WELCOMEWELCOME

EF 105 Fall 2006

EF 105Computer Methods in

Engineering Problem Solving

Week07: Trig Review and ChartsUse of EXCEL

Learning ObjectivesLearn more about functionsLearn how to use Trigonometry functionsLearn to use tables and graphs as problem solving toolsLearn and apply different types of graphs and scalesPrepare graphs in ExcelBe able to edit graphs

Use Excel’s functions

Functions TAKE argumentsFunctions RETURN valuesYou can easily calculate the sum, average, count, etc. of a large number of cells by using a function. A function is a predefined, or built-in, formula for a commonly used calculation. Each Excel function has a name and syntax. The syntax specifies the order in which you must

enter the different parts of the function and the location in which you must insert commas, parentheses, and other punctuation

Arguments are numbers, text, or cell references used by the function to calculate a value

Some arguments are optional

1 2( , ,.. )ny f x x x

Work with the Insert Function button

Excel supplies more than 350 functions organized into 10 categories: Database, Date and Time, Engineering,

Financial, Information, Logical, Lookup, Math, Text and Data, and Statistical functions

You can use the Insert Function button on the Formula bar to select from a list of functions. A series of dialog boxes will assist you in filling in the arguments of the function and this process also enforces the use of proper syntax.

Anatomy of Excel Functions

=FUNCTION(argument1,argument2,..,argumentN,…)

optionalMandatory 1..N-1Name

Define functions, and functions within functions

The SUM function is a very commonly used math function in Excel. A basic formula example to add up a small number of cells is =A1+A2+A3+A4, but that method would be cumbersome if there were 100 cells to add up. Use Excel's SUM function to total the values in a range of cells like this: SUM(A1:A100).You can also use functions within functions. Consider the expression =ROUND(AVERAGE(A1:A100),1). This expression would first compute the average of all

the values from cell A1 through A100 and then round that result to 1 digit to the right of the decimal point

Open the Insert Function dialog box

To get help from Excel to insert a function, first click the cell in which you wish to insert the function. Click the Insert Function button. This action will open the Insert Function dialog box. If you do not see the Insert Function button, you may need to select the appropriate toolbar or add the button to an existing toolbar.

Examine the Insert Function dialog box

This dialog box appears when you click the Insert Function button. It can assist you in defining your function.

Use the Insert Function dialog box to enter function arguments

This figure depicts how you would enter argument values for the PMT function using the Insert Function dialog box.

Recognize optional argumentsIn the preceding figure, note how rate and nper are arguments for each function.For some of the functions, the final two arguments of each function are in brackets. These represent optional arguments, meaning if you do not enter anything, the default values for these arguments will be used. For example, note the PMT function has fv and type as its final two arguments, which are optional. The assumed values, if no others are supplied, are 0 for both

Arguments without brackets do not have default values, so you must supply values or cell references in order for the function to be able to return a value.

Create logical functions

A function that determines whether a condition is true or false is called a logical function. Excel supports several logical functions such as AND, FALSE, IF, NOT, OR and TRUE. A very common function is the IF function, which uses a logical test to determine whether an expression is true or false, and then returns one value if true or another value if false. The logical test is constructed using a comparison operator that compares two expressions to determine if they are equal, not equal, if one is greater than the other, and so forth. The comparison operators are =, >, >=, <, <=, and <>

You can also make comparisons with text strings. You must enclose text strings within quotation marks.

Using the If function

The arguments for the IF function are: IF(logical_test,value_if_true,value_if_false) For example, the function =IF(A1=10,20,30) tests

whether the value in cell A1 is equal to 10 If it is, the function returns the value 20,

otherwise the function returns the value 30 Cell A1 could be empty or contain anything else

besides the value 10 and the logical test would be false; therefore, the function returns the value 30

To insert an IF function, click the Insert Function button and search for the IF function, then click OK. When the Function Arguments dialog box appears, simply fill in the arguments.

