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

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Warm Up Find the missing lengths of the following right Triangles (Hint: use Pythagorean theorem) 1. 2. 3. Find the complementary angles of the following: 4. 𝒎∠𝑨=𝟒𝟓° __________________________ 5. 𝒎∠𝑩=𝟔𝟎° __________________________ 6. 𝒎∠𝑪=𝟑𝟎° __________________________ 𝑐=8.94 𝑎=8.66 45° 𝑥=16.64 30° 60° 3

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

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

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

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

Sine (sin) Sin of angle X= length of opposite leg of ∠𝑥 length of hypotenuse

Cosine (cos) Cos of angle X: length of adjecent leg ∠𝑥 length of hypotenuse

Tangent (tan) Tan of angle X: length of opposite leg ∠𝑥 length of adjacent leg ∠𝑥

HUMAN Example!

SOH CAH TOA A A A 𝑺𝑖𝑛 𝐴= 𝑂 𝐻 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑜𝑓 ∠𝐴 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝐶𝑜𝑠𝐴= 𝐴 𝐻 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑜𝑓∠𝐴 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑇𝑎𝑛𝐴= 𝑂 𝐴 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑜𝑓∠𝐴 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡𝑜𝑓 ∠𝐴

𝑆 𝑂 𝐻 − 𝐶 𝐴 𝐻 𝑇 𝑂 Include Tangent = 𝐸𝐹 𝐷𝐸 = 8 15 ≈0.533 Note: sin and cos are ALWAYS less than 1 Note: Tangent can be either greater or smaller than 1 tan 𝐹= 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑙𝑒𝑔 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑡𝑜 ∠𝐹 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑙𝑒𝑔 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 ∠𝐹 = 𝐷𝐸 𝐸𝐹 = 15 8 ≈1.875

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 19 19.2 𝑆𝑖𝑛 𝐹= 𝑆𝑖𝑛 𝐷= 𝐶𝑜𝑠 𝐹= 𝐶𝑜𝑠 𝐷= 𝑇𝑎𝑛 𝐹= 𝑇𝑎𝑛 𝐷= 𝑆𝑖𝑛 𝑃= 𝑆𝑖𝑛 𝑅= 𝐶𝑜𝑠 𝑃= 𝐶𝑜𝑠 𝑅= 𝑇𝑎𝑛 𝑃= 𝑇𝑎𝑛 𝑅= 5 10.3 3 19.2 Note that these are the same 5 10.3 3 19.2 9 10.3 19 19.2 9/5 19/19.2 5/9 19.2/9

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

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

Practice the Calculator Give THREE decimal places Find the 𝑆𝑖𝑛 40° Find the 𝐶𝑜𝑠 40° Find the Tan 40° =0.643 =0.766 =0.839

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 𝑥 =2.5 3 ℎ =6.12 𝑡 5 = 2.56 This means THREE decimal places 𝑧=12 𝑥=2 ℎ=0.49 z=12.8

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

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= 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑜𝑓 𝐴 𝐻 𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 Angle Value Step 4: Plug in given information sin 11° = 𝑜𝑝𝑝 𝐻𝑦𝑝 = 𝐶𝐵 𝐶𝐴 = 𝑥 12 sin 11° = 𝑥 12

Finding a Missing Leg: Ex. 1 (cont) A. Solve for the length of the Wall. sin 11° = 𝑥 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 11° = 𝑥 12 (12) (12) Include Units in final answer. 12∗sin 11° =𝑥 Step 6: Use calculator to solve everything all at once 𝑇ℎ𝑒 𝑤𝑎𝑙𝑙 𝑖𝑠 2.29 𝑓𝑒𝑒𝑡 𝑡𝑎𝑙𝑙 𝑥=2.29 State answer:

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= 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑜𝑓 𝐴 𝐻 𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 Angle Value Step 4: Plug in given information cos 11° = 𝑎𝑑𝑗 ℎ𝑦𝑝 = 𝐴𝐵 𝐶𝐴 = 𝑦 12 cos 11° = 𝑦 12

