The Basic Trig Functions Software is a free tool for teaching core concepts of trigonometry, plus additional topics like the Mandelbrot Set. The relation and function graphing capability of our software also makes it a great tool for teaching algebra. The software is designed for interactive and dynamic classroom instruction and for teachers who want to create custom instructional content. Each software module presented below includes examples of instructional applications and user interface graphics for quick familiarization.

##### ▷ Demonstrate trigonometric functions on a circle

Module Features

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Graphically demonstrate the basic trigonometric functions (Sin, Cos, Tan) and arc length on a circle with a radius of user defined units.

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Manually increment and decrement through user specified angle steps in units of degrees, radians, or radian decimals. Angle steps can range from 1º to 90º (or radian equivalents). This feature allows teachers to demonstrate relationships between the basic trig functions in a step-by-step format.

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Easily copy graphics to your clipboard to create custom in-class or take-home content.

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Adjust colors to maximize visual impact.

This example shows various trig relationships for a positive second quadrant angle on a unit circle. Trig function relationships are displayed graphically on the circle with corresponding values in a color coded list.

This example shows various trig relationships for a negative second quadrant angle on a circle with a radius of four units. User defined angle increments or decrements can range from 1º to 90º (or radian equivalents).

This example shows the relationship between radius of a circle (10 units in this example), arc length, and radian measure of an angle. This is a good example to show students because many beginning students do not really understand radian measure expressions such as π/6 or -3π/2.

This example shows the angle mode can be set to radian exact form and angles can be incremented or decremented without limit.

This example shows that the angle mode can be set to radian decimal format. Sometimes, students forget that all exact radian angle measurements have decimal approximations.

This example shows the relationship between the radian measure of an angle, arc length, and the radius of a circle when the angle format is radian decimal.

This graphic in one example of various trig diagrams provided in this module that teachers can use to create handouts.

This graphic is a screen shot of the user interface window for the trig functions on a circle module. Layout is simple and intuitive. Controls for selecting radius, angle increment, angle mode, and circle arc are located on the bottom of the window. Users manually increment or decrement through angle values to demonstrate basic trig functions in a step-by-step format. Options for text color are accessed in the Edit Menu.

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##### ▷ Demonstrate the basic trigonometric function equations

Module Features

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Graphically demonstrate the basic trigonometric equations individually or superimposed.

Sin: y = Sin(x) ● y = 2Sin(x) ● y = Sin(2x) ● y = Sin(x/2)

Cos: y = Cos(x) ● y = 2Cos(x) ● y = Cos(2x) ● y = Cos(x/2)

Tan: y = Tan(x) ● y = 2Tan(x) ● y = Tan(2x) ● y = Tan(x/2)

Cot: y = Cot(x) ● y = 2Cot(x) ● y = Cot(2x) ● y = Cot(x/2)

Sec: y = Sec(x) ● y = 2Sec(x) ● y = Sec(2x) ● y = Sec(x/2)

Csc: y = Csc(x) ● y = 2Csc(x) ● y = Csc(2x) ● y = Csc(x/2)

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Manually increment and decrement through user specified angle ranges in units of degrees, radians, or radian decimals. This feature allows teachers to demonstrate relationships between the basic trig function equations in a step-by-step format.

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Easily copy graphics to your clipboard to create custom in-class or take-home content.

This example shows the relationship between following trig equations:

y = Sin(x) for 0 ≤ θ ≤ 360º ● y = Cos(x) for 0 ≤ θ ≤ 90º

As the user increments or decrements the angle, the value of the trig equation for the current angle is displayed under the graph as shown above.

This example shows the relationship between following trig equations:

y = 2Cos(x) for 0 ≤ θ ≤ 360º ● y = Cos(x) for 0 ≤ θ ≤ 360º

This example give a geometric description of the Tan(θ) function in terms of the unit circle. It becomes clear why Tan(θ) is undefined at θ = 90º or π/2 radians.

This example gives a geometric description of the Csc(θ) function in terms of the unit circle and demonstrates the Pythagorean identify 1 + Cot²(θ) = Csc²(θ).

This example shows the relationship between following trig equations:

y = Csc(x) for -2π ≤ θ ≤ 2π ● y = Sin(x) for -2π ≤ θ ≤ 2π

This example shows the relationship between following trig equations:

y = Sin(x) for -360º ≤ θ ≤ 360º ● y = Sin(2x) for -360º ≤ θ ≤ 360º

This graphic is a screen shot of the user interface window for the trig equations graphing module. Layout is simple and intuitive. Controls for selecting equations, angle mode, angle range, y-axis scale, and special features are located on the bottom of the window. Users manually increment or decrement through angle values to demonstrate the relationships between the various basic trig equations in a step-by-step format.

