This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

- Welcome
- Learning Math Strategies (Online)Toggle Dropdown
- Study Skills for MathToggle Dropdown
- Simply Math
- Business MathToggle Dropdown
- Place Value in Decimal Number Systems
- Arithmetic Operations
- Basic Laws
- Operations on Signed numbers
- Order of Operations
- Fractions
- Decimals
- Percents
- Ratios and Proportions
- Exponents
- Statistics
- Factoring
- Rearranging Formulas
- Solving Linear Equations
- Solving Systems of Linear Equations
- Trade and Cash Discounts
- Multiple Rates of Discount
- Payment Terms and Cash Discounts
- Markup
- Simple Interest
- Compound Interest
- Nominal and Effective Interest Rates
- Ordinary Simple Annuities
- Ordinary General Annuities

- Hospitality MathToggle Dropdown
- Place Value in Decimal Number Systems
- Arithmetic Operations
- Order of Operations
- Basic Laws
- Prime Factorisation and Least Common Multiple
- Fractions
- Decimals
- Percents
- Exponents
- Units of Measures
- Fluid Ounces and Ounces
- Metric Measures
- Yield Percent
- Recipe Size Conversion
- Ingredient Ratios
- Food-Service Industry Costs

- Engineering Math
- Basic Laws
- Order of Operations
- Prime Factorisation and Least Common Multiple
- Fractions
- Exponents
- Radicals
- Reducing Radicals
- Factoring
- Rearranging Formulas
- Solving Linear Equations
- Areas and Volumes of Figures
- Congruence and Similarity
- Functions
- Domain and Range of Functions
- Basics of Graphing
- Transformations
- Graphing Linear Functions
- Graphing Quadratic Functions
- Solving Systems of Linear Equations
- Solving Quadratic Equations
- Solving Higher Degree Equations
- Trigonometry
- Graphing Trigonometric Functions
- Graphing Circles and Ellipses
- Exponential and Logarithmic Functions
- Complex Numbers
- Number Bases in Computer Arithmetic
- Linear Algebra
- Calculus
- Set Theory
- Modular Numbers and Cryptography
- Statistics
- Problem Solving Strategies

- Upgrading / Pre-HealthToggle Dropdown
- Basic Laws
- Place Value in Decimal Number Systems
- Decimals
- Significant Digits
- Prime Factorisation and Least Common Multiple
- Fractions
- Percents
- Ratios and Proportions
- Exponents
- Radicals
- Reducing Radicals
- Metric Conversions
- Factoring
- Solving Linear Equations
- Solving Quadratic Equations
- Functions
- Domain and Range of Functions
- Polynomial Long Division
- Exponential and Logarithmic Functions
- Statistics

- Nursing MathToggle Dropdown
- Arithmetic Operations
- Order of Operations
- Place Value in Decimal Number Systems
- Decimals
- Fractions
- Percents
- Ratios and Proportions
- Nutrition Labels
- Interpreting Drug Orders
- Oral Dosages
- Dosage Based on Size of the Patient
- Parenteral Dosages
- Intravenous (IV) Administration
- Infusion Rates for Intravenous Piggyback (IVPB) Bag
- General Dosage Rounding Rules

- Transportation MathToggle Dropdown
- PhysicsToggle Dropdown

The sine and cosine functions link the real numerical values of the x- and y-coordinates of the unit circle points. We write out functions in the form of:

\[ y = sin(x) \]

\[ y = cos(x) \]

In this section we will be discussing how to graph these two trigonometric functions, and the attributes that are responsible for how they appear.

Let us begin with the sine function. We can create a value table, and use several points to help us draw the graph. The table below lists some of the sine function values that can be found on the unit circle:

\(x\) (rad) | \( 0 \) | \( \frac{\pi}{6} \) | \( \frac{\pi}{4} \) | \( \frac{\pi}{3} \) | \( \frac{\pi}{2} \) | \( \frac{2\pi}{3} \) | \( \frac{3\pi}{4} \) | \( \frac{5\pi}{6} \) | \( \pi \) |

\(x\) (°) |
\( 0 \) | \( 30 \) | \( 45 \) | \( 60 \) | \( 90 \) | \( 120 \) | \( 135 \) | \( 150 \) | \( 180 \) |

\( sin(x) \) | \( 0 \) | \( \frac{1}{2} \) | \( \frac{\sqrt{2}}{2} \) | \( \frac{\sqrt{3}}{2} \) | \(1 \) | \( \frac{\sqrt{3}}{2} \) | \( \frac{\sqrt{2}}{2} \) | \( \frac{1}{2} \) | \( 0 \) |

Plotting points from the table and continuing along the positive *x*-axis gives the shape of the sine function:

We can observe the cosine function in a similar way that we did the sine function. We can create a value table, and use several points to help us draw the graph. The table below lists some of the cosine function values that can be found on the unit circle:

\(x\) (rad) | \( 0 \) | \( \frac{\pi}{6} \) | \( \frac{\pi}{4} \) | \( \frac{\pi}{3} \) | \( \frac{\pi}{2} \) | \( \frac{2\pi}{3} \) | \( \frac{3\pi}{4} \) | \( \frac{5\pi}{6} \) | \( \pi \) |

\(x\) (°) |
\( 0 \) | \( 30 \) | \( 45 \) | \( 60 \) | \( 90 \) | \( 120 \) | \( 135 \) | \( 150 \) | \( 180 \) |

\( cos(x) \) | \( 1 \) | \( \frac{\sqrt{3}}{2} \) | \( \frac{\sqrt{2}}{2} \) | \( \frac{1}{2} \) | \(0 \) | \( \frac{-1}{2} \) | \( \frac{-\sqrt{2}}{2} \) | \( \frac{-\sqrt{3}}{2} \) | \( -1 \) |

Plotting the points from the table and continuing along the *x*-axis gives the shape of the cosine function:

A function that has the same general shape as a *sine *or *cosine *function is known as a **sinusoidal function**. The general forms of sinusoidal functions are:

where

**Let's try some examples! **

1. Graph the function \( y = 3sin( \frac{\pi}{4}x - \frac{\pi}{4}) \).

\( \Longrightarrow \) Let us find the *amplitude *of this function.

\( \Longrightarrow amplitude = |A| = |3| = 3\)

\( \Longrightarrow |A| > 1 \) and so we have a *vertical stretch. *

\( \Longrightarrow \) Let us determine the *period*.

\( \Longrightarrow period = \frac{2\pi}{|B|} = \frac{2\pi}{|\frac{\pi}{4}|} = 2\pi \times \frac{4}{\pi} = 8 \)

\( \Longrightarrow |B| < 1 \) and so we have a *horizontal stretch. *

\( \Longrightarrow \) Lastly, we determine the phase shifts. Since there is no "D" value, we only consider a *horizontal shift. *

\( \Longrightarrow \frac{C}{B} = \frac{\frac{\pi}{4}}{\frac{\pi}{4}} = 1 \)

\( \Longrightarrow \frac{C}{B} > 0 \) and so we have a horizontal shift of 1 unit *to the right.*

Before we graph our function, we need to find our \( y-intercept \), and a proper graph will show at least one full period (which is done by adding \( \frac{1}{4} \) of the period to our x-value four times. See the image below for the final result:

2. Graph the function \( y = -2cos(\frac{\pi}{2}x + \pi) + 3 \).

See the video below for the solution:

Try this interactive tool!

Play around with the sliders for each value and observe what happens to both the \(sin(x)\) and \(cos(x)\) functions.

- Last Updated: Sep 22, 2023 2:04 PM
- URL: https://libraryguides.centennialcollege.ca/mathhelp
- Print Page

chat loading...