Angles of Angles of Elevation and Elevation and

DepressionDepressionPlease view this tutorial and Please view this tutorial and

answer the follow-up questions answer the follow-up questions on loose leaf to turn in to your on loose leaf to turn in to your

teacher.teacher.

How to Interpret Word How to Interpret Word Problems Into Right Problems Into Right

TrianglesTriangles 1) Assume trees, buildings, poles, etc. are 1) Assume trees, buildings, poles, etc. are

perpendicular to the ground (forming a 90° perpendicular to the ground (forming a 90° angle)angle)

2) How high or how tall represents the side 2) How high or how tall represents the side perpendicular to the groundperpendicular to the ground

3) Shadows are on the ground3) Shadows are on the ground

4) String of a kite, the sun ray, line of sight, 4) String of a kite, the sun ray, line of sight, a ladder leaning against a building, etc. a ladder leaning against a building, etc. represent the hypotenuserepresent the hypotenuse

ExampleExample

A little boy is flying a kite. The string of the A little boy is flying a kite. The string of the kite makes an angle of 30° with the ground. kite makes an angle of 30° with the ground. If the height of the kite is 9 meters, find the If the height of the kite is 9 meters, find the length of the string that the boy has used.length of the string that the boy has used.

Make a SketchMake a Sketch

Now that we know the important information, try to label the triangle.

Make a SketchMake a Sketch

30°

9 m x

Let’s Find the Length of Let’s Find the Length of the String.the String.

Now that we have our picture, we can use Now that we have our picture, we can use trig ratios to solve for x. trig ratios to solve for x.

What trig ratio should we use based on the What trig ratio should we use based on the given information?given information?

30°

9 m x

Since we have the side opposite of the given angle and we need to find the hypotenuse, we will use SINE to solve for x.

Let’s Find the Length of Let’s Find the Length of the String.the String.

9

x

Sin 30° = Sin 30° =

Sin 30°Sin 30° = = 11

9 9 = = x sin 30° x sin 30°sin 30° sin 30°sin 30° sin 30°

x =x =

X = 18 mX = 18 m

30°

9 m

x

9

x

9

sin 30o

Trig Word Problem Trig Word Problem Special CasesSpecial Cases

Angle of Elevation Angle of Elevation

The angle of elevation to the top of an object The angle of elevation to the top of an object is the angle formed by horizontal and the is the angle formed by horizontal and the line of the sight to the top of the object. line of the sight to the top of the object.

• Angle of DepressionAngle of Depression

The angle of depression to an object is the The angle of depression to an object is the angle formed by the horizontal line of sight angle formed by the horizontal line of sight to the object below. to the object below.

Now let’s review!Now let’s review!

Use the following link to review the terms Use the following link to review the terms angle of elevation and angle of depression angle of elevation and angle of depression as well as view some sample problems.as well as view some sample problems.

Angle of Elevation and Angle of Depression Review

Note: You can also turn back to page 405 in Note: You can also turn back to page 405 in your textbook to review as wellyour textbook to review as well

ExampleExample

An airplane is on the runway strip 200 yards An airplane is on the runway strip 200 yards from the air-traffic control tower. If the from the air-traffic control tower. If the tower is 20 yards high, at what angle would tower is 20 yards high, at what angle would the pilot have to look up to see the top of the pilot have to look up to see the top of the tower?the tower?

ExampleExample

An airplane is on the runway strip An airplane is on the runway strip 200 yards 200 yards from the air-traffic control towerfrom the air-traffic control tower. If . If the the tower is 20 yards hightower is 20 yards high, , at what angle would at what angle would the pilot have to look up to see the top of the pilot have to look up to see the top of the towerthe tower??

Now that we’ve underline the important information in the problem, try to draw a sketch to match it.

How Does Your Sketch How Does Your Sketch Compare?Compare?

20

Now, Let’s Solve for x.Now, Let’s Solve for x.

Tan x =Tan x =

X = X =

X = 5.71° X = 5.71°

20

200

tan−1 20200

⎛⎝⎜

⎞⎠⎟

Another ExampleAnother Example

Bob is standing at the top of a lighthouse Bob is standing at the top of a lighthouse that is 5000 ft high when he notices a boat that is 5000 ft high when he notices a boat in the water. If the boat is 8500 ft from the in the water. If the boat is 8500 ft from the base of the lighthouse. What would be the base of the lighthouse. What would be the angle of depression for Bob to see the boat angle of depression for Bob to see the boat from the top of the lighthouse?from the top of the lighthouse?

