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**Problem Solving with Right Triangles**

Chapter 12 - Lesson 12.2 Problem Solving with Right Triangles HW: 12.2/1-20

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**Finding the Sides of a Triangle**

Remember: SOHCAHTOA

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**First we will find the Sine, Cosine and Tangent**

Review: Trig Ratios First we will find the Sine, Cosine and Tangent ratios for Angle P. P Next we will find the Sine, Cosine, and Tangent ratios for Angle Q 20 12 Adjacent Q 16 Opposite Remember SohCahToa

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**Solving Right Triangles**

Every right triangle has one right angle, two acute angles, one hypotenuse, and two legs. To Solve a Right Triangle means to determine the measures of all six parts.

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**Missing sides r and s, and angle S.**

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**But what if you don’t know either of the acute angles? **

To solve those triangle we must use Inverse Trig Functions

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Missing side….

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**What are the angles of elevation and depression and what is their relationship to right triangles?**

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**Looking down from the horizontal**

ANGLE OF DEPRESSION Looking down from the horizontal observer Eye level Angle of depression cliffs object Sea level

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**Looking up from the horizontal**

ANGLE OF ELEVATION Looking up from the horizontal observer Eye level cliffs Angle of elevation object Sea level

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If an observer sights an object above, the angle between a horizontal line and his or her line of sight is called an angle of elevation. If the line of sight is below the horizontal it is called the angle of depression. Angle of Depression Angle of Elevation

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eye - level line of sight eye - level

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eye - level line of sight eye - level

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**they are alternate interior angles**

The angles are equal – they are alternate interior angles eye - level line of sight eye - level

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**Angles of Elevation and Depression**

Top Horizontal Angle of Depression Line of Sight Angle of Elevation Bottom Horizontal Since the two horizontal lines are parallel, by Alternate Interior Angles the angle of depression must be equal to the angle of elevation.

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**Angles of Elevation and Depression**

line of sight

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**Step 1: Draw a right angled triangle with the given information.**

Example 1 The angle of elevation of building A to building B is The distance between the buildings is 21 meters. Calculate how much taller Building B is than building A. Step 1: Draw a right angled triangle with the given information. Angle of elevation B A 21 m h 250 Step 2: Take care with placement of the angle of elevation Step 3: Set up the trig equation. Step 4: Solve the trig equation.

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**Step 1: Draw a right angled triangle with the given information.**

Example 2 A boat is 60 meters out to sea. Madge is standing on a cliff 80 meters high. What is the angle of depression from the top of the cliff to the boat? Step 1: Draw a right angled triangle with the given information. 60 m 80 m Angle of depression Step 2: Alternate interior angles place inside the triangle. Step 3: Decide which trig ratio to use. Step 4: Use calculator to find the value of the unknown.

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**Step 1: Draw a right angled triangle with the given information. Plane**

Example 3 Marty is standing on level ground when he sees a plane directly overhead. The angle of elevation of the plane after it has travelled 25 km is Calculate the altitude of the plane at this time. Step 1: Draw a right angled triangle with the given information. Plane h 25 km 200 Step 2: Alternate interior angles places 200 inside the triangle. Angle of elevation 200 Step 3: Decide which trig ratio to use. Step 4: Use calculator to find the value of the unknown. (nearest km)

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Examples

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**Therefore, Petra is closer to the tree, since the distance is shorter.**

Example 4 Kate and Petra are on opposite sides of a tree. The angle of elevation to the top of the tree from Kate is 45o and from Petra is 65o. If the tree is 5 m tall, who is closer to the tree, Kate or Petra? K P 450 650 5m k p Kate Petra Answer Therefore, Petra is closer to the tree, since the distance is shorter.

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**The bird is 6.6 m (2 + 4.6) from the ground at that moment.**

Example 5 Maryann is peering outside her window. From her window she sees her car and a bird hovering above her car. The angle of depression of Maryann’s car is 200 whilst the angle of elevation to the bird is If Maryann’s window is 2m off the ground, what is the bird’s altitude at that moment? Step 1: Draw a diagram Step 2: Set up the trig equations in two parts. Find d first, then b. Bird Car 400 200 2 m Step 3: Solve the equations and answer the question. b d Therefore, The bird is 6.6 m ( ) from the ground at that moment.