The TODAY and Now functionsThe TODAY and NOW functions always display the current date and time.You will not normally see the time portion unless you have formatted the cell to display it.If you use the TODAY or NOW function in a cell, the date in the cell is updated to reflect the current date and time of your computer each time you open the workbook.

Let’s open your saved workbooks from last class and add a logical and a date function!

Use a formula to enter the dateIf you wanted a fixed date to remain in a cell , you would enter that date. If you wanted the date in this cell to always reflect the current date and time when you opened the workbook, you would use the expression =NOW() or =TODAY() as shown in the formula bar in the figure.

TRIGONOMETRY FUNCTIONS

When solving trigonometric expressions like sine, cosine and tangent, it is very important to realize that Excel uses radians, not degrees to perform these calculations! If the angle is in degrees you must first convert it to radians. There are two easy ways to do this.

1.Recall that = 180°. Therefore, if the angle is in degrees, multiply it by /180° to convert it to radians. With Excel, this conversion can be written PI( )/180. For example, to convert 45° to radians, the Excel expression would be 45*PI( )/180 which equals 0.7854 radians. 2.Excel has a built-in function known as RADIANS(angle) where angle is the angle in degrees you wish to convert to radians. For example, the Excel expression used to convert 270° to radians would be RADIANS(270) which equals 4.712389 radians

TRIGONOMETRY FUNCTIONS

You can use the DEGREES(angle) function to convert radians into degrees. For example, DEGREES(PI( ) ) equals 180.

Excel uses several built-in trig functions. Those that you will use most often are displayed in the table below. Note that the arguments for the SIN( ), COS( ) and TAN( ) functions are, by default, radians. Also, the functions ASIN( ), ACOS( ) and ATAN( ) return values in terms of radians. (When working with degrees, you will need to properly use the DEGREES( ) and RADIANS( ) functions to convert to the correct unit.)

TRIGONOMETRY FUNCTIONSMathematical

Expression Excel

Expression Excel Examples

sine: sin() SIN(number) SIN(30) equals -0.98803, the sine of 30 radians

SIN(RADIANS(30)) equals 0.5, the sine of 30°

cosine: cos() COS(number) COS(1.5) equals 0.07074, the cosine of 1.5 radians

COS(RADIANS(1.5)) equals 0.99966, the sine of 1.5°

tangent: tan() TAN(number) TAN(2) equals -2.18504, the tangent of 2 radians

TAN(RADIANS(2)) equals 0.03492, the tangent of 2°

arcsine: sin-1(x) ASIN(number) ASIN(0.5) equals 0.523599 radians

DEGREES(ASIN(0.5)) equals 30°, the arcsine of 0.5

arccos: cos-1(x) ACOS(number) ACOS(-0.5) equals 2.09440 radians

DEGREES(ACOS(-0.5)) equals 120°, the arccosine of -0.5

arctangent: tan-

1(x) ATAN(number) ATAN(1) equals 0.785398 radians

DEGREES(ATAN(1)) equals 45°, the arctangent of 1

TRIGONOMETRY FUNCTIONS: Example 1Say, for instance, we want to know the height of the tree in the figure . We know that if we stand 76 m from the base of the tree (x = 76 m) the line of sight to the top of the tree is 32° with respect to the horizon ( = 32°). We know that

Solving for the height of the tree, h, we find

(Make these on a “TRIG” sheet of your workbook.)

TRIGONOMETRY FUNCTIONS: Example 2

In this next example, we wish to know the launch angle, , of the water ski ramp shown. We are given that A = 3.5 m, B = 10.2 m and = 45.0°. To find , we can use the Law of Sines which, in this case can be written

We can rewrite this equation as using the equation

The screen shot below shows how we used Excel to determine that the launch angle of the ramp is 14.04°.

TRIGONOMETRY FUNCTIONS: Example #3

In our final trigonometry example, we will use Excel to examine the trig identity

Notice in the screen shot below that this identity holds true when is given in radians and degrees. Note the units for the angle are placed in different cells than the numbers. If we place the numbers and the units in the same cell, Excel will not be able to decipher the number and therefore we will not be able to reference the cells for use in any equation!

