Trigonometry and Complex Numbers
Collection of Trigonometry and Complex Numbers worksheets.
Table of Contents
- Measuring Angles
- Converting Between Degrees and Radians
- Oriented and Coterminal Angles
- DMS → Decimal Angle Converter
- Decimal → DMS Converter
- Trigonometric Values and Applications
- Interactive Unit Circle - Exact Trig Values
- Graphs of Trigonometric Functions
- Trig Functions and Plane Transformations - Lesson+Exploration
- Triangles
- The Law of Sines
- Complex numbers
- Complex Numbers: Rectangular and Polar Forms
Converting Between Degrees and Radians
Use the sliders to explore angle measures and the conversion formulas between degrees and radians.
Practice Zone
An angle can be measured in radians or degrees.[br]Explain the relationship between these two measurements.
What is the measure in radians of a central angle of 90° in a circle with a radius of 2cm?[br]What about in a circle with a radius of 4cm?[br]What can you conclude?
True or false?[br]The measure of an acute angle in the unit circle is always between [math]0[/math] and [math]\frac{3}{2}\pi[/math].[br]Justify your answer, providing a counterexample if appropriate.
Interactive Unit Circle - Exact Trig Values
Drag the point to explore the position of the main angles in the unit circle, learn the exact values of their sine, cosine, tangent and cotangent, and convert their measure from degrees to radians.
Trig Functions and Plane Transformations - Lesson+Exploration
Explore the plane transformations applied to the sine and cosine functions.[br][br]How does the function graph modify if we apply some "classic" transformations (horizontal and vertical translations, dilations, composition with [math]abs[/math] function, etc) ? [br][br]Before viewing the resulting graph, try to determine the new position of some "convenient" points related to the transformed function (e.g. the intersections with the [i]x[/i] and [i]y[/i] axes), as well as the possible variations of the main characteristics of the function (amplitude, period...).[br][br]Move the sliders to explore dynamically the magnitudes of translations and dilations, and drag point [i]P[/i] along the graph of the parent function to view the graph of the transformed one.[br][br]Deselect the [i]Instructions [/i]checkbox to start exploring transformations.[br]A button displayed on bottom right of the app allows you to delete traces.
The Law of Sines
The [i]law of sines[/i] (or sine rule) relates the lengths of a triangle's sides to the sines of its angles.[br][br]The applet below lets you discover this useful law ([i]Theorem[/i]), learn how it is derived ([i]Proof[/i]) and interact with a dynamic triangle ([i]Explore![/i]) to deepen your understanding of this concept.
Solving a triangle means finding missing sides and angles - Let's solve one.
Solve the given triangle, with [math]a=24[/math], [math]\beta=60°[/math] and [math]\gamma=45°[/math]. [br]
To be or not to be....
Use the Law of Sines to show whether a triangle with:[br][math]a=18[/math], [math]b=36[/math] and [math]\alpha=45°[/math] does exist or not.[br][br]Explain you reasoning below.
Complex Numbers: Rectangular and Polar Forms
Numeric Exploration
Select a conversion type from the drop-down menu, then enter the required values in the input boxes displayed.[br]Press [i]Enter [/i]to confirm and generate the two algebraic representations of the number, along with its position on the Argand-Gauss plane.
Geometric Exploration
Select an algebraic conversion, then drag the complex number in the Argand-Gauss plane.
Practice Zone
Use the app above to represent the complex number [math]z=6+4i[/math].[br]Observe its rectangular and polar forms, then drag the point to represent the opposite of [math]z[/math].[br]How do the coefficients and the angle change?
Use the app above to represent a pure imaginary number.[br]What can you say about the coefficients in its rectangular form and the angle in its polar form?