# UTPA STEM/CBI Courses/Calculus/Applications of Trigonometric and Inverse Trigonometric Functions

Course Title: Pre Calculus with Trigonometry

Lecture Topic: Applications of Trigonometric and Inverse Trigonometric Functions

Instructor: Dr. Andras Balogh

Institution: The University of Texas-Pan American

## Backwards Design[edit]

**Course Objectives**

**Primary Objectives**- By the next class period students will be able to:- Use trigonometric and inverse trigonometric functions to calculate angles and sides of a right triangle

**Sub Objectives**- The objectives will require that students be able to:- Convert between miles per hour and meters per second
- Practice the use of trigonometric and inverse trigonometric functions to calculate sides and angles of a right triangle.

**Difficulties**- Students may have difficulty:- Understanding and visualizing a word problem
- Working with decimal numbers
- Basic values of trigonometric functions

**Real-World Contexts**- There are many ways that students can use this material in the real-world, such as:- Understand the dangers of drive by shootings
- Understand how the U.S. court system works
- Understand how fast 35 miles per hour really is

**Model of Knowledge**

**Concept Map**- trigonometric functions
- inverse trigonometric functions
- trigonometric functions
- applications of trigonometric functions
- graphs of trigonometric functions
- equations with trigonometric functions
- trigonometric identities
- inverse trigonometric functions
- vectors
- sequences and series
- the Binomial Theorem
- conic sections
- parametric equations
- polar equations

**Content Priorities****Enduring Understanding**- trigonometric functions
- inverse trigonometric functions

**Important to Do and Know**- unit conversions
- Pythagorean Theorem

**Worth Being Familiar with**- calculating with decimal numbers
- rounding numbers
- guns
- cars

**Assessment of Learning**

**Formative Assessment**- In Class (groups)
- text
- text

- Homework (individual)
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- text

- In Class (groups)
**Summative Assessment**- worksheet: calculate values of inverse trigonometric functions
- text

## Legacy Cycle[edit]

**OBJECTIVE**

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

- Draw right triangles and identify the different sides based on a word problem.
- Calculate the sides of a right triangle using angles and trigonometric function.
- Calculate the angles of a right triangle using side lengths and inverse trigonometric function.

The objectives will require that students be able to:

- convert between mph and m/s

**THE CHALLENGE**

Sixteen-year old Bobby is charged with premeditated murder. You are the defense lawyer. The prosecution claims that

- Bobby waited in front of his house, 6ft from the street, for his estranged gang members to drive by,
- The passengers of the car were passing by minding their own business, at the maximum allowed speed of 35mph.
- Bobby shot a whole magazine from his semiautomatic hand gun into the passing car through the rear window, hitting the backseat.

One of the passengers died, another was seriously wounded. Bobby claims that the car stopped, the passengers threatened him with guns, and he acted in self-defense. You have to draw a picture for the jury, and try to convince them that there is a reasonable doubt for the prosecution's claim to be true.

**GENERATE IDEAS**
Students will need several days to think about what additional information is needed.

- What kind of hand gun was used?
- What is the fire rate of the gun?
- How many bullets were fired?
- The rear window was closed initially and was broken subsequently by a bullet.
- The bullet ricochets off from the window if the angle is less than 20 degrees.
- Once the car passes the shooter, the bullets will hit the front seat instead of the backseat.

**MULTIPLE PERSPECTIVES**

Different approaches might lead to the same result.

- Try to show that the number of bullets that can be fired into the car passing at 35mph is less than the number really fired,
- Try to show how slow the car must have been if the given number of bullets were fired.
- Try to show at what angle the first bullet would have hit the window if the all the bullets were fired into the car passing by 35mph.

**RESEARCH & REVISE**

After students figure out the geometric picture, explain how trigonometric and inverse trigonometric functions can be used. They have to figure which one to choose and when.

**TEST YOUR METTLE**

Do the calculations using right triangles, trigonometric and inverse trigonometric functions.

**GO PUBLIC**

Students should be able to calculate sides and angles of right triangles using different trigonometric and inverse trigonometric functions. Students have to make group presentations in a courtroom setting, explaining the significance of their findings to the jury. The rest of the class members have to come to a decision about the guilt beyond a reasonable doubt of the defendant. Votes will be cast by clickers.

## Pre-Lesson Quiz[edit]

- A car travels at 35 mph.
- How far will a car travels in 10 seconds?
- How much time does it take for the car to travel 100ft?

- In a right triangle with shorter sides of length 6ft and 15ft
- How long is the shots per hypothenuse?
- What are the angles of the triangle?

## Test Your Mettle Quiz[edit]

- Bobby used a Glock 17, which has firing rate of about 1,100 / minute. He fired the whole magazine into the passing car that had speed 35 mph. Bobby was standing at a distance of 6ft from the street.
- How much time should it take to empty the magazine?
- How much distance would the car have travelled during that time?
- At what angle must the bullet have hit the car?

- Draw a figure of the above scenario and describe the reason why the above story is not reasonable.
- Recalculate the answers to the above questions assuming that the car stopped in front of Bobby, initially, and then accelerated at a constant rate to 40 mph in 10 seconds.

Bobby used a Glock 17, which has firing rate of about 1,100 per minute.
That equates to over 18 rounds per second with this handgun. The prosecutors story is not possible.