Proper Use of Tables & Graphs

Engineers record and present data in two primary formats: Tables and Graphs

Height H (m)

Temp T (C)

Pressure P (kPa)

0 15.0 101.3 300 12.8 97.7 600 11.1 94.2

0

2

4

6

8

10

12

14

16

0 500 1000 1500Height (m)

Tem

pera

ture

(C)

0

20

40

60

80

100

120

140

160

Pre

ssur

e (k

Pa)

Temp

Pressure

(Make these on a “TCG” sheet of your workbook.)

Tables

Tables should always have: Title Column headings with brief descriptive name,

symbol and appropriate units.Numerical data in the table should be written to the proper number of significant digits. The decimal points in a column should be

aligned. Tables should always be referenced and discussed (at least briefly) in the body of the text of the document containing the table.

Table Example

Temperature and Pressure ofWidgets at Various Heights

Height H (m)

Temp T (C)

Pressure P (kPa)

0 15.0 101.3 300 12.8 97.7 600 11.1 94.2

ExerciseEnter the following table in Excel (Label a sheet in the workbook you’ve been using in class.)

You can make your tables look nice by formatting text and borders

Independent Variable, x

Dependent Variable, y1

Dependent Variable, y2

1 1 1 2500 10 50 5000 100 100 7500 1000 150 10000 10000 200

Graphs

Proper graphing of data involves several steps: Select appropriate graph type Select scale and gradation of axes, and

completely label axes Plot data points, then plot or fit curves Add titles, notes, and or legend

Graphs - Types

Travel Expenses

02468

1012141618

Food

Gas

Mot

el

Expe

nses

($)

1. Pie Chart 2. Bar Graph

Graphs - Types

4. Line Graph

0

2

4

6

8

10

12

14

16

0 500 1000 1500Height (m)

Tem

pera

ture

(C)

0

20

40

60

80

100

120

140

160

Pre

ssur

e (k

Pa)

Temp

Pressure

1234

5

6

7

0123456

78

02468101214

Object 1Object 2

Bod

y Te

mpe

ratu

re (0 C

Speed (m/s) Dist

ance

(m)

3. 3-D Graph

Graphs

Each graph must include: A descriptive title which provides a clear and

concise statement of the information being presented

A legend defining point symbols or line types used for curves needs to be included

Labeled axes Graphs should always be referenced/discussed in the body of the text of the document containing the table.

Titles and Legends

Each graph must be identified with a descriptive titleThe title should include clear and concise statement of the information being presentedA legend defining point symbols or line types used for curves needs to be included

Axis Labels

Each axis must be labeledThe axis label should contain the name of the variable and its units. The units can be enclosed in parentheses, or separated from the label by a comma.

Length (km)

Gradation

Scale gradations should be selected so that the smallest division of the axis is an integer power of 10 times 1, 2, or 5. Exception is units of time.

Scale Graduations,Smallest Division=1

Acceptable

Scale Graduations,Smallest Division=3.33

Not Acceptable

Data Points and Curves

Data Points are plotted using symbols The symbol size must be large enough to

easily distinguish them A different symbol is used for each data setData Points are often connected with lines A different line style is often used for each

data set

Example

Velocity of Three RunnersDuring a 5 km Race

Building a Graph In ExcelSelect the data that you want to include in the chart by dragging through it with the mouse.Then click the Chart Wizard

Choose XY (Scatter), with data connected by lines if desired.Click “Next”

Building the Graph

Building the Graph

Make sure that the series is listed in columns, since your data is presented in columns.Click the Series tab to enter a name for the data set, if desired.Choose “Next”

Building the GraphFill in Title and Axis information“Next”

Building a ChartSelect “As new sheet” to create the chart on it’s own sheet in your Excel file, or “As object in” to create the chart on an existing sheet“Finish”

Creating a Secondary Axis

This is useful when the data sets cover very different ranges.Right click on the line (data series) on the chart that you want to associate with a secondary axis.Select “format data series”Select the Axis tab, then “Plot series on secondary axis” as shown.“OK”

Editing/Adding Labels

Now you can go back to the “chart options” to add labels Click the chart in a blank area, then either right click and

select chart options or choose chart options from the “Chart” menu

Fill in or edit the axes labels, title, etc.Click “OK”

Result

A Baseball Problem

A runner is on 3rd base, 90 ft from home plate. He can run with an average speed of 27 ft/s. A ball is hit to the center fielder who catches it 310 ft from home plate. The center fielder can throw the ball no faster than 110 ft/s. The runner tags up and runs for home plate.

