This Packet Belongs to Student Name Topic 6

This Packet Belongs to ____________ (Student Name) Topic 6: Trigonometry Unit 5– Trigonometry Module 13: Trigonometry with Right Triangles 13. 1 13. 2 13. 3 13. 4 Tangent Ratio Sine and Cosine Ratios Special Right Triangles Problem Solving with Trigonometry 1

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

Trigonometry • Trigonometry: the study of the relationships between the _____ and the ______ of triangles. Focusing specifically on right triangles. 5

Right Triangle and its Parts Remember! The hypotenuse is ALWAYS opposite to the right angle and also the largest side. 6

The BIG Three Trig • Sine (sin) – like a “sign” • Cosine (cos) “co-sign” • Tangent (tan) 7

Sine (sin) • 8

Cosine (cos) • 9

Tangent (tan) • 10

SOH CAH TOA A 12

Include Tangent Note: sin and cos are ALWAYS less than 1 Note: Tangent can be either greater or smaller than 1 13

YOUR TURN Directions: Find the THREE trig ratios (sin, cosine, tangent) for the two angles (not including the right angle). Keep as fraction. 14

Using Complementary Angles Sin A = Cos B Sin A = Cos (90 -A) 1) How Many Degrees make-up a triangle? 2) Now subtract the right angle 15

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Using the Calculator 1. Make sure your Calculator is in “Degrees” How to set your calculator to ‘Degree’…. . -MODE (next to 2 nd button) -Degree (third line down… highlight it) -2 nd -Quit 2. Then, type the angle measure number value 3. Finally click “sin”, “cos”, or “tan” 4. The number you get is value of the trig value with that angle TIP! Plug everything in BACKWARDS in school calculator 18

Practice the Calculator Give THREE decimal places 19

**SKIP!! Warm-Up This means THREE decimal places 20

1. DRAW a Visual Picture and Label. 2. Decide which trig function involves the given angle and the given side Finding a Missing Leg: The Steps 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 21

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 Step 4: Plug in given information 22

Finding a Missing Leg: Ex. 1 (cont) A. Solve for the length of the Wall. 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 Include Units in final answer. Step 6: Use calculator to solve everything all at once State answer: 23

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 Angle Value Step 4: Plug in given information 24

Finding a Missing Leg: Ex. 2 (cont) A. Solve for the length of the floor 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 Step 6: Use calculator to solve everything all at once State answer: Include Units in final answer. 25

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

Mixed Practice Round answer to the nearest hundredth. 2. 1. 5. 3. 4. 27

YOUR TURN (a) (b) (c) (d) 28

Finding a Missing Leg What are they asking for? Step 3: Set-up the Ratio: SOH CAH TOA Step 4 -6: Solve for missing Variable THEN plug in Step 1: is Done Step 2: Decide the trig function to use: Sin Cos Tan 29

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Word Problems…How to… 1. Read the whole question 2. Draw a picture 3. Read sentence by sentence • • Cross out sentences that are useless Do each calculation sentence by sentence 4. Check your answer…Does it make sense? ! 31

Example 2. Choose Ratio: 3. Set-up and Solve: 1. Draw

Practice A ladder 7 m long stands on level ground and makes a 73° 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: 3. Set-up and Solve: 33

Practice A guy wire is anchored 12 feet from the base of a pole. The wire makes a 58° angle with the ground. How long is the wire? 1. Draw 2. Choose Ratio: 3. Set-up and Solve: 34

Practice To see the top of a building 1000 feet away, you look up 24° from the horizontal. What is the height of the building? 1. Draw 2. Choose Ratio: 3. Set-up and Solve: 35

What is Radians? •

Radian Practice SWITCH CALCULATOR MODE!!! 37

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 TIP! Degrees to Radians 38

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: 40

Angle of Elevation VS Depression 1. 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. 41

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Missing Angle/Inverse These questions do not ask for the side length, they give you the sides but they ask for the These questions require using the INVERSE of sin, cos or tan. The inverse gives us the value of the angle measure A 43

Finding the Missing Angle: The Steps 44

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Using the Calculator for ANGLE 1. Make sure your Calculator is in “Degrees” 2. Then, type the angle measure number value 3. Click “SHIFT” 4. Choose and click Sin, Cos, Tan TIP! For ANGLE measure always use SHIFT, function 46

Warm-Up This means TWO decimal places 47

Word Problems…How to… 1. Read the whole question 2. Draw a picture 3. Read sentence by sentence • • Cross out sentences that are useless Do each calculation sentence by sentence 4. Check your answer…Does it make sense? ! 48

Example 1. Draw 2. Choose Ratio: 3. Set-up and Solve: If you know that you are looking for an ANGLE, jump to this step. SHIFT, TAN 49

Pythagorean Theorem 1. Draw Several options available 50

Space to work on Previous Problem

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Mixed Practice

Objectives Vocabulary N/A Assignments: 57

Special Right Triangles There are TWO special Right Triangles that have special lengths for the hypotenuse and the two legs Remember! 58

45°-90° Special Right Triangle • Example: 45° Leg Hypotenuse 5 cm X X 45° 5 cm X Leg 45° 59

30°-60°-90° Special Right Triangle • Hypotenuse 30° Example: 2 X Long Leg 30° X 5 cm 10 cm 60° Short Leg 5 cm 60 X

1. Take a Square 2. Find the Diagonal 3. Find its Lengths x What kind of angles are made by the diagonals in a square? Moody d x 61

o o o 45 -90 leg le 62

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

1. Take an Isosceles Triangle 2. Find the Draw the altitude 3. Find its Lengths 2 x d Moody 2 x 2 x What kind of angles are made by the diagonals in a square? 71

Short Leg o o o 30 -60 -90 Hyp o t enu e s Long 72

o o o 30 -60 -90 Short Leg 73

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

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