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Your Turn 1: You sight a rock climber on a cliff at a 32o angle of elevation. The horizontal ground distance to the cliff is 1000 ft. Find the line of sight distance to the rock climber. x 1000 ft

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Your Turn 2: An airplane pilots sights a life raft at a 26o angle of depression. The airplane’s altitude is 3 km. What is the airplane’s surface distance d from the raft? 3 km d

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FYI: Theodolite Theodolite is a precision instrument for measuring angles in the horizontal and vertical planes. Theodolites are mainly used for surveying applications, and have been adapted for specialized purposes in fields like meteorology and rocket launch technology. When the telescope is pointed at a target object, the angle of each of these axes can be measured with great precision Theodolites are still used today for ultra high precision optical alignment and measurement

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Your Turn 3: A surveyor stands 200 ft from a building to measure its height with a 5-ft tall theodolite. The angle of elevation to the top of the building is 35°. How tall is the building? x = 200 • tan 35° tan 35° = x 200 So x ≈ 140 To find the height of the building, add the height of the Theodolite, which is 5 ft tall. The building is about 140 ft + 5 ft, or 145 ft tall.

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**Your Turn 4: x miles ≈ 18.9939 miles ≈ 19 miles**

An airplane flying 3500 ft above ground begins a 2° descent to land at an airport. How many miles from the airport is the airplane when it starts its descent? sin 2° = 3500 x ft x ft = 3500 ft sin 2° x ft ≈ 100, ft x miles ≈ miles ≈ 19 miles The airplane is about 19 mi from the airport when it starts its descent.

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Your Turn 5: A 6-ft man stands 12 ft from the base of a tree. The angle of elevation from his eyes to the top of the tree is 40°. 1. About how tall is the tree? 2. If the man releases a pigeon that flies directly to the top of the tree, about how far will it fly? 3. What is the angle of depression from the treetop to the man’s eyes? about 16 ft about 15.7 ft 40°

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Your Turn 6: CIRCUS ACTS. At the circus, a person in the audience watches the high-wire routine. A 5-foot-6-inch tall acrobat is standing on a platform that is 25 feet off the ground. How far is the audience member from the base of the platform, if the angle of elevation from the audience member’s line of sight to the top of the acrobat is 27°? 27° Step 1: Draw a triangle to fit problem x 30.5 = Step 2: Label sides from angle’s view adj opp Step 3: Identify trig function to use S O / H C A / H T O / A Step 4: Set up equation 30.5 tan 27° = x Step 5: Solve for variable x tan 27° = 30.5 x = (30.5) / (tan 27°) x = 59.9

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Your Turn 7: DIVING At a diving competition, a 6-foot-tall diver stands atop the 32-foot platform. The front edge of the platform projects 5 feet beyond the ends of the pool. The pool itself is 50 feet in length. A camera is set up at the opposite end of the pool even with the pool’s edge. If the camera is angled so that its line of sight extends to the top of the diver’s head, what is the camera’s angle of elevation to the nearest degree? 37 = 45 = x°

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Your Turn 8: From a point 80m from the base of a tower, the angle of elevation is 28˚. How tall is the tower to the nearest meter? x 28˚ 80m tan 28˚ = 80 (tan 28˚) = x 80 (.5317) = x x ≈ 42.5m is the height of the tower

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Your Turn 8: A ladder that is 20 ft is leaning against the side of a building. If the angle formed between the ladder and ground is 75˚, how far is the bottom of the ladder from the base of the building? 20 building ladder 75˚ x cos 75˚ = 20 (cos 75˚) = x 20 (.2588) = x x ≈ 5.2ft from the base of the building

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Your Turn 9: When the sun is 62˚ above the horizon, a building casts a shadow 18m long. How tall is the building? x 62˚ 18 shadow tan 62˚ = 18 (tan 62˚) = x 18 (1.8807) = x x ≈ 33.9m is the height of the building

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Your Turn 10: A kite is flying at an angle of elevation of about 55˚. Ignoring the sag in the string, find the height of the kite if 85m of string have been let out. kite 85 x string 55˚ sin 55˚ = 85 (sin 55˚) = x 85 (.8192) = x x ≈ 69.6m is the height of the kite

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Your Turn 11: A 5.5 foot person standing 10 feet from a street light casts a 14 foot shadow. What is the height of the streetlight? 5.5 x˚ 10 14 shadow 1st find the angle of elevation 2nd use the angle to find the height of the light. tan x˚ = x° ≈ ° height = 9.43 feet

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Your Turn 12: The angle of depression from the top of a tower to a boulder on the ground is 38º. If the tower is 25m high, how far from the base of the tower is the boulder? 38º angle of depression 25 Alternate Interior Angles are congruent 38º x

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