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Module 13: Trigonometry with Right Triangles

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1 Module 13: Trigonometry with Right Triangles
This Packet Belongs to ________________________ (Student Name) Topic 6: Trigonometry Unit 5– Trigonometry Module 13: Trigonometry with Right Triangles 13.1 Tangent Ratio 13.2 Sine and Cosine Ratios 13.3 Special Right Triangles 13.4 Problem Solving with Trigonometry

2 Quick Access to Module Lesson. Click on desired Lesson

3 Warm Up Find the missing lengths of the following right Triangles (Hint: use Pythagorean theorem) Find the complementary angles of the following: 4. π’Žβˆ π‘¨=πŸ’πŸ“Β° __________________________ 5. π’Žβˆ π‘©=πŸ”πŸŽΒ° __________________________ 6. π’Žβˆ π‘ͺ=πŸ‘πŸŽΒ° __________________________ 𝑐=8.94 π‘Ž=8.66 45Β° π‘₯=16.64 30Β° 60Β° 3

4 Objectives Vocabulary: Assignments: Use Trigonometry ratios
-Trigonometry, Sine, Cosine, Tangent Assignments: 4

5 Trigonometry Trigonometry: the study of the relationships between the sides and the angles of triangles. Focusing specifically on right triangles.

6 Right Triangle and its Parts
SIX Parts 3 angles 1 right angle (90Β°) 2 acute 3 Sides 1 Hypotenuse 2 legs The hypotenuse is ALWAYS opposite to the right angle and also the largest side. Remember!

7 The BIG Three Trig Sine (sin) – like a β€œsign” Cosine (cos) β€œco-sign”
Tangent (tan)

8 Sine (sin) Sin of angle X= length of opposite leg of ∠π‘₯ length of hypotenuse

9 Cosine (cos) Cos of angle X: length of adjecent leg ∠π‘₯ length of hypotenuse

10 Tangent (tan) Tan of angle X: length of opposite leg ∠π‘₯ length of adjacent leg ∠π‘₯

11 HUMAN Example!

12 SOH CAH TOA A A A 𝑺𝑖𝑛 𝐴= 𝑂 𝐻 = π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ π‘œπ‘“ ∠𝐴 β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’
πΆπ‘œπ‘ π΄= 𝐴 𝐻 = π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘œπ‘“βˆ π΄ β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ π‘‡π‘Žπ‘›π΄= 𝑂 𝐴 = π‘‚π‘π‘π‘œπ‘ π‘–π‘‘π‘’ π‘œπ‘“βˆ π΄ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘π‘œπ‘“ ∠𝐴

13 𝑆 𝑂 𝐻 βˆ’ 𝐢 𝐴 𝐻 𝑇 𝑂 Include Tangent
= 𝐸𝐹 𝐷𝐸 = 8 15 β‰ˆ0.533 Note: sin and cos are ALWAYS less than 1 Note: Tangent can be either greater or smaller than 1 tan 𝐹= π‘™π‘’π‘›π‘”π‘‘β„Ž π‘œπ‘“ 𝑙𝑒𝑔 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ π‘‘π‘œ ∠𝐹 π‘™π‘’π‘›π‘”π‘‘β„Ž π‘œπ‘“ 𝑙𝑒𝑔 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ ∠𝐹 = 𝐷𝐸 𝐸𝐹 = 15 8 β‰ˆ1.875

14 Note that these are the same
YOUR TURN Directions: Find the THREE trig ratios (sin, cosine, tangent) for the two angles (not including the right angle). Keep as fraction. 9 10.3 𝑆𝑖𝑛 𝐹= 𝑆𝑖𝑛 𝐷= πΆπ‘œπ‘  𝐹= πΆπ‘œπ‘  𝐷= π‘‡π‘Žπ‘› 𝐹= π‘‡π‘Žπ‘› 𝐷= 𝑆𝑖𝑛 𝑃= 𝑆𝑖𝑛 𝑅= πΆπ‘œπ‘  𝑃= πΆπ‘œπ‘  𝑅= π‘‡π‘Žπ‘› 𝑃= π‘‡π‘Žπ‘› 𝑅= 5 10.3 3 19.2 Note that these are the same 5 10.3 3 19.2 9 10.3 9/5 19/19.2 5/9 19.2/9