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##### ▷ Graph x-y relations and solve one variable equations

Module Features

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Simultaneously graph up to six implicitly or explicitly user defined x-y variable relations. If one side of an equation is linear, the program attempts to transform the equation into an explicit function of one variable by solving for a variable in the linear part. This makes it easy for users to apply equation transformation rules in a consistent manner, and the software can use a faster algorithm to graph the equation.

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After the graph is drawn, the user can plot open or closed circle points on the graph(s) and/or line segments anywhere in the plot area to groom the graph or demonstrate concepts. In addition, relative minimum and maximum points, x-intercepts, a trace mark, and intersections of the graphs can be displayed with simple mouse controls.

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Use this module to demonstrate and explore the following topics plus much more:

● Odd, even, and one-to-one functions

● General relations

● Inverse of a relation

● Equation of a line

● Linear or exponential growth and decay modeling

● Linear and rational polynomial inequalities

● Inequalities involving the absolute function

● Trigonometric and algebraic identities

● Piecewise defined functions with or without discontinuities

● Equation transformation rules

● Limit and continuity of a function at a point

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Easily copy graph outputs to your clipboard to create custom in-class or take-home content.

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Adjust colors, line weights, plot densities, scan resolution, axis range and scale, and all output labeling to maximize visual impact. Leave the background of the plot area blank or fill it with lattice points or grid lines.

This example shows the intersection of the following linear inequalities:

x + y ≤ 10 ● 3x + y ≥ -12 ● -2x + 3y ≥ -15 ● y < 6.5

Notice that y < 6.5 is automatically drawn with a dashed line.

This example shows the following trig function equations:

y = 2Cos(x) ● y = 2Sec(x) ● Cos(x) = 0

When an x-variable equation such as Cos(x) = 1/2 or x³ + 2x² = 27x + 54 is graphed, all solutions from xmin to xmax are displayed in a list to the right of the graph. Solutions are formatted as rational multiples of π if the domain parameters are rational multiples of π.

This example shows the relationship between the following trig equations:

inverse of y = Sin (x) ● y = ArcSin(x)

Color and line weight for each graph was adjusted to maximize contrast and effectively demonstrate the relationship between the equations.

This example shows the geometric relationship between the graph of an equation (y = X² + 2 in this example) and the graph of its inverse. After plotting the equations, corresponding point pairs (e.g. (0,2) and (2,0)) were manually drawn, then connected with line segments to demonstrate that the segments are perpendicular to the graph of the equation y = x.

This example shows an odd function (y = xCos(x) in this example) and its symmetry about the origin. After plotting the equation, corresponding point pairs (e.g. (-3,3) and (3,-3)) were manually drawn, then connected with line segments to demonstrate that the segments pass through the origin.

This example shows the following implicitly defined relations:

Sin(xy) = 0.5 ● xy = π/6 ● xy = 5π/6

Notice that the relation Sin(xy) = 0.5 is a family of hyperbolas or inverse variation graphs.

This example shows the following functions with removable discontinuities:

y = Sin(3x)/x ● y = 0.5(x³ -8)/(x-2)-4

This example shows the following function with jump discontinuities:

y = Floor(x)

Use the "draw dot graph" option to graph functions such as Floor(x), Ceil(x), Round(x), and Int(x). To finish the graph, open and closed circle points were manually drawn using simple mouse controls that are summarized in the Mouse Help menu.

This example shows how to graph a piecewise defined function with jump discontinuities:

y = (2)(x < -4) ● y = (x - 1)(-4 ≤ x ≤ 2) ● y = (0.5(x - 5)² - 6)(x > 2)

Notice that each part of the function is the product of a math expression and domain description which uses standard math inequality format and syntax. To finish the graph, open and closed circle points were manually drawn using simple mouse controls that are summarized in the Mouse Help menu.

This example shows how to apply equation transformation rules in a consistent manner. Students can begin to understand that most of the graphs they will encounter are the result of applying a series of simple equation transformation rules, in a specific order, to a basic equation.