Finding a Missing Leg: Ex. 2 (cont) A. Solve for the length of the Wall. cos 11° = 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 11° = 𝑦 12 (12) (12) Include Units in final answer. 12∗sin 12° =𝑥 Step 6: Use calculator to solve everything all at once 𝑥=11.78 State answer: 𝑇ℎ𝑒 𝑤𝑎𝑙𝑙 𝑖𝑠 11.78 𝑓𝑒𝑒𝑡 𝑡𝑎𝑙𝑙

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

Mixed Practice 2. 1. 5. 4. 3. My Answers: 1. 140.04 2. 4.00 3. 57.65 4. 6.18 5. 1.45 Mixed Practice Round answer to the nearest hundredth. 2. 1. 𝑇𝑎𝑛 55°= 200 𝑥 𝑆𝑖𝑛 30°= 𝑥 8 5. 3. 4. 𝑆𝑖𝑛 70°= 11 𝑥 𝑆𝑖𝑛 14°= 𝑥 6 𝑆𝑖𝑛 31°= 𝑥 12

YOUR TURN (a) (b) (c) (d) 𝑇𝑎𝑛 30°= 𝑜𝑝𝑝 𝑎𝑑𝑗 = ℎ 50 ℎ=28.87 𝑓𝑡

What are they asking for? Finding a Missing Leg What are they asking for? Step 3: Set-up the Ratio: SOH CAH TOA Given: 𝐴𝐵=6 ∠𝐴=76° 𝑇𝑎𝑛 76°= 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑜𝑓 ∠𝐴 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑜𝑓∠𝐴 = 𝐶𝐵 𝐴𝐵 = 𝐶𝐵 6 Step 4-6: Solve for missing Variable THEN plug in 𝑇𝑎𝑛 76°= 𝐶𝐵 6 (6) (6) 𝟔∗𝑻𝒂𝒏 𝟕𝟔°=𝑪𝑩 𝐶𝐵=26.06 Step 1: is Done Step 2: Decide the trig function to use: Sin Cos Tan 𝑇ℎ𝑒 𝑑𝑖𝑎𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑔𝑟𝑜𝑢𝑛𝑑 𝑡𝑜 𝑡ℎ𝑒 𝑒𝑎𝑣𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑜𝑜𝑓 𝑖𝑠 26.06 𝑓𝑒𝑒𝑡.

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?!

Example 2. Choose Ratio: 𝑇𝑎𝑛gent 1. Draw 3. Set-up and Solve: 𝑇𝑎𝑛 24°= 𝑜𝑝𝑝 𝑎𝑑𝑗 = 𝑎 1200 a=534.27 𝑦𝑎𝑟𝑑𝑠

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: 𝐶𝑜𝑡𝑎𝑛𝑔𝑒𝑛𝑡 3. Set-up and Solve: 𝐶𝑜𝑠 53°= 𝑎𝑑𝑗 ℎ𝑦𝑝 = 𝑔 1200 g=2.05 𝑚

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: 𝐶𝑜𝑡𝑎𝑛𝑔𝑒𝑛𝑡 3. Set-up and Solve: 𝐶𝑜𝑠 58°= 𝑎𝑑𝑗 ℎ𝑦𝑝 = 𝑤 12 𝑤=22.65 𝑓𝑡

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: 𝑇𝑎𝑛𝑔𝑒𝑛𝑡 3. Set-up and Solve: 𝑇𝑎𝑛 24°= 𝑜𝑝𝑝 𝑎𝑑𝑗 = 𝑏 1000 𝑏=445.23 𝑓𝑡

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

SWITCH CALCULATOR MODE!!! Radian Practice SWITCH CALCULATOR MODE!!! 𝑇𝑎𝑛 2𝜋 7 = 𝑥 4 Cos 𝜋 5 = 10 𝑥

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 Multiply by 𝜋 180 Plug in 180 where you see 𝜋 & vice-versa TIP!

13.1-13.2 Classwork/Homework Starts on Page 582 Handout

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:

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.

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 −1 𝑜𝑝𝑝 ℎ𝑦𝑝 =𝐴 𝐜𝐨𝐬 −1 𝐚𝐝𝐣 ℎ𝑦𝑝 =𝐴 𝐭𝐚 𝐧 −𝟏 𝑜𝑝𝑝 𝐚𝐝𝐣 =𝐴

Finding the Missing Angle: The Steps

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

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

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?!

