Angle of Elevation By Karen Borst

Goal Develop an understanding of the trigonometric ratios and their real life applications SOH CAH TOA sin = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 cos= 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 tan= 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

Objectives Given materials, students will successfully build a clinometer which is: An object used to measure angles to find the height of an object Using the clinometer, students will find the height of objects that are unmeasurable using trig ratios. Using the clinometer, students will understand the application of trig ratios. Tangent used in these cases

Grade Level Generally this would be a middle school or high school lesson. Building and reading the clinometer: elementary level Using tangent to find the height: middle/high school students

Standards High School Geometry Similarity, Right Triangles, & Trigonometry Define trigonometric ratios and solve problems involving right triangles 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Previous Class Introduce right triangle trig SOH CAH TOA Compare to Pythagorean Theorem Can only be used to find sides Determine how to solve for a side or an angle

Example To measure the height of an inaccessible TV tower, a surveyor paces out a base line of 200 meters and measures the angle of elevation to the top of the tower to be 60°. How high is the tower? tan= 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 tan(60)= 𝑥 200 x 200 tan 60 ≈346 𝑚𝑒𝑡𝑒𝑟𝑠 60° 200 meters

Building my Clinometer Protractor Ruler String Scissors Tape Button Cardboard Straw

protractor Finished Product straw cardboard String with button

How do you use it? Need two people Measure the distance from the base of the object to where you are standing. Look at the object you are measuring Have partner look at what degree value is shown on clinometer Use trig ratios to determine the height of the object DON’T FORGET! Add the height from the ground to your eye level since that is a part of the height of the object.

What I did using a Clinometer Emergency Blue Light pole 15, 25, and 35 paces away 1 pace ≈ 10.6 inches 159, 265, and 371 inches away Found the degree values to be 18°, 10°, and 8° respectively The height from the ground to my eye = 62 inches

Too small – must add my height Eye to ground since my measurement Using tangent, I estimated the heights to be ≈51.7, 51.5, 𝑎𝑛𝑑 52.1 𝑖𝑛𝑐ℎ𝑒𝑠 ℎ𝑖𝑔ℎ . Too small – must add my height Eye to ground since my measurement was from my eyes 62 inches Real heights for the emergency blue light pole 113.7, 113.5, and 114.1 inches high Actual Height = 114 inches

Percent Error 113.5−114 114 ∗100=.43859%

Example 2 – No Parking Sign Paces Conversion to Inches Degrees Height Final Height (+62) 10 106.299 18 ≈34.538 ≈96.538 20 212.598 ≈37.487 ≈99.486 30 319.897 5 ≈33.625 ≈95.625 Actual Height = 99 inches Approximate Average Percent Error: 2.12%

Example 3 – Lamp Post ~ Chose something I couldn’t measure using a ruler being that I knew my method was generally accurate Inches Conversion Degree Value Height Final Height (+62) 120 44 ≈115.88 ≈177.88 240 28 ≈127.61 ≈189.61 360 21 ≈138.19 ≈200.19 Average Approximate Height: 189.27 inches

Example 4 - Stairs

For Students Have students stand at different intervals from the base (10, 20, 30 feet away) Record the degree values by looking through the straw on the clinometer. Find the height for each trial. Average those values to determine the approximate height of the object.

Possible Errors Taping the string to the cardboard Error in measurements Shaking hands Not look at same area Under/Over estimating degree values

Challenge Problem If you know the height of the object and the angle of elevation, how would you be able to determine how far away from the object you are? If you know the height of the object and how far away from it you are, how would you determine the angle of elevation?

What they will learn next… After angle of elevation, students will look at the angle of depression and determine how they would be able to the find the height of objects smaller than them and how this process would be different than using the clinometer.

Thanks to my helpers!

Reference http://centraledesmaths.uregina.ca/RR/database/RR.09.97/bracken1.pdf