1: y = 2√(x) is the original equation in this example

2: y = 2√(-x) reflects the graph over the y-axis

3: y = 2√(-(x-5)) slides the graph horizontally 5 units in the positive direction

4: -y = 2√(-(x-5)) reflects the graph over the x-axis

5: -(y + 4) = -2√(-(x-5)) slides the graph vertically 4 units in the negative direction

This graphic is a screen shot of the user interface window for the x-y equation and relation graphing module. Layout is simple and intuitive. Sample equations and key press equivalents are provided for the user to explore and become familiar with the interface quickly. Additional graph output options, such as line weight and color, are accessed in the Edit Menu of the software's main window.

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##### ▷ Graph polar functions

Module Features

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Simultaneously graph up to six explicitly user defined polar functions of angle θ or radius r.

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If the polar equation is of the form r = f(θ), users can use the mouse to plot polar points on the graph of polar equation 1. After a polar point is graphed, the program displays the polar function output value r and shows which polar point, (r, θ) or (-r, θ), was plotted.

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After plotting a polar graph, add a trace mark to the graph with simple mouse controls. As the trace mark is manually incremented or decremented along the graph, corresponding values of angle θ and radius r are displayed in real time to the right of the graph.

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Easily copy graph outputs to your clipboard to create custom in-class or take-home content.

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Adjust colors, line weights, radius values, angle plot steps, angle range, and display options for polar circles and rays to best suit the purpose. The polar axes can also be toggled on or off to provide more control over the graphic output.

This example shows the graph of the following polar function:

θ = ± 2π/5

This example shows that a polar equation can have one ± symbol. Explicit polar functions of r must be of the form θ = k, where the constant k represents the measure of an angle in radians. The line weight of the graph was set to 2.

This example shows the graph of the following polar function:

r = θ

This graph is the famous Spiral of Archimedes when θ ranges from 0 to 4π radians. Notice how polar function output is graphed when θ = 11π/6 radians. The line weight of the graph was set to 2.

This example shows the graph of the following polar function:

r = √(-100Sin(2θ))

This is the graph of a lemniscate when θ ranges from 0 to 360º. Notice how polar function output is handled when θ = 225º. It is interesting to see how output of polar plot points is handled for different values of θ. The line weight of the graph was set to 1.

This example shows the graph of the following polar function:

r = 9Sin(19θ/20)

This is the graph of a spider web when θ ranges from 0 to 40π radians with a plot step of π/4 radians. The polar axes are turned off to isolate the graphic. The line weight of the graph was set to 1.

This example shows the graph of the following polar function:

r = e^Cos(θ) - 2Cos(4θ) + Sin(θ/4)³

This is the graph of a butterfly when θ ranges from 0 to 8π radians with a plot step of π/90 radians. The line weight of the graph was set to 1.

Rollover the animation for viewing controls.

This loop animation demonstrates the trace mark feature on the following polar function:

r = 8Sin(5θ), a five petal rose

Users can add a trace mark to the graph with simple mouse controls, then manually increment or decrement the mark. The value of angle θ and radius r for each point are displayed in real time to the right of the graph as shown above. The rate of advancement or decrement of the trace mark can be increased by holding down the <a> and <b> keys, respectively. The trace mark radius can be toggled on or off. This is a great tool for visually demonstrating how polar equations are graphed and instantaneous rate of change to students.

This graphic is a screen shot of the user interface window for the polar function graphing module, which gives users complete control over the appearance of the polar graph. Layout is simple and intuitive. Notice that the symbol for θ is entered by pressing the "x" or "X" key. Polar equations are fun and interesting to graph. Users are encouraged to explore and experiment with the traditional polar equations found in trig text books. Additional graph output options, such as line weight and color, are accessed in the Edit Menu of the software's main window.

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##### ▷ Create trigonometric circle diagrams

Module Features

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Create trigonometric circle diagrams by selecting Pythagorean triples (e.g. 3 - 4 - 5 ) or polar points (r, θ) to define one or two unique points on the terminal side of a standard trigonometric angle in quadrant I, II, III, or IV. Swap x-y coordinates or apply scaling factors to the terminal points to create a wider variety of circle diagrams.

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Choose color and labeling options for x-y axes intercepts, wrapping function arcs, angles, and x-y points to best suit the purpose.

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Diagrams are automatically copied to your clip board for easy creation of custom in-class or take-home content.

In this example, Pythagorean triples (12-35-37 and 33-56-65) were chosen to define the terminal points for two angles in quadrants II and IV. From this diagram, students could be asked find radii of circles, trig function values, values of angles α and β in degrees, and arc lengths λ and ω.

Similar to the previous example, Pythagorean triples (7-24-25 and 13-84-85) were chosen to define the terminal points for two angles in quadrants I and IV. Scale factors of 1.75 and 0.75 were applied to the triples, respectively, to create custom terminal points for the diagram.