15 Using Complementary Angles
Sin A = Cos B 𝐡=90βˆ’π΄ Sin A = Cos (90-A) How Many Degrees make-up a triangle? Now subtract the right angle

16

17

18 Using the Calculator TIP!
Make sure your Calculator is in β€œDegrees” Then, type the angle measure number value Finally click β€œsin”, β€œcos”, or β€œtan” The number you get is value of the trig value with that angle How to set your calculator to β€˜Degree’….. -MODE (next to 2nd button) -Degree (third line down… highlight it) -2nd -Quit Plug everything in BACKWARDS in school calculator TIP!

19 Practice the Calculator
Give THREE decimal places Find the 𝑆𝑖𝑛 40Β° Find the πΆπ‘œπ‘  40Β° Find the Tan 40Β° =0.643 =0.766 =0.839

20 This means THREE decimal places
Warm-Up Directions: Find the value of the missing variable. Round to the nearest 100th place. 6= 𝑧 2 5 π‘₯ =2.5 3 β„Ž =6.12 𝑑 5 = 2.56 This means THREE decimal places 𝑧=12 π‘₯=2 β„Ž=0.49 z=12.8

21 Finding a Missing Leg: The Steps
1. DRAW a Visual Picture and Label. 2. Decide which trig function involves the given angle and the given side 3. Set-up the Trig ratio 4. Plug-in the given information 5. Solve for the missing variable. 6. Use the calculator to find the value of the trig function with the given angle 7. Solve Finding a Missing Leg: The Steps

22 Finding a Missing Leg: Ex. 1
A. Solve for the length of the Wall. Step 1: is Done Step 2: Decide the trig function to use: Sin Cos Tan Step 3: Set-up the Ratio: SOH CAH TOA sin 𝐴= π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ π‘œπ‘“ 𝐴 𝐻 π‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ Angle Value Step 4: Plug in given information sin 11Β° = π‘œπ‘π‘ 𝐻𝑦𝑝 = 𝐢𝐡 𝐢𝐴 = π‘₯ 12 sin 11Β° = π‘₯ 12

23 Finding a Missing Leg: Ex. 1 (cont)
A. Solve for the length of the Wall. sin 11Β° = π‘₯ 12 Note on Calculator: try to do all the calculator work all at once and round AT THE END Step 5 Solve for missing variable value sin 11Β° = π‘₯ 12 (12) (12) Include Units in final answer. 12βˆ—sin 11Β° =π‘₯ Step 6: Use calculator to solve everything all at once π‘‡β„Žπ‘’ π‘€π‘Žπ‘™π‘™ 𝑖𝑠 2.29 𝑓𝑒𝑒𝑑 π‘‘π‘Žπ‘™π‘™ π‘₯=2.29 State answer:

24 Finding a Missing Leg: Ex. 2
A. Solve for the length of the Floor Step 1: is Done Step 2: Decide the trig function to use: Sin Cos Tan Step 3: Set-up the Ratio: SOH CAH TOA cos 𝐴= π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘œπ‘“ 𝐴 𝐻 π‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ Angle Value Step 4: Plug in given information cos 11Β° = π‘Žπ‘‘π‘— β„Žπ‘¦π‘ = 𝐴𝐡 𝐢𝐴 = 𝑦 12 cos 11Β° = 𝑦 12

25 Finding a Missing Leg: Ex. 2 (cont)
A. Solve for the length of the Wall. cos 11Β° = 𝑦 12 Note on Calculator: try to do all the calculator work all at once and round AT THE END Step 5 Solve for missing variable value cos 11Β° = 𝑦 12 (12) (12) Include Units in final answer. 12βˆ—sin 12Β° =π‘₯ Step 6: Use calculator to solve everything all at once π‘₯=11.78 State answer: π‘‡β„Žπ‘’ π‘€π‘Žπ‘™π‘™ 𝑖𝑠 𝑓𝑒𝑒𝑑 π‘‘π‘Žπ‘™π‘™

