Lesson 2 High Tide Solidify Understanding

Jump Start

1.

Find the measure of angle $θ$ in the following diagram. Represent your solution process using equations.

2.

Find the measure of angle $θ$ in the given diagram. Represent your solution process using equations.

Learning Focus

Model periodic contexts that do not involve circular motion using trigonometric functions.

Solve trigonometric equations.

How can we write functions to represent other types of periodic behavior such as the relationship between the length of time of sunlight as a function of the day of the year, or the period of vibration of a string as a function of the length of a string on a musical instrument, or even the height of water in a lake as a function of the time of day?

Open Up the Math: Launch, Explore, Discuss

To prepare for the science fair, your team is planning a field trip to the beach to conduct some experiments that will require setting up lab equipment on a portion of the beach that is sometimes underneath the tide waters and sometimes exposed to the hot sun. In order to know when to remove the equipment before it is washed away in the tide, you have found that the water level on the beach is given by this equation:

$f (t) = 20 \sin (\frac{π}{6} t)$ .

In this equation, $f (t)$ represents how far the waterline is above or below its average position. The distance is measured in feet, and $t$ represents the elapsed time (in hours) since midnight.

1.

What is the highest up the beach (compared to its average position) that the waterline will be during the day? (This is called high tide.) What is the lowest that the waterline will be during the day? (This is called low tide.)

2.

Suppose you plan to set up the lab equipment on the average waterline just as the water has moved below that line. How much time will you have to conduct experiments before the equipment will need to be removed because of the incoming tide?

3.

Suppose you want to place the lab equipment $10$ feet below the average waterline to take advantage of the damp sand. What is the maximum amount of time you will have before you need to remove the equipment? How can you convince your team that your answer is correct?

4.

Suppose you decide to place the lab equipment $15$ feet above the average waterline to give you more time to conduct your work. What is the maximum amount of time you will have to carry out your experiments? How can you convince your team that your answer is correct?

5.

You may have answered the previous problems using a graph of the tide function. Is there a way you could use algebra and the inverse sine function to solve these problems? If so, show your work.

a.

Algebraic work for problem 3:

b.

Algebraic work for problem 4:

6.

Suppose your team decides you only need two hours to conduct experiments. What is the lowest point on the beach where you can place the lab equipment? How can you convince your team that your answer is correct?

Ready for More?

Look up data for the average daily temperature or the length of the day between sunrise and sunset as functions of the day of the year for the location in which you live. Are these relationships periodic? If so, write equations to model the daily temperature or the average amount of the sunlight for your area.

Takeaways

Solving trigonometric equations requires:

Trigonometric functions can be used to model periodic behavior, such as:

Answering problems that involve using inverse trigonometric functions to find specific instances in time in such contexts may involve:

Lesson Summary

In this lesson, we applied trigonometric functions to model a context that was periodic, but not about circular motion. We used graphs and the unit circle to interpret the meaning of values obtained when solving the equation for time using inverse trigonometric functions.

Retrieval

1.

Recall that the right triangle definition of the tangent ratio is: $\tan A = \frac{length of side opposite angle A}{length of side adjacent to angle A}$ .

Solve for $y$ . Then find $\tan A$ and $\tan B$ .

$y =$

$\tan A =$

$\tan B =$

2.

Use the given solutions of a quadratic function and the $y$ -intercept to find the original equation.

$x = - 4 \pm 3 i$ with $y$ -intercept $(0, 75)$