Objectives:

       At the end of the lesson, the students are expected to:

              a. define the following terms:

                     Angle of elevation

                     Angle of depression

                     Line of sight

b. apply sine, cosine and tangent ratios to find angles of elevation and depression;

c. measure lengths and use measurements to determine angle measures;

              d. solve word problems regarding angle of elevation and depression.

Angles of Elevation / Inclination and

     Angles of Depression / Declination

The angle of elevation of an object as seen by an observer is the angle between the horizontal and the line from the object to the observer's eye (the line of sight).

If the object is below the level of the observer, then the angle between the horizontal and the observer's line of sight is called the angle of depression.

The picture below illustrates an example of an angle of depression and an angle of elevation 

Images that shown "Angle of Elevation"

Images that shown "Angle of Depression"

Here below are some examples to illustrate the concepts and application as stated above:

	Example 1: Given- you are standing on the top of the building, where from the angle of elevation of top of a 120 ft tower is 10 degrees. From a window 6ft below the top of the building, the angle of depression of the base of the tower is 30 degrees. Find out the height of the building and the distance between the tower and the building. Step 1: Draw the sketch NTS) and insert information given (below): Step 2: Note that (a + b + 6) =120 Step 3: tan 10 = a/d, and tan 30= b/d Step 4: From above, a = d* tan 10° and b= d* tan 30 Plug these values of a and b, in expression at step 2 above. Step 5: It simplifies to: (d * tan 10+ d * tan 30° + 6) = 120 Thus d = 114/( tan 10+tan 30)= 151 ft, as the final answer.(values of tan 10 & tan 30 are obtained from trig tables or use calculator).
	Example 2: Given- if a plane that is flying at an altitude of 35,000 feet wants to land at JFK, it must begin its descent so that the angle of depression to the airport is 8. Find out the how many miles from the airport must the plane start descending? Step 1: The altitude is 35000 ft and angle of depression is 8(Given) Step 2: tan 8= 35000/d, (the distance is assumed d, from air port to the point where from plane starts descending.) From the trig tables tan 8= 0.1405, therefore, d= 35000/ 0.1405= 249110 ft or 249110/5280 = 47.2 miles (use measurement conversion- 1 mile = 5280 ft), as the final answer.
	Example 3: Given- Is the angle of elevation and angle of depression numerically are equal? Step 1: Recall and notice that the angle of elevation and the angle of depression are the interior alternate angles of two horizontal parallel lines. Step 2: From the theorem learn earlier for parallel lines and transversal, you know that if a transversal intersects two parallel lines; the interior alternate angles are equal. Therefore, the angle of elevation and angle of depression numerically are equal, is the final answer. Example 4: Given- Harry (2 m tall) stands on horizontal ground 20 m from a tree. The angle of elevation of the top of the tree from his eyes is 26°. Calculate the height of the tree. Step 1: Say, the height of the tree be ‘h’. Sketch a diagram to insert the given information. Step 2: tan 26= (h - 2)/20. Simplifying it gives, (h - 2) = 20 * tan 26=  20 * 0.4877 = 9.75 m (value obtained from trig table for tan or you may use calculator to find it.) Thus the height of the tree is 9.75 m (rounded to two decimal places), as the final answer.
	Example 5: Given- a ranger's tower is located 50m from a tall tree. The angle of elevation to the top of the tree is 10°from the top of the tower, and the angle of depression to the base of the tree is 20°. How tall is the tree? Step 1: Draw the sketch and insert the given information. Step 2: tan 20= Tower Ht/50. So tower height = tan 20* 50 = 0.3640 * 50= 18.2 m Step 3: Say, height of the tree is h m. So, (h - 18.2)/50 = tan 10 (0.1763 from tri table). Thus, h -18.2 = 50 * 0.1763 = 8.82. h = (8.82 + 18.2) = 27 m (rounded) The height of the tree is 27 m, as the final answer.

More Sample Problems?
http://www.purplemath.com/modules/incldecl.htm

http://www.winpossible.com/lessons/Geometry_Angles_of_Elevation_and_Depression_-_Geometry.html

The videos below will explain more in detail about Angles of Elevation and Depression, and how to apply the concepts in solving real-world problems. This is explained with the help of several examples and done watching video. This helps you to deal with solving problems and help doing the Trigonometry home work.

