TEF - Modeling with Trigonometric Expressions and Functions Module Overview

Modeling with Trigonometric Expressions and Functions Module Overview

The input becomes the output; the output becomes the input sin(angle) = ration and sin to the neg 1(ratio) = angle

Trigonometry is a major part of Precalculus and integral in your mathematical preparation. In this module, we will connect what you know about trigonometry (sine, cosine, and tangent) to special right triangles and the coordinate plane! Precalculus is when many different parts of math finally connect! Hopefully you are excited to begin this journey. Then we will explore graphing trigonometric functions. We will take the ratios we've learned on the Unit Circle, and use those to model the graphs of sine, cosine and tangent. Finally, you will learn how to utilize those sinusoidal functions to model real world phenomena like the tides or the temperature!

Once we explored the Unit Circle, common trigonometric ratios and how to graph the functions of sine, cosine and tangent, we will explore the inverse functions. The inverse functions can help us solve equations and use sinusoidal equations in real-world applications!

Essential Questions

How do I think about an angle as the rotation of a ray around its endpoint?
What is meant by the radian measure of an angle?
What is the connection between the radian measure of an angle and the length of the arc on the unit circle the angle intercepts?
What does it mean to prove a trigonometric identity?
What does the Unit Circle have to do with Trigonometric Functions?
How are the amplitude, midline, period, and phase shift of a trigonometric function related to the transformation of the parent graph?
If I know the characteristics of the graphs of a sinusoidal function, how can I write an equation of that graph?
How can we model a real-world situation with a trigonometric function?
How does symmetry help us extend our knowledge of the Unit Circle to an infinite number of angles?
How do I utilize technology to find all solutions for a trig equation?
How do inverse trigonometric functions help us solve equations?

Key Terms

The following key terms will help you understand the content in this module.

Coterminal Angles - share the same initial side and the same terminal side of angles of rotation.

Radian - the measure of the central angle of a circle subtended by an arc of equal length to the radius

Initial Side - the "beginning" side of an angle of rotation, usually on the positive x-axis

Terminal Side - the "ending" side of an angle of rotation

Standard Position - an angle is in standard position when the vertex is at the origin and the initial side lies on the positive x-axis

Negative Angle - an angle in standard position is negative when the location of the terminal side results from a clockwise rotation

Positive Angle - an angle in standard position is positive when the location of the terminal side results from a counterclockwise rotation

Reference Angle - the measure of the acute angle formed by the terminal side and the x-axis

Identity - an equation that is true for all values of the variable for which the expressions in the equation are defined

Unit Circle - a circle with a radius of 1 and center at the origin

Sinusoidal Function - a function is considered sinusoidal if its graph has a shape of equation image indicator $LaTeX: y=\sin x$ $y=\sin x$ or a transformation of $LaTeX: y=\sin x$ $y=\sin x$ .

Midline - a horizontal line located halfway between the maximum and minimum values.

Amplitude - the distance from the midline to either the maximum or minimum value; ½ the distance between the maximum and minimum values.

Period - the horizontal length of one complete cycle; the distance between any two repeating points on the function.

Frequency - the number of cycles the function completes in a given interval; the reciprocal of the period.

Asymptote - a line that continually approaches a given curve but does not meet it at any finite distance.

Phase Shift - a change in the phase of a waveform.

Even Functions - A function f is even if the graph of f is symmetric with respect to the y-axis. Algebraically, f is even if and only if equation image indicator $LaTeX: f\left(-x\right)=f\left(x\right)$ $f\left(-x\right)=f\left(x\right)$ for all x in the domain of f.

Odd Functions - A function f is even if the graph of f is symmetric with respect to the origin. Algebraically, f is odd if and only if equation image indicator $LaTeX: f\left(-x\right)=-f\left(x\right)$ $f\left(-x\right)=-f\left(x\right)$ for all x in the domain of f.

Inverse Function - An inverse function is a function that undoes the action of another function. A function g is the inverse of a function f if whenever y then x. In other words, applying f and then g is the same thing as doing nothing.

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