If you know that you are looking for an ANGLE, jump to this step. 1. Draw Example 2. Choose Ratio: 𝑇𝑎𝑛𝑔𝑒𝑛𝑡 3. Set-up and Solve: 𝑇𝑎𝑛 𝜃= 𝑜𝑝𝑝 𝑎𝑑𝑗 = 40 36 If you know that you are looking for an ANGLE, jump to this step. 𝑇𝑎𝑛 𝜃= 40 30 𝑇𝑎 𝑛 −1 ( ) 𝑇𝑎 𝑛 −1 ( ) 𝜃=𝑇𝑎 𝑛 −1 40 30 𝜃=48° SHIFT, TAN

1. Draw 𝐵 𝐶 2 +𝐴 𝐶 2 =𝐴 𝐵 2 𝐵 𝐶 2 + 7 2 = 9 2 𝐵 𝐶 2 +49=81 𝐵 𝐶 2 =32 Pythagorean Theorem 𝐵 𝐶 2 +𝐴 𝐶 2 =𝐴 𝐵 2 𝐵 𝐶 2 + 7 2 = 9 2 𝐵 𝐶 2 +49=81 𝐵 𝐶 2 =32 𝐵 𝐶 2 = 32 ≈5.66 Several options available 𝑚∠𝐵=51° 𝑚∠𝐴=39°

Space to work on Previous Problem

Mixed Practice

Objectives Vocabulary Assignments: - Understand the relationships between 45°−45°−90° and 30°−60°−90° triangles Vocabulary N/A Assignments:

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 30° or one 60° , it is the 30-60-90 Remember!

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

30°-60°-90° Special Right Triangle In a triangle 30°-60°-90° , 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 30° 2X Long Leg 30° 10 cm X 5 cm 60° 60° X Short Leg 5 cm

What kind of angles are made by the diagonals in a square? Take a Square Find the Diagonal Find its Lengths d 𝑥 2 + 𝑥 2 = 𝑑 2 x 2 𝑥 2 = 𝑑 2 2 𝑥 2 = 𝑑 2 2 𝑥 2 = 𝑑 2 2 ∙𝑥=𝑑 x What kind of angles are made by the diagonals in a square? Moody Mathematics

45o-45o-90o leg∙ 2 leg leg

Example: Find the value of a and b. b = 7 cm 45° 7 cm 45° x b x 45 ° 45° a = 7 cm a x Step 1: Find the missing angle measure. 45° Step 2: Decide which special right triangle applies. 45°-45°-90° Step 3: Match the 45°-45°-90° 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

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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 + 𝑥 2 = (2𝑥) 2 d 𝑑 2 + 𝑥 2 = 4𝑥 2 𝑑 2 =3 𝑥 2 𝑑= 3 ∙ 𝑥 2 2x 𝑑= 3 ∙𝑥 𝒙 𝒙 What kind of angles are made by the diagonals in a square? Moody Mathematics

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

30o-60o-90o Short Leg 60° 2∙Short Leg 30° Short Leg . 3

Example: Find the value of a and b. b = 14 cm 60° 7 cm 30° 2x b 30 ° 60° a = cm a x Step 1: Find the missing angle measure. 30° Step 2: Decide which special right triangle applies. 30°-60°-90° Step 3: Match the 30°-60°-90° 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

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30o-60o-90o Moody Mathematics

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

30°-60°-90° Special Right Triangle In a triangle 30°-60°-90° , 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 30° 2X 10 cm 5 cm 5 3 cm Long Leg 30° X 60° X Short Leg 60°

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This means that: sin 30° = 𝑜𝑝𝑝 ℎ𝑦𝑝 = 5 3 10 3 = 5 10 = 1 2 10 3 15 This means that: sin 30° = 𝑜𝑝𝑝 ℎ𝑦𝑝 = 5 3 10 3 = 5 10 = 1 2 What is sin 60° ? This means that: cos 30° = 𝑎𝑑𝑗 ℎ𝑦𝑝 = 15 10 3 = What is cos 60° ?