Check Your Understanding

 Work Sheet

  1. A balloon at a height of 31 meters from the ground level is attached to a string inclined at 57^o to the horizontal. Find the length of the string to the nearest meter.

	a.		37 meters
	b.		57 meters
	c.		40 meters
	d.		34 meters

 2. A ladder with its foot on a horizontal flat surface rests against a wall. It makes an angle of 30° with the horizontal. The foot of the ladder is 41 ft from the base of the wall. Find the height of the point where the ladder touches the wall.

	a.		24 ft
	b.		23 ft
	c.		26 ft
	d.		29 ft

 3. Simon stands at 170 m away from the base of a building of height 79.22 m. Find the angle of depression of Simon from the top of the building.

	a.		30°
	b.		25°
	c.		46°
	d.		35°

4. The angle of elevation of the top of a dowel from a point on the ground is 36°. After walking 36 ft towards the dowel, the angle of elevation becomes 49°. What is the height of the dowel rounded to the nearest foot?

	a.		76 ft
	b.		71 ft
	c.		74 ft
	d.		69 ft

 5. A 37 ft high tree standing vertically upwards is broken by the wind in such a way that its top just touches the ground and makes an angle of 52° with the ground. At what height from the ground did the tree break?

	a.		20 ft
	b.		16 ft
	c.		14 ft
	d.		18 ft

 6. A man on the deck of a ship is 15 ft above sea level. He observes that the angle of elevation of the top of a cliff is 70° and the angle of depression of its base at sea level is 50°. Find the height of the cliff and its distance from the ship.

	a.		23 ft and 23 ft
	b.		238 ft and 41ft
	c.		50 ft and 13 ft.
	d.		33 ft and 20 ft.

 7. A jet when 3400 m high passes vertically above another jet at an instant when the angles of elevation of the two jets from the same point on the ground are35° and 20° respectively. Find the vertical distance between the two jets to the nearest meter.

	a.		1630 m
	b.		1638 m
	c.		1633 m
	d.		1643 m

 8. Two trees of equal height stand on either side of a roadway, which is 50 ft wide. At a point on the roadway between the trees, the elevations of the tops of the trees are 40° and 25°. Find the height of the trees.

	a.		22 ft
	b.		15 ft
	c.		18 ft
	d.		20 ft

 9. The angle of elevation of the top of a tree is 30^o from a point 28 ft away from the foot of the tree. Find the height of the tree rounded to the nearest feet.

	a.		23 ft
	b.		10 ft
	c.		16 ft
	d.		8 ft

 10. A flagstaff stands on the top of a building. At a distance of 48 ft away from the foot of the building, the angle of elevation of the top of a flagstaff is 60° and the angle of elevation of the top of a building is 46°. Find the height of the flagstaff to the nearest feet.

	a.		22 ft
	b.		17 ft
	c.		16 ft
	d.		34 ft

 11. The angle of elevation of the top of a hill from the foot of a tower is 65° and the angle of elevation of the top of the tower from the foot of the hill is 50°. If the distance between the foot of the tower and the foot of the hill is 75 ft, then find the height of the tower and the height of the hill rounded to the nearest feet.

	a.		161 ft & 89 ft
	b.		90 ft & 162 ft
	c.		89 ft & 161 ft
	d.		162 ft & 90 ft

 12. The angle of elevation of the top of a tower from a point on the ground is 60°. If the height of the tower is 131 m, then find the distance of the point from the foot of the tower.

	a.		1313√ m
	b.		131 m
	c.		3√131 m
	d.		1313√ m

 13. From the top of a spire of height 50 ft, the angles of depression of two cars on a straight road at the same level as that of the base of the spire and on the same side of it are 25° and 40°. Calculate the distance between the two cars.

	a.		47 ft
	b.		7.013 ft.
	c.		40.523 ft.
	d.		27.786 ft

 14. A boy is standing on the ground and flying a kite with a string of length 210 ft at an angle of elevation of 30°.Another boy is standing on the roof of a 85 ft high building and is flying his kite at an angle of elevation of 55°. Both the boys are facing each other. What shall be the length of the string of the kite of the second boy so that the two kites would touch?

