Unit Circle

2 Key excerpts on "Unit Circle"

• 

  eBook - ePub

  Pre-Calculus Workbook For Dummies

  • Mary Jane Sterling(Author)
  • 2019(Publication Date)
  • For Dummies

    (Publisher)

  Part 2

  Trig Is the Key: Basic Review, the Unit Circle, and Graphs

  IN THIS PART …

  You should be familiar with the basics of trigonometry from earlier math classes — right triangles, trig ratios, and angles, for example. But your Algebra II course may or may not have expanded on those ideas to prepare you for the direction that pre-calculus is going to take you. For this reason, it's assumed that you’ve never seen this stuff before. You won't be left behind when continuing your mathematics journey.

  This part begins with trig ratios and word problems and then moves on to the Unit Circle: how to build it and how to use it. You’ll some trig equations and make and measure arcs. Graphing trig functions is a major component of pre-calculus, so you'll see how to graph each of the six functions.

  Passage contains an image Chapter 6

  Basic Trigonometry and the Unit Circle

  IN THIS CHAPTER

  Working with the six trigonometric ratios

  Making use of right triangles to solve word problems

  Using the Unit Circle to find points, angles, and right triangle ratios

  Isolating trig terms to solve trig equations

  Calculating arc lengths

  Ah … trigonometry, the math of triangles! Invented by the ancient Greeks, trigonometry is used to solve problems in navigation, astronomy, and surveying. Think of a sailor lost at sea. All he has to do is triangulate his position against two other objects, such as two stars, and calculate his position using — you guessed it — trigonometry!

  In this chapter, the basics of right triangle trigonometry are reviewed. Then you see how to apply that knowledge to the Unit Circle, a very useful tool for graphically representing trigonometric ratios and relationships. From there, you can solve trig equations. Finally, these concepts are combined so that you can apply them to arcs. The ancient Greeks didn’t know what they started with trigonometry, but the modern applications are endless!

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  eBook - ePub

  The Equations World

  • Boris Pritsker(Author)
  • 2019(Publication Date)
  • Dover Publications

    (Publisher)

  2 α = 1.

  The most important properties of the trigonometric functions f (x ) = sin x , f (x ) = cosx , f (x ) = tan x , f (x ) = cot x are derived from their definitions.

  Trigonometric functions are periodic functions, which repeat their values at a specific point in regular intervals or periods. The period for the trigonometric functions sine and cosine is 2π radians or 360◦ ; the period for the trigonometric functions tangent and cotangent is π radians or 180◦ . It is very important to understand when you solve trigonometric equations that as you get one solution, you must include all the other solutions repeated in the interval period. The domain of the functions f (x ) = sin x and f (x ) = cos x is all real numbers. Since | sin x | ≤ 1 and | cos x | ≤ 1, the range of the functions f (x ) = sin x and f (x ) = cos x consists of real numbers from − 1 through 1, including the end points. The ranges of the functions f (x ) = tan x and f (x ) = cot x are all real numbers. The domain of f (x ) = tan x is all real numbers except and the domain of f (x ) = cot x is all real numbers except x = πn , where n is any integer. In order not to repeat it every time in the solutions of all the following equations, please remember that when we write the answer and add the period multiplied by some number n , that number is assumed to be any integer, which will be written as n ∈ Z . The function f (x ) = cos x is an even function, i.e., cos(− x ) = cos x , for all real x ; f (x ) = sin x is an odd function, i.e., sin(− x )= − sin x , for all real x ; f (x ) = tan x is an odd function, i.e., tan(− x )= − tan x , for all real f (x ) = cot x is an odd function, i.e., cot(− x ) = − cot x , for all real x ≠ πn , n ∈ Z

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