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Chapter 5: Section 5.4 - Applications of Trigonometric Functions

Sometime you will use trigonometric functions to represent periodic applications; such as, the tide depths, sunrise, sunset, ferris wheels, wheels and anything that is periodical (cyclical). Here the period will be in integers (without ). The functions will be written in the form:

Notice that the b value is now written as . This means that when you find the period using the formula

we’ve already learned , that the period is the denominator of the b value if it is written in the

this form with in the numerator. The period of these functions will be rational numbers and the units will be related to the question. As long as you have in the formula, the period will be in rational numbers

Examples: Find the period of the following functions:

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To graph these functions with a calculator see the following example:

Practice Examples: Graph these on the grid provided using your graphing calculator:

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With application questions, you may be asked to evaluate the function for a particular value.

For example: The height of an object in metres is given by the equation where time is in seconds.

a) At what height with the object be after 2.5 seconds?

b) When is the first time that the object reaches a height of 16.3 metres?

c) How long is the object above 15.9 metres?

Steps for Graphing these functions without the Graphing Calculator:

1. Determine the amplitude, vertical displacement, period and phase shift2. Adjust the scale on the x-axis according to the period – the scale is in integers3. Adjust the scale on the y-axis according to the amplitude and vertical displacement (use the Max and

Min)4. Determine your new midline on the graph.5. Determine your starting point on the graph from the sine or cosine and phaseshift and vertical

displacement6. Divide the period by 4 and graph the 4 sections of the curve ending at the phaseshift plus the period

considering your max and min.

Examples: Graph the following functions.

a)

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Practice Examples:Graph the following on a separate sheet of graph paper:

More Application Problems:

The classic Ferris Wheel problem

a)

b) cosine equation

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Alternatively, you can write a sine equation adjusting the phaseshift:

c) t = 52 seconds. Substitute this into your equation and solve for the height

d) Graph your equation in and a horizontal line where and find the intercepts

Practice Example:

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Answer:

Other periodic application questions:

a) cosine equation

Alternatively, sine equation

b)

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c) Using a calendar, October 4th is the 277th day of the year.

d) graph the function and the line y=23 and find the 2 intersection points; subtract the last intersection day and first intersection day to get the total number days.

a) cosine equation

Alternatively, sine equation

b) c) Using a calendar, April 6th is the 96th day of the year.

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d) graph the function and the line y=20 and find the 2 intersection points; add the first intersection of days with the difference between 365 and the last intersection to get the total number days then divide by 365 and convert to percentage.

Practice Example:

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Answer:

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