26 Finding a Missing Leg: Ex. 3
This time, solve for the missing value as an expression The more decimals the more accurate

27 Mixed Practice 2. 1. 5. 4. 3. My Answers: 1. 140.04 2. 4.00 3. 57.65
Mixed Practice Round answer to the nearest hundredth. 2. 1. π‘‡π‘Žπ‘› 55Β°= 200 π‘₯ 𝑆𝑖𝑛 30Β°= π‘₯ 8 5. 3. 4. 𝑆𝑖𝑛 70Β°= 11 π‘₯ 𝑆𝑖𝑛 14Β°= π‘₯ 6 𝑆𝑖𝑛 31Β°= π‘₯ 12

28 YOUR TURN (a) (b) (c) (d) π‘‡π‘Žπ‘› 30Β°= π‘œπ‘π‘ π‘Žπ‘‘π‘— = β„Ž 50 β„Ž=28.87 𝑓𝑑

29 What are they asking for?
Finding a Missing Leg What are they asking for? Step 3: Set-up the Ratio: SOH CAH TOA Given: 𝐴𝐡=6 ∠𝐴=76Β° π‘‡π‘Žπ‘› 76Β°= π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ π‘œπ‘“ ∠𝐴 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘œπ‘“βˆ π΄ = 𝐢𝐡 𝐴𝐡 = 𝐢𝐡 6 Step 4-6: Solve for missing Variable THEN plug in π‘‡π‘Žπ‘› 76Β°= 𝐢𝐡 6 (6) (6) πŸ”βˆ—π‘»π’‚π’ πŸ•πŸ”Β°=π‘ͺ𝑩 𝐢𝐡=26.06 Step 1: is Done Step 2: Decide the trig function to use: Sin Cos Tan π‘‡β„Žπ‘’ π‘‘π‘–π‘Žπ‘‘π‘Žπ‘›π‘π‘’ π‘œπ‘“ π‘‘β„Žπ‘’ π‘”π‘Ÿπ‘œπ‘’π‘›π‘‘ π‘‘π‘œ π‘‘β„Žπ‘’ π‘’π‘Žπ‘£π‘’ π‘œπ‘“ π‘‘β„Žπ‘’ π‘Ÿπ‘œπ‘œπ‘“ 𝑖𝑠 𝑓𝑒𝑒𝑑.

30

31 Word Problems…How to… Read the whole question Draw a picture
Read sentence by sentence Cross out sentences that are useless Do each calculation sentence by sentence Check your answer…Does it make sense?!

32 Example 2. Choose Ratio: π‘‡π‘Žπ‘›gent 1. Draw 3. Set-up and Solve: π‘‡π‘Žπ‘› 24Β°= π‘œπ‘π‘ π‘Žπ‘‘π‘— = π‘Ž 1200 a= π‘¦π‘Žπ‘Ÿπ‘‘π‘ 

33 Practice A ladder 7 m long stands on level ground and makes a 73Β° angle with the ground as it rests against a wall. How far from the wall is the base of the ladder? 1. Draw 2. Choose Ratio: πΆπ‘œπ‘‘π‘Žπ‘›π‘”π‘’π‘›π‘‘ 3. Set-up and Solve: πΆπ‘œπ‘  53Β°= π‘Žπ‘‘π‘— β„Žπ‘¦π‘ = 𝑔 1200 g=2.05 π‘š

34 Practice A guy wire is anchored 12 feet from the base of a pole. The wire makes a 58Β° angle with the ground. How long is the wire? 1. Draw 2. Choose Ratio: πΆπ‘œπ‘‘π‘Žπ‘›π‘”π‘’π‘›π‘‘ 3. Set-up and Solve: πΆπ‘œπ‘  58Β°= π‘Žπ‘‘π‘— β„Žπ‘¦π‘ = 𝑀 12 𝑀=22.65 𝑓𝑑