	a.		26 ft
	b.		24 ft
	c.		28 ft
	d.		29 ft

 15. The angle of elevation of a cloud observed from a point at a height 170 ft above the level of water in a lake is 54°. The angle of depression of its image in the lake from the same point is 67°. Find the height of the cloud above the lake.

	a.		804 ft
	b.		814 ft
	c.		806 ft
	d.		809 ft

 16. From the top of a building of height h meters in a street, the angles of elevation and depression of the top and the foot of another building on the opposite side of the street are θ° and φ° respectively. Find the height of the opposite building.[Given h = 70, φ = 35, and θ = 50.] 

	a.		214 meters
	b.		224 meters
	c.		229 meters
	d.		219 meters

 17. From the top of a tower of height h meters the angle of depression of a red car is φ° and the angle of depression of a blue car which is on the opposite side of the tower is θ°. If the distance between the foot of the tower and the red car is y meters, then find the height of the tower and the distanc between the two cars. 
Assume that the points of location of the cars and the foot of the tower are collinear.[Given y = 115, θ = 55° and φ = 40°.]

	a.		96 m & 182 m
	b.		98 m & 183 m
	c.		96 m & 184 m
	d.		101 m & 182 m

 18. Two towers are constructed on a plot of land. The angle of depression of the top of the second tower when seen from the top of the first tower is 40°. If the height of the first tower is 135 ft and the height of the second tower is 78 ft, then find the horizontal distance between the two towers.

	a.		74 ft
	b.		69 ft
	c.		79 ft
	d.		94 ft

 19. A flagpole of 10 m length is installed on the top of an office building 29 m high. The building and the pole subtend equal angles at a point outside the building which is at a height of 39 m. What is the distance of this point from the top of the pole?

	a.		19 m
	b.		16 m
	c.		14 m
	d.		11 m

 20. The angle of elevation of an unfinished tower from a point 120 m away from its base is 25°. How much higher the tower be raised so that its angle of elevation from the same point will be 40°?

	a.		50 m
	b.		47 m
	c.		48 m
	d.		45 m

1. A photographer using a camera photographs a rare bird roosting on a high branch of a tree at an angle of elevation of φ°. The distance between the camera lens and the bird is l ft. In order to get a clear picture, the photographer moves closer to the base of the tree. The angle of elevation of the bird is now θ°. Find the new distance between the camera lens and the bird. [Given l = 90, φ = 30° and θ = 55°.]

	a.		57 ft
	b.		60 ft
	c.		62 ft
	d.		55 ft

 22. From a shipmast head 135 meters high, the angle of depression of a boat is observed to be 50°. Find the distance of the boat from the ship.

	a.		290 m
	b.		113 m
	c.		161 m
	d.		130 m

 23. A telegraph pole is 120 m high. Its shadow is207.972 m in length. Find the angle of elevation of the sun.

	a.		33°
	b.		20°
	c.		40°
	d.		30°

 24. The angle of elevation of the top of a cliff from the point Q on the ground is 30°. On moving a distance of20 m towards the foot of the cliff the angle of elevation increases to φ°. If the height of the cliff is17.3 m, then find φ.

	a.		45°
	b.		60°
	c.		120°
	d.		30°

 25. Tom and Sam are on the opposite sides of a tower of160 meters height. They measure the angle of elevation of the top of the tower as 40° and 55°respectively. Find the distance between Tom and Sam.

	a.		308 m
	b.		313 m
	c.		303 m
	d.		306 m

 26.  A man on the deck of a ship is 13 ft above water level. He observes that the angle of elevation of the top of a cliff is 40° and the angle of depression of the base is 20°. Find the distance of the cliff from the ship and the height of the cliff if the base of the cliff is at sea level.

	a.		36 ft and 45 ft.
	b.		36 ft and 43 ft.
	c.		18 ft and 43 ft.
	d.		18 ft and 21 ft

 27. What is the angle of depression in the figure?

a.

Ð

 28.  Is the angle of elevation numerically equal to the angle of depression?

	a.		Yes
	b.		No

 29. The angles of elevation of the top of a tower from the top and bottom of a 60 m high building are 30^o and 60^o. What is the height of the tower?

	a.		80 m
	b.		90 m
	c.		55 m
	d.		60 m

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