**Topics: **

• The inverse trig functions , , and : definitions and notation

• Solving right triangles using the inverse trig functions

**Definitions and how to read the notation:**

Note: the superscript -1 in these functions’ names represents the inverse function. It is very important that you remember that **this is not the -1 power**.

is read as “the inverse sine of x” or “the arcsine of x” (which literally means “the arc whose sine is x”). I like to read it as “the angle whose sine is x”.

An infinite number of angles (arcs on the unit circle) can have the same sine x. The inverse sine picks out the one angle which is between and (inclusive).

Definition: and

Note that the sine of an angle must be between -1 and 1 (inclusive), therefore is only defined for x which is between -1 and 1 (inclusive). If you try to find , for example, on your calculator, you will get an error message because it is undefined.

similarly:

is read as “the inverse cosine of x” or “the arc-cosine of x” (which literally means “the arc whose cosine is x”). I like to read it as “the angle whose cosine is x”.

An infinite number of angles (arcs on the unit circle) can have the same cosine x. The inverse cosine picks out the one angle which is between and (inclusive).

Definition: and

Note that the cosine of an angle must be between -1 and 1 (inclusive), therefore is only defined for x which is between -1 and 1 (inclusive). If you try to find , for example, on your calculator, you will get an error message because it is undefined.

is read as “the inverse tangent of x” or “the arctangent of x” (which literally means “the arc whose tangent is x”). I like to read it as “the angle whose tangent is x”.

An infinite number of angles (arcs on the unit circle) can have the same tangent x. The inverse tangent picks out the one angle which is between and (not inclusive).

Definition: and

can be any real number, therefor there is no restriction on the values of x for .

**Homework:**

• Review the example we discussed in class. (My WeBWorK problem)

• Do the WeBWorK assignments. It is recommended that you do them in the following order:

(1) Unit Circle (from last time)

(2) Graphing Sine Cosine (also from last time)

(3) Solving Right Triangles – Inverse Trig

• Here are problems (including the promised problems from Coordinate Plane Definition) from the textbook which you may put on the board at the start of class next time:

On the coordinate plane definition of the trig functions:

Find the values of the six trig functions of if this point is on the terminal side of the angle:

(1) (-6, 8)

(2) (5, -12)

(3) (-7, -7)

From the Trig textbook:

Remember that if you get stuck on any of the problems or have a question about any of the material, you can post a question to the Piazza discussion board.

In case you prefer to view videos on any of these topics, Here is a copy of the Course Outline with links to video resources for each topic.

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Here are the review self-tests:

And here are the answers:

MAT1275Test3ReviewAnswersSpring2017

There will also be a problem on trig, using the coordinate plane. Practice on the WeBWorK

“CoordinatePlaneTrig” problems.

Here are some resources for some of the other problems: also please see the relevant WeBWorK assignments, which will show you solutions to similar problems.

For problem 1, solving a 3 by 3 linear system:

3 by 3 linear systems of equations:

Paul’s Notes

Paul’s notes practice problems with solutions

For problem 5, solving a nonlinear system of equations:

Non-linear systems of equations:

Paul’s Notes

Paul’s notes practice problems with solutions

For the problems on circles:

Paul’s notes

Paul’s notes practice problems with solutions

Remember that if you get stuck on any of the problems or have a question about any of the material, you can post a question to the Piazza discussion board. I have posted several of the review problems so that you could post your solutions for a homework point if you wish. If you post by taking a photo (or better, a scan) of your work, please make sure that it is clearly legible.

Also, you may post a solution to any other problem from the review by posting the problem itself as a Question on Piazza and then posting your solution as the student solution to that question.

There are also several student questions which you could post a solution to if you wish.

Genius scan is a very nice app for taking scans with a smartphone.

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Here are the review self-tests:

And here are the answers:

MAT1275Test3ReviewAnswersSpring2017

There will also be a problem on trig, using the coordinate plane. Practice on the WeBWorK

“CoordinatePlaneTrig” problems.

Remember that if you get stuck on any of the problems or have a question about any of the material, you can post a question to the Piazza discussion board. I have posted several of the review problems so that you could post your solutions for a homework point if you wish. If you post by taking a photo (or better, a scan) of your work, please make sure that it is clearly legible.

