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**Angles of Elevation and Depression**

Please view this tutorial and answer the follow-up questions on loose leaf to turn in to your teacher.

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**How to Interpret Word Problems Into Right Triangles**

1) Assume trees, buildings, poles, etc. are perpendicular to the ground (forming a 90° angle) 2) How high or how tall represents the side perpendicular to the ground 3) Shadows are on the ground 4) String of a kite, the sun ray, line of sight, a ladder leaning against a building, etc. represent the hypotenuse

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

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Make a Sketch A little boy is flying a kite. The string of the kite makes an angle of 30° with the ground. If the height of the kite is 9 meters, find the length of the string that the boy has used. Now that we know the important information, try to label the triangle.

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

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

Now that we have our picture, we can use trig ratios to solve for x. What trig ratio should we use based on the given information? 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. 9 m x 30°

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

Sin 30° = 1 = x sin 30° sin 30° sin 30° x = X = 18 m x 9 m 30°

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**Trig Word Problem Special Cases**

Angle of Elevation The angle of elevation to the top of an object is the angle formed by horizontal and the line of the sight to the top of the object. Angle of Depression The angle of depression to an object is the angle formed by the horizontal line of sight to the object below.

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Now let’s review! Use the following link to review the terms angle of elevation and angle of depression 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 your textbook to review as well

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

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Example An airplane is on the runway strip 200 yards from the air- traffic control tower. If the tower is 20 yards high, at what angle would the pilot have to look up to see the top of the tower? Now that we’ve underline the important information in the problem, try to draw a sketch to match it.

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**How Does Your Sketch Compare?**

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Now, Let’s Solve for x. Tan x = X = X = 5.71°

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

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

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How Did We Do? Do not forget to draw the second triangle in an angle of depression problem! 8500 ft Since the figure is a rectangle, we know that opposite sides are the same length 5000 ft 5000 ft 8500 ft

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Let’s Solve For X Tan X = X = X = °

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

Ronnie is 3 m tall and is standing 40 m from the base of a tower. If Ronnie is looking up at the top of the tower with an angle of 67°, what is the height of the tower? Remember to draw and label a sketch to help solve the problem.

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**Check your sketch with the one below**

Do you notice any difference in his problem? x Notice that for the first time the angle is not level with the ground. 67° 3m 40 m

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**Let’s Solve the Problem**

Since we know the side adjacent to the angle and we need to find the side opposite of it, we use tangent. tan (67) = x = 40 tan (67) x = m

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

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**The Total Height of the Tower**

The TOTAL height of the tower from the ground is = m. 94.23 67° 3m 3m 40 m

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Time For Practice! Use what you’ve just reviewed to help you answer the following questions. Complete the following problems and make sure to turn in all work to your teacher when finished. Be sure to include a sketch if not given, to help solve the problem correctly. GOOD LUCK!

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Problem 1 A ladder leans against a building. The foot of the ladder is 6 feet from the building. The ladder reaches a height of 14 feet on the building. Find the length of the ladder to the nearest foot. Find to the nearest degree, the angle the ladder makes with the ground.

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Problem 2 Find the distance from the tree to the airplane

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Problem 3 The angle of elevation from a point on the ground to the top of a tree is 28°. If the tree is 43 feet high, find the distance from this point to the base of the tree.

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

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