Can the center fielder throw him out? To do so, he must get the ball to the catcher at an appropriate height before the runner can get to home plate.

If so, at what angle and what velocity does he need to throw the ball in order to put the runner out?

(Make these on a “Baseball” sheet of your workbook.)

Graphic Translation

Runner

90 ft

tvttr

sft27)(

Center Fielder

310 ft

jgttVty

20 2

1)sin()(

itVtx

)cos()( 0

V0

j

i

Solving with Excel-Iteration Method

Open an Excel spreadsheet and create column heads like the example.Rows 1 - 6 are for constants. Remember to use the $ notation when reference absolute address

Solutions - Building a Table

Rows 7 and above can be used to calculate the x and y positions at different times t using the formula for projectile motion. For example, under x(t) in Cell B8 enter the formula:

= $B$4*cos($C$2)*A8What formula would be entered for y(t) (height)and r(t) (runner position)?Is there an easy way to enter the values for time beginning in Cell A8?

Solution - iterations

Notice how changing Cell B2 effects the rest of the spreadsheet, especially x(t) and y(t) columns.

By watching the results in those columns, you can get arbitrarily close to 310 and 0. Also Cell B4 can be changed for even finer tuning.

Solution - Using a Chart

Another way to solve this problem is with a graph. This method will use the data generated on the previous slides but will use a chart to show the result.The next slide shows a completed chart. Notice that the line shows the ball position reaches 310 ft before the runner has traveled 90 ft.

Building a Chart (Step 1)

Select the data that you want to include in the chart by dragging through it with the mouse.

Building a Chart (Step 2) click the chart wizard.

Choose XY (Scatter) Then choose “Next”

Building a Chart (Step 3)

Building a Chart (Step 4)

Make sure that the series is listed in columns.Choose “Next”

Building a Chart (Step 5)

Fill in Title and Axis information

“Next”

Building a Chart (Step 6)

Choose “As new sheet”, then “Finish”

Building a Chart (Step 7)

Creating a Secondary axis. Right click on the data

series that you want to associate with a secondary axis.

Right click and choose “format data series.

Select “Plot series on secondary axis”

Building a Chart (Step 8)

Select “Chart”, then “Chart Options”Fill in the title for secondary value (Y) axis.Click “OK”This should complete the chart.

Using Solver

Select and copy the first 8 rows of the first 4 columns of the spread sheet.Remember that Row 8 contains the formulas for calculating the x, y and r positions.

Using Solver

Select another worksheet from the bottom of the spreadsheetRight click on its label and rename if desired.Select Cell A1Paste.

Using Solver

Pull down Tools, then select solver.

Set Target CellDesired ValueManipulated

Cells Constraints

Select Solve

Using Solver

Solver arrives at a solution that is within the constraints. = 28.31 degreesV0 = 110 ft/st = 3.20 seconds.The ball is at home plate two feet off the ground while the runner is still 3.58 feet away.

Helpful Hint

Note that any cell can be assigned a name. This can be done by first clicking on the cell (say B4) and then typing the name in the name box (above the column A label). This can be very useful when that cell is used as an absolute address.

Helpful HintThe name can then be used when typing formulas. This creates a formula that looks more like the actual equation making it easier to type and to verify. In this example the cell names, Vo_solver, theta_solver, and time were used instead of $B$4, $C$2, $A$8.

Next,

1. Do the Tutorial, Part 2on your own

2. Solve the following on your own: 

A. What is the length of horizontal base of the triangle? (cm)B. What is the area of this triangle? (sq. cm)