Another ExampleAnother Example

Bob is standing on top of Bob is standing on top of a lighthouse that a lighthouse that is 5000 ft high is 5000 ft high when he notices a boat in the when he notices a boat in the water. If the water. If the boat is 8500 ft from the base boat is 8500 ft from the base of the lighthouseof the lighthouse.. What would be the angle What would be the angle of depressionof depression for Bob to see the boat from for Bob to see the boat from the top of the lighthouse?the top of the lighthouse?

How Did We Do?How Did We Do?

Do not forget to draw the second triangle in Do not forget to draw the second triangle in an angle of depression problem!an angle of depression problem!

8500 ft

5000 ft

Since the figure is a rectangle, we know that opposite sides are the same length

8500 ft

5000 ft

Let’s Solve For XLet’s Solve For X

Tan X =Tan X =

X = X =

X = 30. 47°X = 30. 47°

5000

8500

tan−1 50008500

⎛⎝⎜

⎞⎠⎟

Let’s try one more problem Let’s try one more problem before you try some on your before you try some on your

own!own! Ronnie is 3 m tall and is standing 40 m from Ronnie is 3 m tall and is standing 40 m from

the base of a tower. If Ronnie is looking up the base of a tower. If Ronnie is looking up at the top of the tower with an angle of 67°, at the top of the tower with an angle of 67°, what is the height of the tower? Remember what is the height of the tower? Remember to draw and label a sketch to help solve the to draw and label a sketch to help solve the problem.problem.

Check your sketch with Check your sketch with the one belowthe one below

Do you noticeany difference in his problem?

Notice that for the first time the angle is not level with the ground.

40 m

67°

3m

x

Let’s Solve the Let’s Solve the ProblemProblem

Since we know the side adjacent to the Since we know the side adjacent to the angle and we need to find the side opposite angle and we need to find the side opposite of it, we use tangent.of it, we use tangent.

tan (67) =tan (67) =

x = 40 tan (67)x = 40 tan (67)

x = 94.23 m x = 94.23 m

x

40⎛⎝⎜

⎞⎠⎟

Did we forget Did we forget anything?anything?

Remember that we were given the height of Remember that we were given the height of Ronnie and that the angle was not forming Ronnie and that the angle was not forming with the ground. Therefore, we need to with the ground. Therefore, we need to remember to add on his height to the remember to add on his height to the previous answer to get the total height of previous answer to get the total height of the tower.the tower.

The Total Height of the The Total Height of the TowerTower

40 m

67°3m

94.23

3m

The TOTAL height of the tower from the ground is 94.23 + 3 = 97.23 m.

Time For Practice!Time For Practice!

Use what you’ve just reviewed to help you Use what you’ve just reviewed to help you answer the following questions.answer the following questions.

Complete the following problems and make Complete the following problems and make sure to turn in all work to your teacher sure to turn in all work to your teacher when finished. when finished.

Be sure to include a sketch if not given, to Be sure to include a sketch if not given, to help solve the problem correctly.help solve the problem correctly.

GOOD LUCK!GOOD LUCK!

Problem 1Problem 1

A ladder leans against a building. The foot A ladder leans against a building. The foot of the ladder is 6 feet from the building. The of the ladder is 6 feet from the building. The ladder reaches a height of 14 feet on the ladder reaches a height of 14 feet on the building. Find the length of the ladder to building. Find the length of the ladder to the nearest foot. Find to the nearest the nearest foot. Find to the nearest degree, the angle the ladder makes with the degree, the angle the ladder makes with the ground. ground.

Problem 2Problem 2 Find the distance from the tree to the airplane

Problem 3Problem 3

The angle of elevation from a point on the The angle of elevation from a point on the ground to the top of a tree is 28°. If the tree ground to the top of a tree is 28°. If the tree is 43 feet high, find the distance from this is 43 feet high, find the distance from this point to the base of the tree. point to the base of the tree.

Problem 4Problem 4

Tom is flying a kite at an angle of elevation Tom is flying a kite at an angle of elevation of 42°. All 70 meters of string have been let of 42°. All 70 meters of string have been let out. If Tom is 4 meters tall, find the height out. If Tom is 4 meters tall, find the height of the kite.of the kite.