Geometry: Angle of Elevation and Depression
This is a free lesson from our course in Geometry
n this lesson you’ll be introduced to the basics and concepts of Angles of Elevation and Depression. Once the review of trigonometric ratios is done, the important relationship that can be used for solving the real world problems will be explained and also the steps to be taken to do it will be discussed. The presentation covering such content will be done by the instructor in own handwriting, using video and with the help of several examples and solution. This will help you understand approach to solve home word problems and also draw help for Geometry home work.
You may recall from earlier learning how to apply sine, cosine and tangent ratios to find angles. Also review how to measure lengths and use measurements to determine angle measures. (More text below video...)
<h2> Angles of Elevation and Depression - Geometry - Watch video (Geometry)</h2> <p> angles, elevation, geometry, depression, method, angles of elevation and depression, find, problem, angle of elevation, angle of depression, example, solution, geometry help, practice questions, quizzes </p> <p> It covers the steps or method to find angle of elevation or angle of depression when the distance is known</p>
People who saw this lesson also found the following lessons useful:
The Tangent Ratio
Inverse Tangent
Inverse Sine and Cosine
(Continued from above)Angles of elevation and depression () are formed by the horizontal lines that a viewer’s lines of sight form to an object. If a person is looking up, the angle is an elevation angle. If a person is looking down, the angle is a depression angle (Figure below). Notice that: 
• The angle of elevation and the angle of depression both are measured with respect to the horizontal line. 
• Tangents are frequently used to solve problems involving angles of elevation and depression.
Now you’ll explore some practical applications of trigonometry for calculating the angle of elevation and angle of depression.
Solving Trigonometry Word Problems may be done generally in three major steps process: 
• draw a sketch/diagram and label the given information or variable(s) for giving understanding of the problem. 
• write the relationship or equation 0f the known information, and to find out. 
• simplify the relationship or solve the equation. Verify the answer given for its logic or correctness.

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 learnt 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.
The video above 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 Geometry home work.
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