Unit 7 Trigonometric Functions, Equations, and Identities

Lesson 1

Learning Focus

Shift the graphs of trigonometric functions horizontally.

Lesson Summary

In this lesson, we examined the horizontal shift of a trigonometric function, which is also referred to as a phase shift. Different forms of the equation representing the horizontal shift led to two different interpretations in the context of the Ferris wheel: shifting the position of the rider on the wheel at time $t = 0$ , or shifting the time when we start timing the position of the rider on the wheel.

Lesson 2

Learning Focus

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

Solve trigonometric equations.

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.

Lesson 3

Learning Focus

Write equivalent sine and cosine equations.

Find the complete set of solutions for a trigonometric equation.

Model periodic contexts.

Lesson Summary

In this lesson, we reviewed writing trigonometric functions and solving trigonometric equations to model situations in a context. We observed that equivalent sine and cosine functions can be written to model the same context, and that equivalent forms of equations that represent a horizontal translation of a trigonometric function emphasize changing different quantities in the context, such as the initial position or the start time.

Lesson 4

Learning Focus

Define and identify key features of the tangent graph.

Lesson Summary

In this lesson, we extended the definition of the tangent ratio for right triangles to include all angles of rotation. Using this definition, we were able to find values for the tangent of angles that are multiples of $\frac{π}{6}$ or $\frac{π}{4}$ on the unit circle and we were able to determine the key features of the tangent function graph.

Lesson 5

Learning Focus

Derive and justify trigonometric identities.

Lesson Summary

In this lesson, we identified and explained some fundamental trigonometric identities—trigonometric statements that are true for all angles. Trigonometric identities will allow us to change the form of a trigonometric expression, when needed. One of the identities, $\sin^{2} θ + \cos^{2} θ = 1$ , looks complicated and may seem like it shouldn’t be true, but it can be verified using the Pythagorean theorem.

Lesson 6

Learning Focus

Solve trigonometric equations using identities and graphs.

Lesson Summary

In this lesson, we extended the process for solving trigonometric equations to include looking for trigonometric identities that might change the trigonometric expressions to simpler forms. We also identified ways to determine how many solutions to the trigonometric equation occur in the interval from 0 to $2 π$ .

Lesson 7

Learning Focus

Derive trigonometric identities for the sum or difference of two angles.

Lesson Summary

In this lesson, we expanded our list of trigonometric identities to include identities for finding the sine or cosine of angles that are the result of adding two angles together or subtracting one angle from another one. If the angles are the same size, then we can use these sum identities to find the sine and cosine of an angle that is twice as big as a given angle.

Lesson 8

Learning Focus

Define the inverse sine, inverse cosine, and inverse tangent functions.

Lesson Summary

In this lesson, we defined the inverse sine, the inverse cosine, and the inverse tangent functions by first restricting the domains of the sine, cosine, and tangent functions to an interval over which the inverse function could be defined. These restricted domains become the range of the inverse trigonometric functions. Knowing how the inverse trigonometric functions are defined helps interpret the results produced on a calculator when it is used to solve trigonometric equations.

Lesson 9

Learning Focus

Solve trigonometric equations strategically.

Lesson Summary

In this lesson, we extended our strategies for solving trigonometric equations to include equations with different multiples of the variable inside trigonometric expressions, equations that could be solved exactly using the unit circle diagram, and those that required using the inverse trigonometric features of the calculator, and trigonometric equations that behaved like quadratic equations, requiring expressions to be factored.

Lesson 10

Learning Focus

Represent complex numbers using polar coordinates.

Multiply complex numbers written in polar form.

Lesson Summary

In this lesson, we learned how to write complex numbers in polar form. We used the polar form to multiply and divide complex numbers and to raise complex numbers to powers. We also learned that every complex number has $n th$ roots that are easy to find in polar form.