In this example, polar points ((10, 0.8 radians) and (10, 1.4 radians) with scaling factors applied) were chosen to define the terminal points for two angles in quadrants II and III. If the teacher provides the angles and radii, students could be asked to find the coordinates of (x1, y1) and (x2, y2), trig function values, values of angles α and β in degrees, and arc lengths λ and ω.

In this example, the teacher could inform students that α = 70°, β = -230°, and provide the radius of each circle. Students could be asked to find the coordinates of (x1, y1) and (x2, y2), trig function values, values of angles α and β in radians, and arc lengths of λ and ω.

This graphic is a screen shot of the user interface window for the trigonometric circle diagram module. Layout is simple and intuitive. Angle measure, radius, and arc length values are displayed in this window for the selected terminal point(s). Output options for text color and size are accessed in the Edit Menu of the software's main window.

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##### ▷ Graph powers of a complex number

Module Features

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Graph the power of a complex number c to reveal important and interesting properties of complex numbers.

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Enter a complex number c in standard a + bi form or polar form, then output the list of powers of c in a + bi form or polar form for a maximum of 20 powers.

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Choose labeling and display options for x-y axes and intercepts, powers, radii of powers, and lists of powers of c to best suit the purpose.

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Graphs and output lists are easily copied to your clipboard for creation of custom in-class or take-home content.

This example shows the graph of the first twelve powers of c = 1 + i. Values of the powers are provided in an output list as shown above. The output list can be in either a + bi form (shown above) or rCiS(θ) form. Colors for the graph and output list can be changed to suit the purpose. Using this example, teachers could ask students to calculate the first six powers of c = 1 + i prior to presenting this topic in class.

This example shows the graph of the first twenty powers of c = 1.15{Cos(40) + iSin(40)] , which could be used as a test question. Teachers could give students the graph of the powers and have students calculate four or five specific powers of c = 1.15[Cos(40) + iSin(40)] in polar and/or a + bi form. The output list shown above is in rCiS(θ) form.

This example shows the graph of the first twenty powers of c = 0.9{Cos(50) + iSin(50)] , which could be used as a test question. Teachers could give students the graph of the powers and have students calculate four or five specific powers of c = 0.9[Cos(50) + iSin(50)] in polar and/or a + bi form.

This example shows the graph of the first five powers of c = √3 - i, which could be used as a test question. Teachers could give students the graph of the powers and have students calculate the five powers of c = √3 - i in polar and a + bi form. If the polar angle is expressed as a negative angle, the pattern becomes very clear.

This example shows the graph of the first twenty powers of c = √2/2 - √2/2i, which could be used as a test question. Teachers could give students the graph of the powers and have students calculate the first four or five powers of c = √2/2 - √2/2i in polar and/or a + bi form.

This example shows the graph of the first eight powers of c = 1.2 + 0i = -1.2, which could be used to show the geometry of the powers of a negative real number.

This graphic is a screen shot of the user interface window for the powers of a complex number module. Layout is simple and intuitive. Additional color options for graph labeling and the output list are accessed in the Edit Menu of the software's main window.

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##### ▷ Graph roots of a complex number

Module Features

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Graph the N roots of a complex number c.

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Enter a complex number c in standard a + bi form or polar form, then graph the roots on a circle for a maximum of 20 roots.

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Choose labeling and display options for x-y axes and intercepts, powers, radii of roots, and lists of roots to best suit the purpose.

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Graphs and output lists are easily copied to your clipboard for creation of custom in-class or take-home content.

This example shows the graph of the ten roots of x¹⁰ = 1,024i, entered in polar form as c = 1,024[Cos(90) + iSin(90)]. Root values are provided in an output list as shown above. The output list can be in either a + bi form or rCiS(θ) form (shown above). Colors for the graph and output list can be changed to suit the purpose. Teachers could use this example in a classroom presentation. How are the polar angles of the roots calculated? How is the radius of a root calculated? The teacher could ask the students to find the first four roots in a + bi form. If students have a graphing calculator, they could check to see if these roots are really the roots of x¹⁰ = 1,024i.

This example shows the graph of the five roots of x⁵ = -32, entered in polar form as c = 32[Cos(180) + iSin(180)]. Root labels on the graph can be omitted or presented in a + bi form, rCiS(θ) form (shown above), or as radii (r1, r2, r3, etc). Teachers could use this example in a classroom presentation, asking questions similar to the previous example.

