Back To Course

ACT Prep: Help and Review44 chapters | 435 lessons | 26 flashcard sets

Instructor:
*Sharon Linde*

This lesson shows a step by step process for graphing the 1 cos 2x, which is a periodic function. We'll show you how to find the maximum and minimum values, and the values of x when y is equal to zero.

Leo has always been a wiz with engines. When he was 9, he fixed the weed-eater engine his dad was too frustrated to keep working on. He hangs out at the local mechanic down the street from his house, and he's even started to make some money off of a side business fixing lawnmowers, hedge-trimmers, and other small engines.

Ordinarily, Leo wasn't the biggest fan of the math classes he had to take in high school, but when his teacher mentioned the cosine function was useful in engine design and repair, he definitely perked up. How could this math, that looked both boring and difficult, help him understand engines better? So, Leo did something he almost never did in math class - he raised his hand and asked a question.

'Mr. Grand, you said something about engines, which I love to work on, but then I look up and you have some funny looking squiggle drawn on the board. How does that squiggle represent anything about an engine?'

Mr. Grand explains that the sine and cosine functions can represent the repetitive movement of the cylinders in a combustion engine or the rotation of an electrically driven engine. The **amplitude**, or height of the graph, indicate how much the engine is moving each cycle, and the **period**, or distance between peaks on the graph, describe how quickly the parts are rotating. He draws a 'squiggly line' on the board to explain what he is talking about:

For the sine and cosine functions, you need to know a few things to be able to graph any particular function:

- Maximum and minimum values of y
- Value of y when x=0
- Value of x when y=0
- Distance between repetitions of the cycle

These concepts were stated differently than Leo normally thought of them, but he recognized the usefulness of knowing these things when he was working on engines. He watched closely as Mr. Grand filled in a chart related to graphing the function:

*y* = 1 cos (2*x*).

Leo thought of *y* as the distance a single piston in his engine was from a neutral position and *x* as the number of degrees the crankshaft was turning.

x |
2x |
cos (2x) |
1 cos (2x) |
---|---|---|---|

0 | 0 | 1 | 1 |

45 | 90 | 0 | 0 |

90 | 180 | -1 | -1 |

135 | 270 | 0 | 0 |

180 | 360 | 1 | 1 |

Using the information that Mr. Grand had put on the board Leo is able to draw the graph of *y* = 1 cos (2*x*):

Since Leo now sees the applications of the sine and cosine functions to his engines, he's interested to know what will happen to the amplitude and period if you adjust the numbers in the equation:

*y* = Acos(B*x*), where A and B are real numbers.

Since Leo knows that the maximum and minimum value of any cosine function is 1 and -1, he knows the amplitude of *y* = Acos(B*x*) is simply going to be A. But, what effect does B have on graphing this function? They already did one example in class, so he fills his own charts in with B equal to 1 and 3.

x |
1x |
cos (1x) |
1 cos (1x) |
---|---|---|---|

0 | 0 | 1 | 1 |

90 | 90 | 0 | 0 |

180 | 180 | -1 | -1 |

270 | 270 | 0 | 0 |

360 | 360 | 1 | 0 |

Leo can see from this that the standard cosine function has a period of 360 degrees. He knows that the period was only 180 degrees when Mr. Grand graphed *y* = 1cos(2*x*).

x |
3x |
cos (3x) |
1 cos (3x) |
---|---|---|---|

0 | 0 | 1 | 1 |

30 | 90 | 0 | 0 |

60 | 180 | -1 | -1 |

90 | 270 | 0 | 0 |

120 | 360 | 1 | 1 |

He makes another chart to see if he can ferret out the relationship of B and period

B | period |
---|---|

1 | 360 |

2 | 180 |

3 | 120 |

After staring at the above chart for a few minutes, it comes to Leo that the period of the cosine function is just equal to 360/B. If you double B, the period will halve, and if you triple B, the period will become 1/3 of what it was originally.

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
10 in chapter 31 of the course:

Back To Course

ACT Prep: Help and Review44 chapters | 435 lessons | 26 flashcard sets

- Trigonometry: Sine and Cosine 7:26
- Converting Between Radians and Degrees 7:15
- How to Solve Trigonometric Equations for X 4:57
- List of the Basic Trig Identities 7:11
- Trigonometry and the Pythagorean Theorem 4:14
- Trigonometry Functions & Exponentials on a Calculator 6:45
- Trigonometry Functions & Exponentials on the CLEP Calculator 5:40
- Radians & Degrees on a Calculator 4:25
- Radians & Degrees on the CLEP Scientific Calculator 7:01
- How to Graph 1 cos 2x: Steps & Tutorial
- Go to ACT Math - Trigonometry: Help and Review

- Computer Science 109: Introduction to Programming
- Introduction to HTML & CSS
- Introduction to JavaScript
- Computer Science 332: Cybersecurity Policies and Management
- Introduction to SQL
- Progressive Politics & American Imperialism
- Reconstruction, Westward Expansion, Industrialization & Urbanization
- North America & the 13 Colonies
- The Renaissance & The Age of Exploration
- Algorithmic Analysis, Sorting & Searching
- CEOE Test Cost
- PHR Exam Registration Information
- Claiming a Tax Deduction for Your Study.com Teacher Edition
- What is the PHR Exam?
- Anti-Bullying Survey Finds Teachers Lack the Support They Need
- What is the ASCP Exam?
- ASCPI vs ASCP

- Convergent Sequence: Definition, Formula & Examples
- Mauryan Empire Art & Culture
- Multi-Dimensional Arrays in C Programming: Definition & Example
- Tests for Identifying Common Gases
- Singing Lesson Plan
- Arrays & Strings in JavaScript: Conversion & String Methods
- Heuristic Methods in AI: Definition, Uses & Examples
- Quiz & Worksheet - Average & Instantaneous Rates of Change
- Quiz & Worksheet - Speed, Velocity & Acceleration
- Quiz & Worksheet - Functions & Parameters Overview
- Quiz & Worksheet - Incremental & Radical Change
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies
- Reading Comprehension Strategies & Resources for All Teachers
- Anti-Bullying Guide | Stop Bullying in Schools

- AP English Language Syllabus Resource & Lesson Plans
- Essay Writing: Help & Tutorial
- Research Methods in Psychology for Teachers: Professional Development
- Financial Accounting Syllabus Resource & Lesson Plans
- Mitosis: Help & Review
- NMTA Essential Academic Skills Math: Mathematical Logic
- AEPA Middle Grades Math: Simplifying Expressions
- Quiz & Worksheet - Technical Instructions Audience Needs
- Quiz & Worksheet - Standard Enthalpy of Formation
- Quiz & Worksheet - Review of Trigonometry for Physics
- Quiz & Worksheet - Synonyms & Antonyms
- Quiz & Worksheet - Using Transitions Between PowerPoint Slides

- Monotonic Function: Definition & Examples
- Secondary Impact Syndrome: Symptoms & Complications
- In-School Resources for Teacher Professional Development
- How Much Does It Cost to Homeschool?
- SAT Subject Test Registration Information
- ELM Test Dates
- Average PSAT Score
- ACT Exam Prep Information
- How Does Tuition Reimbursement Work with Financial Aid?
- Is the PSAT Hard?
- Scholarships for Study Abroad
- Where Can I Find SAT Chemistry Practice Tests?

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject