UTPA STEM/CBI Courses/Calculus/Trigonometry/Angles of Elevation and Depression

Course Title: Trigonometry

Lecture Topic: Angles of Elevation and Depression

Instructor: Dagoberto Guerrero Jr.

Institution: Laredo Community College

Backwards Design

Course Objectives

• Primary Objectives- By the next class period students will be able to:
  • Apply their knowledge of trigonometric ratios in order to solve meaningful real-world problems

• Sub Objectives- The objectives will require that students be able to:
  • To evaluate trigonometric ratios (SOHCAHTOA).
  • Find sides of a right triangle.
  • Understand the concepts of the angle of elevation and depression.
  • Analyze lengths of sides to find what it is being asked.

• Difficulties- Students may have difficulty:
  • It is important for students to realize that the angles of elevation and depression may not always be part of the triangle being solved.
  • Identify the proper trigonometric ratio needed to be used on the problem depending on given data. (SOHCAHTOA)

• Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
  • Engineering
  • Biology
  • Business and Economy
  • Chemistry ...ets
  • Example : Elevation angle of a ramp

Model of Knowledge

• Concept Map
  • Information to look for what angle of what vertex is being used (Angle of elevation or depression).
  • Sides given from the triangle.
  • Identifying the proper trigonometric function.
  • Solutions.

• Content Priorities
  • Enduring Understanding
    • Analyze the scenario given to identify the proper trigonometric function.
    • Determine if trigonometric ratios can be used in a non - right triangles.
  • Important to Do and Know
    • Basic algebraic operations such as proportions.
    • Understand how to use the calculator when it comes to find the sine, cosine, or tangent of a particular degree angle.
    • Reading abilities to interpret word problems.
  • Worth Being Familiar with
    • Formulas related to the given problem.
    • How to find Derivatives and what it means

Assessment of Learning

• Formative Assessment
  • In Class (groups)
    • Distribute the handout: Exploring Trigonometric Ratios to each student.
    • Have students work in groups.
    • Assign a word problem to each group.
    • The students have to present in front of the class (Go Public). See attachment.
    • Make sure the student has a calculator.
  • Homework (individual)
    • Practice problems
• Summative Assessment
  • Students will take an exam of what was covered during class.
  • Students from each group will have to present the problem assigned.

Legacy Cycle

OBJECTIVE

By the next class period, students will be able to:

• Apply their knowledge of trigonometric ratios in order to solve meaningful real-world problems

The objectives will require that students be able to:

• To evaluate trigonometric ratios (SOHCAHTOA).
• Find sides of a right triangle.
• Understand the concepts of the angle of elevation and depression.
• Analyze lengths of sides to find what it is being asked.

THE CHALLENGE

The owner of Nokia Cellular Company has a mansion on the beach in Puerto Vallarta, Jalisco, Mexico. The mansion is 50 meters from sea level and it is perpendicular to the ground. The mansion is located at the tip of a cliff. He also owns two beautiful Xoloitzcuintle dogs that he bought in the area of Puerto Vallarta. One day while he was preparing dinner, he called the dogs in, but he realized that the dogs where taking a nap in the sand. When he was on the balcony facing the beach, he saw that there was a distance between the two dogs. The line of sight to the first dog has an angle of depression of 50 degrees and the second dog had an angle of depression of 25. What is the distance that existed between the two dogs?

GENERATE IDEAS

The students will be asked to find ways to determine the distance between the two dogs.

• Idea 1: Ask the owner to use a measuring tape.
• Idea 2: Look for patterns that define the situation.
• Idea 3: Use the Pythagorean Theorem to help generate solutions.
• Idea 4: Apply the newly learned tools of Trigonometry.

MULTIPLE PERSPECTIVES

Students are asked to gather in groups of four. They will discuss whether their idea was the easiest way to find the distance between the two dogs. The optimal solution would not involve use of the measuring tape.

RESEARCH & REVISE

• The instructor will introduce the concepts of angle of elevation and depression.
• The instructor will also use the knowledge taught in the previous class about trigonometric ratios. That knowledge will be used to make a connection to the concept of angles of elevation/depression.

TEST YOUR METTLE

When surveyor’s measure distances on land that slopes significantly, the distance measured along the ground is longer than the horizontal distance to be drawn on maps. Trigonometry is used to calculate this horizontal distance. Suppose the top edge of Dry Creek is 45.4 meters from the bottom and the land slopes downward at an angle of 33 degrees. How far is the horizontal distance to the bottom of Dry Creek?

GO PUBLIC

• Students will be given a handout entitled Applications of Trigonometry and given the opportunity to practice analyzing word problems.
• The purpose of this activity is to give students an opportunity to apply their knowledge of trigonometric ratios in a problem-solving setting.

Pre-Lesson Quiz

Previous lessons will lead into this topic. Therefore a pre-lesson quiz will not be administered.

Test Your Mettle Quiz

1. Commercial airliners fly at an altitude of about 10 kilometers. They start descending toward the airport when they are still far away in order to lessen the angle they must drop. a. If the pilot wants the path to make a 5 degrees angle with the ground, how far out must the plane begin to descend? b. If he starts descending 350 kilometers from the airport, what angle will the plane’s path make with the ground?
2. A new rope must be ordered for the flagpole in front of the school. Before ordering the rope,the height of the pole must be determined. It is observed that the flagpole casts a shadow 10.5 meters long when the sun is at an angle of elevation of 33 degrees. How tall is the flagpole?
3. A cat is trapped on a tree branch 18.5 feet above the ground. The ladder is only 20 feet long.If you place the ladder’s tip on the branch, what angle must the ladder make with the ground?
4. San Francisco has very steep streets. Sue decides to determine the angle the street she lives on makes with the horizontal. On the wall of her house she measures horizontal and vertical distances of 33 centimeters and 5 centimeters, respectively. What angle does her street make with the horizontal?
5. From a point on the North Rim of the Grand Canyon to a point on the South Rim, a surveyor measures an angle of depression of 3 degrees. The horizontal distance between the two points is 10 miles. How many feet is the South Rim below the North Rim?

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