√3

MATH 2 Unit 5 Notes: DAY 3 – Trig Ratios

Warm-up

11

Day 3 Notes: Trig RatiosTrigonometry: the study of ______________________ measurement.

We use SOH CAH TOA to help us set up ratios so that we can solve problems.

12

SOHCAHTOA is used to help find missing _____________ and ______________ in a right triangle when Pythagorean Theorem does not work!

S (sine) O (opposite) H (hypotenuse) à

C (cosine) A (adjacent) H (hypotenuse) à

T (tangent) O (opposite) A (adjacent) à

How to write a ratio:

1. Identify the non-right angle given.2. Label the sides…. ______________________________________________3. Determine which ratio to use4. Write ratio

Ex 1) What are the tangent ratios for <T and <U?

Examples:

13

****To solve…make sure calculator is in

DEGREE MODE!!

Setting up Trigonometry Ratios and Solving for Sides

1. _____________________________ (NOT the right angle)14

2. ____________________ (Opposite, Adjacent, Hypotenuse)3. _____________________________:

1. ________ if we have the opposite and hypotenuse2. ________ if we have the adjacent and the hypotenuse3. ________ if we have the opposite and the adjacent

4. Set up the proportion and solve for x!

Example 1 Example 2

15

46°

X

16

Inverse SOHCAHTOA

SOH CAH TOA

I. Setting up Trigonometry Ratios and Solving for Anglesi. Select a given angle (NOT the right angle)ii. Label your sides (Opposite, Adjacent, Hypotenuse)iii. Decide which trig function you can use:

SOH if we have the opposite and hypotenuse CAH if we have the adjacent and the hypotenuse TOA if we have the opposite and the adjacent

iv. Solve the equation … remember to you your inverses!

Example:

Find the measure of angle A.

II. Find the measure of both missing angles:

1.

17

2.

3.

MATH 2 Unit 5 Notes: DAY 4 – Trig Practice

Warm-up

18

Day 4 Notes: Practice DayPractice (SOHCAHTOA) Trigonometry

Write the ratios for sin X, cos X, and tan X.

19

1. 2. 3.

Find the value of x. Round to the nearest tenth.

8. An escalator at a shopping center is 200 ft 9 in. long, and rises at anangle of 15°. What is the vertical rise of the escalator? Round to thenearest inch.

9. A 12-ft-long ladder is leaning against a wall and makes a 77° anglewith the ground. How high does the ladder reach on the wall?Round to the nearest inch.

10. A straight ramp rises at an angle of 25.5° and has a base 30 ft long. How high isthe ramp? Round to the nearest foot.

Find the value of x.Round to the nearest degree.

20

4. 5.

6.

7.

11. 12.

14.

MATH 2 Unit 5 Notes: DAY 5 – Angles of Elevation & Depression

Warm-up

21

13.

Day 5 Notes: Angles of Elevation and Depression I. Angles of Elevation and Depression

a. The angle of elevation is the angle formed by a _________________ and the line of sight _____________________.

22

b. The angle of depression is the angle formed by a _______________ and the line of sight ____________________.

c. Notice … the angle of elevation and the angle of depression are _____________________________ when in the same picture!

Angles of Elevation & DepressionFind all values to the nearest tenth.

1. A man flies a kite with a 100 foot string. The angle of elevation of the string is 52 o . How high off the ground is the kite?

23

2. From the top of a vertical cliff 40 m high, the angle of depression of an object that is level with the base of the cliff is 34º.  How far is the object from the base of the cliff?

3. An airplane takes off 200 yards in front of a 60 foot building. At what angle of elevation must the plane take off in order to avoid crashing into the building? Assume that the airplane flies in a straight line and the angle of elevation remains constant until the airplane flies over the building.

4. A 14 foot ladder is used to scale a 13 foot wall. At what angle of elevation must the ladder be situated in order to reach the top of the wall?

24

5. A person stands at the window of a building so that his eyes are 12.6 m above the level ground. An object is on the ground 58.5 m away from the building on a line directly beneath the person. Compute the angle of depression of the person’s line of sight to the object on the ground.

6. A ramp is needed to allow vehicles to climb a 2 foot wall. The angle of elevation in order for the vehicles to

safely go up must be 30 o or less, and the longest ramp available is 5 feet long. Can this ramp be used safely?

25