Also, you may post a solution to any other problem from the review by posting the problem itself as a Question on Piazza and then posting your solution as the student solution to that question.

There are also several student questions which you could post a solution to if you wish.

Genius scan is a very nice app for taking scans with a smartphone.

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#25-31 odd, 45, 47, 55-63 odd, 64

• Angles in the coordinate plane in standard position

• The coordinate plane definitions of the trig functions

• Coterminal angles

• Radian measure for angles (very important!)

Important note: There will be one more definition of the trig functions, which comes from this, and it is the most important one, so make sure you learn these right away!

**You must learn the two special right triangles and the definitions of the trig functions by heart!**

VOCABULARY TO KNOW:

angle (as a rotation)

angle in standard position

positive angle

negative angle

initial side

terminal side

coterminal angles

THE COMPLETE DEFINITIONS OF THE 6 TRIGONOMETRIC FUNCTIONS IN THE COORDINATE PLANE:

is an angle in standard position, is a point on the terminal side of , and is the distance from the origin to :

Then define the six trigonometric functions as:

sine:

cosine:

tangent: sine:

secant: sine: (the reciprocal of cosine)

cosecant: (the reciprocal of sine)

cotangent: (the reciprocal of tangent)

**Homework:**

• There are two new WeBWorK assignments. It is recommended that you do them in the following order;

(1) Coordinate plane trig

(2) Angle measure – radians

• I will post some problems from the textbook which you may put on the board at the start of class when I have a chance to look through them. Stay tuned!

Remember that if you get stuck on any of the problems or have a question about any of the material, you can post a question to the Piazza discussion board.

In case you prefer to view videos on any of these topics, Here is a copy of the Course Outline with links to video resources for each topic.

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• Solving a 3 by 3 system of linear equations – for short, this is called a 3 by 3 linear system

• Solving nonlinear systems of equations

It is useful to look at the graphs of these systems, in Desmos for instance, so you can know how many solutions you expect to have. Highly recommended!

• Two important special right triangles

• Right triangle trigonometry: the definitions of the six trig ratios.

Important note: These definitions are for the trig ratios as they are used in right triangles. There will be two more definitions, which come from these, later in the course and you will need to know all three ways of defining the trig functions, so make sure you learn these right away!

**You must learn the two special right triangles and the definitions of the trig ratios by heart!**

**Homework:**

• There are several WeBWorK assignments. It is recommended that you do them in the following order;

(1) 3×3-Systems

(2) NonlinearSystems

(3) SpecialTriangles

(4) TrigonometryRatios

• Here are some problems from the textbook which you may put on the board at the start of class: p. 782: 23-37 odd, 49

Remember that if you get stuck on any of the problems or have a question about any of the material, you can post a question to the Piazza discussion board.

In case you prefer to view videos on any of these topics, Here is a copy of the Course Outline with links to video resources for each topic.

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p.760: 5,9,11,13,23-31 odd,39,41,45,61,63,65,69,75

NOTE: I am not entirely sure about the page number. These are problems that come at the end of Section 9.1 in any case.

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There is a Test 2 Review assignment on WeBWorK.

This assignment is absolutely due at the stated time, and there will be no extensions. You will be able to view the solutions once the assignment has closed.

Test 2 review Self-Test here:

Answers and partial solutions here:

MAT1275Test2ReviewAnswersFall2018

Please use the Piazza discussion board if you have any questions and/or (especially!) if you think you have found any typographical or other errors in these!

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• Perfect square trinomials

• The Square Root Property

• Completing the square to solve a quadratic equation

[link to YouTube]

It is very important to practice using the perfect square patterns so that you can recognize them when you see them, and this will also help you to understand completing the square. We rarely use completing the square to solve quadratic equations, but we will need to know how to complete squares for a later topic in this course. Also, it is used to prove the Quadratic Formula.

**Homework:**

• View the video on completing the square, if you want to.

• Complete the WeBWorK assignment we started in class. There will not be extensions of this assignment, so make sure to ask questions on Piazza or view the video or seek help in some other way if you get stuck.

• Next topic: The Quadratic Formula and complex number solutions to quadratic equations.

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