35 Practice To see the top of a building 1000 feet away, you look up 24Β° from the horizontal. What is the height of the building? 1. Draw 2. Choose Ratio: π‘‡π‘Žπ‘›π‘”π‘’π‘›π‘‘ 3. Set-up and Solve: π‘‡π‘Žπ‘› 24Β°= π‘œπ‘π‘ π‘Žπ‘‘π‘— = 𝑏 1000 𝑏= 𝑓𝑑

36 What is Radians? Radians are like degrees except with a different unit
Conversion like feet to inches The unit it Pi πœ‹=180Β° Questions with radians will ask you to set up the ratio without solving

37 SWITCH CALCULATOR MODE!!!
Radian Practice SWITCH CALCULATOR MODE!!! π‘‡π‘Žπ‘› 2πœ‹ 7 = π‘₯ 4 Cos πœ‹ 5 = 10 π‘₯

38 Dealing with Radians Two Options:
1. Change mode of calculator to radians. 2. Convert radians to degrees, then complete calculation using degrees. How to Convert Radian to Degrees (not going to be tested, but you may have to do it to complete a question): Radians to Degrees Degrees to Radians Multiply by 180 πœ‹ Multiply by πœ‹ 180 Plug in 180 where you see πœ‹ & vice-versa TIP!

39 13.1-13.2 Classwork/Homework Starts on Page 582
Handout

40 Using Trigonometry Ratios to find Missing Angles
Objectives -Find the angle of Elevation or Depression in a Right Triangle Vocabulary: - Angle of Elevation and Angle of Depression Assignments:

41 Angle of Elevation VS Depression
Definition of Angle of Elevation. The word "elevation" means "rise" or "move up". Angle of elevation is the angle between the horizontal and the line of sight to an object above the horizontal 2. Definition of Angle of Depression. The word "depression" means "fall" or "drop". Angle of depression is the angle between the horizontal and the line of sight to an object beneath the horizontal. Take a look at the example below.

42

43 Missing Angle/Inverse
These questions do not ask for the side length, they give you the sides but they ask for the angle measure These questions require using the INVERSE of sin, cos or tan. The inverse gives us the value of the angle measure A sin βˆ’1 π‘œπ‘π‘ β„Žπ‘¦π‘ =𝐴 𝐜𝐨𝐬 βˆ’1 𝐚𝐝𝐣 β„Žπ‘¦π‘ =𝐴 𝐭𝐚 𝐧 βˆ’πŸ π‘œπ‘π‘ 𝐚𝐝𝐣 =𝐴

44 Finding the Missing Angle: The Steps

45

46 Using the Calculator for ANGLE
Make sure your Calculator is in β€œDegrees” Then, type the angle measure number value Click β€œSHIFT” Choose and click Sin, Cos, Tan For ANGLE measure always use SHIFT, function TIP!

47 This means TWO decimal places
Warm-Up Directions: Find the value of the missing variable. Round to the nearest 100th place. sin βˆ’1 (0.54) cos βˆ’1 (0.1234) tan βˆ’1 (1.76) sin βˆ’1 (0.135) This means TWO decimal places =32.68 =82.92 =60.40 =7.76

48 Word Problems…How to… Read the whole question Draw a picture
Read sentence by sentence Cross out sentences that are useless Do each calculation sentence by sentence Check your answer…Does it make sense?!