This example shows the graph of the five roots of x⁵ = -32, entered in polar form as c = 32[Cos(180) + iSin(180)]. Root labels on the graph can be omitted or presented in a + bi form (shown above), rCiS(θ) form, or as radii (r1, r2, r3, etc). Teachers could use this example in a classroom presentation, asking questions similar to the first example.

This example shows the graph of the six roots of x⁶ = -64i, entered in polar form as c = 64[Cos(270) + iSin(270)]. Root labels on the graph are omitted. Teachers could use this example as an exercise by showing only the graph of the roots and ask students to find the six roots in polar and/or a + bi form.

This example shows the graph of the eight roots of x⁸ = 100 - 100i, entered in a + bi form as c = 100 - 100i. Root labels on the graph are omitted. Similar to the previous example, teachers could show only the graph of the roots and ask students to find the eight roots in polar and/or a + bi form.

This example shows the graph of the five roots of x⁵ = -16 + 16√3i, entered in a + bi form as c = -16 + 27.71281292i. Root labels on the graph are omitted. Similar to the previous example, teachers could show only the graph of the roots and ask students to find the five roots in polar and/or a + bi form.

This graphic is a screen shot of the user interface window for the roots of a complex number module. Layout is simple and intuitive. Additional color options for graph labeling and the output list are accessed in the Edit Menu of the software's main window.

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##### ▷ Explore the Mandelbrot Set

Module Features

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Explore the famous Mandelbrot Set, which most mathematicians consider to be most complex object in all of mathematics.

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Test whether or not the orbit of a complex number c, in the form a + bi, is included in the Mandelbrot Set after 10 million iterations.

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Draw the Mandelbrot orbit of a + bi and the Julia Set in user specified draw iterations, and display a list of the orbit values.

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Click inside the Mandelbrot Set to view the orbit of a + bi, or click and drag a rectangle to zoom-in on the region.

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Graphics and output lists are easily copied to your clipboard for creation of custom in-class or take-home content.

This example shows orbit values of -0.43 + 53i for the orbit range of 620-640. At iteration 620, the orbit becomes periodic with a period of 5. The Julia Set of -0.43 +53i, displayed in the upper left corner of the graphic, appears to be connected.

This example shows orbit values of 0.3. At iteration 12, the orbit starts to escape to infinity. Iteration values for the orbit range are displayed in list format as shown above.

This example shows orbit values of -1.0. At iteration 3, the orbit of -1.0 becomes periodic with a period of 2. Users should experiment with orbit values of -2.0, -1.5, and -0.5.

This example shows orbit values of -0.71 + 0.25i. At iteration 39, the orbit starts to escape to infinity. The Julia Set of -0.71 + 0.25i appears to be connected.

This example shows orbit values of -0.36+ 0.6334i for the orbit range of 1,146 - 1,166. At iteration 1,164, the orbit starts to escape to infinity. The Julia Set of -0.36+ 0.6334i appears to be connected.

This example is the magnified region around around the complex number -0.36 + 0.6334i from the previous example. Users can zoom-in on any region of the Mandelbrot Set by clicking and dragging a rectangle.

This graphic is a screen shot of the user interface window for the Mandelbrot Set module. Layout is simple and intuitive. Interesting orbits are provided as a starting point for exploration. Before the orbit values are displayed, the user can test the range of orbit values for inclusion in the Mandelbrot Set before they are displayed.

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Version 3.8

Released 5/1/16

(1.54 MB file size)

Download the Basic Trig Functions Software installation package by clicking on the icon to the right, and install the software on your computer or external hard drive by executing the installer package. This software is FREE - no strings attached. We are excited to see how creative math teachers everywhere implement the software to generate custom instructional content and present mathematical concepts in their classrooms. Note that the software is licensed for educational purposes only. Please review the License Agreement before downloading and using the software.

- The software requires Microsoft® Vista, 7, 8, or 10 operating system. Sorry, Mac® OS is not supported at this time.
- The software license does not expire.
- Use with confidence; our software contains no advertising, malware, spyware, or other subversive code.
- The software installation package can be downloaded anytime and without restriction on number of downloads.
- Technical support is provided for questions regarding use and functionality of the software.

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We hope that you thoroughly enjoy using our software. If you find the Basic Trig Functions Software useful, consider making a donation. We are constantly working to improve its functionality and usability while developing new software. We are 100% supported by donations, so your support goes further than you may realize. Thanks so much!

Release Notes

Version 3.8 fixes a minor bug.

✓ Release Notes

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