UTPA STEM/CBI Courses/Calculus/Inverse Trigonometric Functions

Course Title: Precalculus

Lecture Topic: Inverse trigonometric Functions

Instructor: Dr. Constantin Onica

Institution: University of Texas - Pan American

Backwards Design

Course Objectives

• Primary Objectives- By the next class period students will be able to:
  • Define the inverse sine, inverse cosine, and inverse tangent functions.
  • Evaluate inverse trigonometric functions without a calculator.
  • Evaluate inverse trigonometric functions using a calculator.

• Sub Objectives- The objectives will require that students be able to:
  • Know the domain and range for each inverse trigonometric function.
  • Know the inverse function identities.

• Difficulties- Students may have difficulty:
  • Remembering the domains and ranges of inverse trigonometric functions ( they tend to interchange some of them).
  • Applying the inverse trigonometric identities.

• Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
  • the height of a ballon, view from a satellite, surfing the perfect wave, etc.

Model of Knowledge

• Concept Map
  • Definition of inverse trigonometric functions
  • Evaluation of inverse trigonometric functions

• Content Priorities
  • Enduring Understanding
    • Inverse sine, inverse cosine, inverse tangent functions.
    • Evaluate inverse trigonometric functions for some classical numbers.
    • Use the calculator to evaluate inverse trigonometric functions for arbitrary numbers.
  • Important to Do and Know
    • Know the domain and range for each inverse trigonometric function.
    • Inverse trigonometric function identities.
  • Worth Being Familiar with
    • Composing a trigonometric function and an inverse.
    • Graphs of inverse trigonometric functions and their relationship with the graphs of trigonometric functions.

Assessment of Learning

• Formative Assessment
  • In Class (groups)
    • Lecture
  • Homework (individual)
    • From textbook
    • Online homework on Webassign
• Summative Assessment
  • Weekly quiz
  • Exam

Legacy Cycle

OBJECTIVE

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

• Define the inverse sine, inverse cosine and inverse tangent functions.
• Evaluate inverse trigonometric functions without a calculator.
• Evaluate inverse trigonometric functions using a calculator.

The objectives will require that students be able to:

• Know the domain and range for each inverse trigonometric function.
• Know the inverse function identities.

THE CHALLENGE

Where to sit at the movies?

Here is a particular problem concerning the above question that will be asked during class:

The screen in a theatre is 20 ft. high and is positioned 9 ft. above the floor, which is flat. The first row of seats is 8 ft. from the screen and the rows are 3 ft. apart. Suppose your eyes are 4 ft. above the floor. Find the row in which you should sit to maximize the view.

GENERATE IDEAS

Present the problem and ask for opinions without telling them what "maximizing the view" means.

MULTIPLE PERSPECTIVES

Next I will present what maximizing the view means (see below) and ask the students to group up and discuss the problem.

The apparent size of an object depends on its distance from the viewer. This apparent size is determined by the angle the object subtends at the eye of the viewer. With that in mind we want to answer the following question: How far from the screen of a movie theatre should you sit in order to maximize the view?

RESEARCH & REVISE

Teach the material referring to the challenge problem from time to time.

TEST YOUR METTLE

• Homework from textbook covering the above topics will be assigned.
• Online homework will be due next class for immediate feedback.
• Weekly quiz containing some problems involving inverse trigonometric functions.

GO PUBLIC

Students will solve the problem given as challenge giving all details and explaining the conclusion.

Pre-Lesson Quiz

1. Draw the graphs of sine and cosine on $[-\pi ,\pi ]$ and tangent on $\left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]$.
2. Fill in the blanks: $\sin$______$=1/2$, $\cos$______$=-{\sqrt {3}}/2$, $\tan$______$=-1$.