49 If you know that you are looking for an ANGLE, jump to this step.
1. Draw Example 2. Choose Ratio: π‘‡π‘Žπ‘›π‘”π‘’π‘›π‘‘ 3. Set-up and Solve: π‘‡π‘Žπ‘› πœƒ= π‘œπ‘π‘ π‘Žπ‘‘π‘— = 40 36 If you know that you are looking for an ANGLE, jump to this step. π‘‡π‘Žπ‘› πœƒ= π‘‡π‘Ž 𝑛 βˆ’1 ( ) π‘‡π‘Ž 𝑛 βˆ’1 ( ) πœƒ=π‘‡π‘Ž 𝑛 βˆ’ πœƒ=48Β° SHIFT, TAN

50 1. Draw 𝐡 𝐢 2 +𝐴 𝐢 2 =𝐴 𝐡 2 𝐡 𝐢 2 + 7 2 = 9 2 𝐡 𝐢 2 +49=81 𝐡 𝐢 2 =32
Pythagorean Theorem 𝐡 𝐢 2 +𝐴 𝐢 2 =𝐴 𝐡 2 𝐡 𝐢 = 9 2 𝐡 𝐢 2 +49=81 𝐡 𝐢 2 =32 𝐡 𝐢 2 = 32 β‰ˆ5.66 Several options available π‘šβˆ π΅=51Β° π‘šβˆ π΄=39Β°

51

52 Space to work on Previous Problem

53

54

55

56 Mixed Practice

57 Objectives Vocabulary Assignments:
- Understand the relationships between 45Β°βˆ’45Β°βˆ’90Β° and 30Β°βˆ’60Β°βˆ’90Β° triangles Vocabulary N/A Assignments:

58 Special Right Triangles
There are TWO special Right Triangles that have special lengths for the hypotenuse and the two legs If the triangle shows whether one 30Β° or one 60Β° , it is the Remember!

59 45Β°-45Β°-90Β° Special Right Triangle
In a triangle 45Β°-45Β°-90Β° , the hypotenuse is 2 times as long as a leg. Example: 45Β° 45Β° cm Hypotenuse 5 cm Leg X X 45Β° 5 cm 45Β° Leg X

60 30Β°-60Β°-90Β° Special Right Triangle
In a triangle 30Β°-60Β°-90Β° , the hypotenuse is twice as long as the shorter leg, and the longer leg is 3 times as long as the shorter leg. Example: Hypotenuse 30Β° 2X Long Leg 30Β° 10 cm X cm 60Β° 60Β° X Short Leg 5 cm

61 What kind of angles are made by the diagonals in a square?
Take a Square Find the Diagonal Find its Lengths d π‘₯ 2 + π‘₯ 2 = 𝑑 2 x 2 π‘₯ 2 = 𝑑 2 2 π‘₯ 2 = 𝑑 2 2 π‘₯ 2 = 𝑑 2 2 βˆ™π‘₯=𝑑 x What kind of angles are made by the diagonals in a square? Moody Mathematics

62 45o-45o-90o legβˆ™ 2 leg leg

63 Example: Find the value of a and b.
b = cm 45Β° 7 cm 45Β° x b x 45 Β° 45Β° a = 7 cm a x Step 1: Find the missing angle measure. 45Β° Step 2: Decide which special right triangle applies. 45Β°-45Β°-90Β° Step 3: Match the 45Β°-45Β°-90Β° pattern with the problem. Step 4: From the pattern, we know that x = 7 , a = x, and b = x . Step 5: Solve for a and b

64 45o-45o-90o 8 8 8 Moody Mathematics

65 45o-45o-90o 5 5 5 Moody Mathematics

66 45o-45o-90o 10 10 10 Moody Mathematics

67 45o-45o-90o 2 Moody Mathematics

68 45o-45o-90o Moody Mathematics

69 45o- 45o-90o Moody Mathematics

70 45o-45o-90o Moody Mathematics

71 What kind of angles are made by the diagonals in a square?
Take an Isosceles Triangle Find the Draw the altitude Find its Lengths 2x 2x 𝑑 2 + π‘₯ 2 = (2π‘₯) 2 d 𝑑 2 + π‘₯ 2 = 4π‘₯ 2 𝑑 2 =3 π‘₯ 2 𝑑= 3 βˆ™ π‘₯ 2 2x 𝑑= 3 βˆ™π‘₯ 𝒙 𝒙 What kind of angles are made by the diagonals in a square? Moody Mathematics

72 30o-60o-90o Short Leg Hypotenuse 60Β° 30Β° Long leg

73 30o-60o-90o Short Leg 60Β° 2βˆ™Short Leg 30Β° Short Leg . 3

74 Example: Find the value of a and b.
b = 14 cm 60Β° 7 cm 30Β° 2x b 30 Β° 60Β° a = cm a x Step 1: Find the missing angle measure. 30Β° Step 2: Decide which special right triangle applies. 30Β°-60Β°-90Β° Step 3: Match the 30Β°-60Β°-90Β° pattern with the problem. Step 4: From the pattern, we know that x = 7 , b = 2x, and a = x . Step 5: Solve for a and b

75 30o-60o-90o Moody Mathematics

76 30o-60o-90o Moody Mathematics

77 30o-60o-90o Moody Mathematics

78 30o-60o-90o Moody Mathematics

79 30o-60o-90o Moody Mathematics

80 30o-60o-90o Moody Mathematics

81 30o-60o-90o Moody Mathematics

82 30o-60o-90o Moody Mathematics

83 30o-60o-90o Moody Mathematics

84 30o-60o-90o Moody Mathematics

85 45Β°-45Β°-90Β° Special Right Triangle
In a triangle 45Β°-45Β°-90Β° , the hypotenuse is 2 times as long as a leg. Example: Leg 45Β° X 45Β° 5 cm cm Hypotenuse X Leg X 45Β°

86 30Β°-60Β°-90Β° Special Right Triangle
In a triangle 30Β°-60Β°-90Β° , the hypotenuse is twice as long as the shorter leg, and the longer leg is 3 times as long as the shorter leg. Note: sometimes the triangle is rotated! Example: Hypotenuse 30Β° 2X 10 cm 5 cm cm Long Leg 30Β° X 60Β° X Short Leg 60Β°

87 Mixed Practice: Moody Mathematics

88 Moody Mathematics

89 Moody Mathematics

90 8 8 8 Moody Mathematics

91 10 10 10 Moody Mathematics

92 Moody Mathematics

93 Moody Mathematics

94 Moody Mathematics

95 Moody Mathematics

96 Moody Mathematics

97 Moody Mathematics

98 Moody Mathematics

99 Moody Mathematics

100 Moody Mathematics

101 Moody Mathematics

102 4 4 Moody Mathematics

103 2 Moody Mathematics

104 Moody Mathematics

105 This means that: sin 30Β° = π‘œπ‘π‘ β„Žπ‘¦π‘ = 5 3 10 3 = 5 10 = 1 2
10 3 15 This means that: sin 30Β° = π‘œπ‘π‘ β„Žπ‘¦π‘ = = 5 10 = 1 2 What is sin 60Β° ? This means that: cos 30Β° = π‘Žπ‘‘π‘— β„Žπ‘¦π‘ = = What is cos 60Β° ?

106

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1 WARM UP 1)Find the altitude a 1)Find the missing legs. 3) m

1 WARM UP 1)Find the altitude a 1)Find the missing legs. 3) m
THE NATURE OF GEOMETRY Copyright Β© Cengage Learning. All rights reserved. 7.

THE NATURE OF GEOMETRY Copyright Β© Cengage Learning. All rights reserved. 7.
SECTION 8.4 TRIGONOMETRY. The word trigonometry comes from two greek terms, trigon, meaning triangle, and metron, meaning measure. a trigonometric ratio.

SECTION 8.4 TRIGONOMETRY. The word trigonometry comes from two greek terms, trigon, meaning triangle, and metron, meaning measure. a trigonometric ratio.
Splash Screen. Lesson Menu Five-Minute Check (over Lesson 8–3) CCSS Then/Now New Vocabulary Key Concept: Trigonometric Ratios Example 1: Find Sine, Cosine,

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 8–3) CCSS Then/Now New Vocabulary Key Concept: Trigonometric Ratios Example 1: Find Sine, Cosine,
Right Triangle Trigonometry

Right Triangle Trigonometry
Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b.

Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b.

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