Trigonometric functions define the angles in a triangle in terms of the relationship of two sides in that triangle. These tutorials cover the angle functions, with step-by-step instructions for working them out (and the opportunity to use your own numbers).

Atmosphere Applet: This program lets you study how the properties of the atmosphere change with altitude. You can study the atmosphere of either the Earth or Mars. The equations used in this program are taken from the ICAO standard day model for the Earth and from some curve fits of the Martian atmosphere gathered by the Global Surveyor spacecraft. Using the airplane graphic you can select an altitude, or you can type an altitude into the input box.

The program instantly outputs a selected property and displays the local temperature and pressure on gauges You can output the temperature, pressure, density, local speed of sound, Mach number for specified velocity, or the ratio of aircraft lift to the lift on Earth at sea level. Input and output can be given in either English or metric units.

The textbook "Calculus" by Gilbert Strang, is a modern calculus text written in a human-friendly style. Published in 1991 and still in print from Wellesley-Cambridge Press, the book is a useful resource for educators and self-learners alike. It is well organized, covers single variable and multivariable calculus in depth, and is rich with applications. There is also an online Instructor's Manual and a student Study Guide.

This digital textbook was reviewed for its alignment with California content standards.

The University of California, Irvine Extension, supported by generous grants from the William and Flora Hewlett Foundation and The Boeing Company, is developing online courses to prepare science and mathematics teachers for the California Subject Examinations for Teachers (CSET).

UC Irvine Extension's online test-preparation courses correspond with the 10 CSET science subtests and three CSET mathematics subtests.

Accuracy of measurement in navigation depends very much on the situation. If a sailor's target is an island 200 km wide, sailing off center by 10 or 20 km is not a major problem. But, if the island were only 1 km wide, it would be missed if off just the smallest bit. Many of the measurements made while navigating involve angles, and a small error in the angle can translate to a much larger error in position when traveling long distances.

Accuracy of measurement in navigation depends very much on the situation. If a sailor's target is an island 200 km wide, sailing off center by 10 or 20 km is not a major problem. But, if the island were only 1 km wide, it would be missed if off just the smallest bit. Many of the measurements made while navigating involve angles, and a small error in the angle can translate to a much larger error in position when traveling long distances.

Students find and calculate the angle that light is transmitted through a holographic diffraction grating using trigonometry. After finding this angle, student teams design and build their own spectrographs, researching and designing a ground- or space-based mission using their creation. At project end, teams present their findings to the class, as if they were making an engineering conference presentation. Student must have completed the associated Building a Fancy Spectrograph activity before attempting this activity. This activity is best completed over four 60-minutes sessions.

This exercise has students determine how fast a dinosaur was moving based on the tracks it made. It allows students with minimal quantitative background to become motivated and begin to develop an appreciation for dimensional analysis as they see whether or not they could outrun the track-making dinosaurs. Measurements from any dinosaur track site can be used in this activity. Learning goals, context for use, teaching tips, materials, assessment tips and related resources are provided.

EngineSim was developed at the NASA Glenn Research Center in an effort to foster hands-on, inquiry-based learning in science and math. EngineSim is a simulator that models the design and testing of jet engines. The program works in two modes: Design Mode or Tunnel Test Mode. In the Design Mode, you can change design variables including the flight conditions, the engine size, the inlet performance, the turbo machinery compressor and turbine performance, the combustors or burner performance, or the nozzle performance. For a turbofan engine design you can also vary the fan performance and the bypass ratio. When you have a design that you like, you can switch to the Tunnel Test Mode, where you can vary only the flight conditions (airspeed, altitude, and throttle setting).

RangeGames uses jet engines to present a variety of math and physics problems. RangeGames includes a "Play Mode," in which nothing is recorded; a "Learn Mode," in which you will get several tries at determining the correct answer; and an "Exam Mode," where you get one chance and your answer is recorded for your teacher.

Rate and force problems of different levels of difficulty are included. These are targeted for the high school student. The rate problems deal with how far and how long an airplane can fly on a given amount of fuel. The force problems deal with Newton's laws of motion on takeoff. Graphical feedback is given for each answer choice.

FoilSim II computes the theoretical lift of a variety of airfoil shapes. The user can control the shape, size, and inclination of the airfoil and the atmospheric conditions in which the airfoil is flying. The program includes a stall model for the airfoil, a model of the Martian atmosphere, and the ability to specify a variety of fluids for lift comparisons. The program has graphical and numerical output, including an interactive probe which you can use to investigate the details of flow around an airfoil.
Using the CurveBall applet, students learn more about aerodynamics by controlling conditions of a big league baseball pitch, including altitude(location), speed of pitch, and spin of pitch. The latest version has the ability to specify atmospheric conditions for the baseball game.

This is a series of computer animations which demonstrate all the possible combinations of the ideal gas law or equation of state. Gases have various properties which we can observe with our senses, including the gas pressure, temperature, mass, and the volume which contains the gas. Careful, scientific observation has determined that these variables are related to one another and the values of these properties determine the state of the gas. In a scientific manner, we can fix any two of the four primary properties and study the nature of the relationship between the other two by varying one and observing the variation of the other. The variations are demonstrated using computer graphics in the animated gas lab.

The main goal of Geometry is for students to develop a Euclidean geometric structure and apply the resulting theorems and formulas to address meaningful problems. Students will use experimentation and inductive reasoning to construct geometric concepts, discover geometric relationships, and formulate conjectures. Students will employ deductive logic to prove theorems and justify conclusions. Students will extend their pre-existing experiences with algebra and geometry to trigonometry, coordinate geometry, and probability. Students will use GeoGebra, compass and straightedge applications, Voice Threads, and other tools to investigate and explore mathematical ideas and relationships and develop multiple strategies for analyzing complex situations. Students will apply mathematical skills and make meaningful connections to lifeŐs experiences.

The main goal of Geometry is for students to develop a Euclidean geometric structure and apply the resulting theorems and formulas to address meaningful problems. Students will use experimentation and inductive reasoning to construct geometric concepts, discover geometric relationships, and formulate conjectures. Students will employ deductive logic to prove theorems and justify conclusions. Students will extend their pre-existing experiences with algebra and geometry to trigonometry, coordinate geometry, and probability. Students will use GeoGebra, compass and straightedge applications, Voice Threads, and other tools to investigate and explore mathematical ideas and relationships and develop multiple strategies for analyzing complex situations. Students will apply mathematical skills and make meaningful connections to lifeŐs experiences.

In this lesson, students will investigate error. As shown in earlier activities from navigation lessons 1 through 3, without an understanding of how errors can affect your position, you cannot navigate well. Introducing accuracy and precision will develop these concepts further. Also, students will learn how computers can help in navigation. Often, the calculations needed to navigate accurately are time consuming and complex. By using the power of computers to do calculations and repetitive tasks, one can quickly see how changing parameters likes angles and distances and introducing errors will affect their overall result.

ZeGenie is a On-Line Interactive Learning Engine designed to help the student through Math material as if with a real human tutor.

All courses are available to students free of charge.

Math materials cover:

Algebra

1. Algebraic Expressions
2. Exponents
3. Quadratic Functions
4. Linear Relations
5. The Pythagorean Theorem
6. The function Basics
7. Functions
8. Absolute Function
9. Square Root Function
10. Step Functions
11. Exponential & Logarithms
12. Factoring
13. System of Equations
14. Conics

Geometry

1. Basics of Geometry
2. Lines, Division Points, Line Equations
3. Triangles and Similarity
4. Circle Theorems
5. Isometries

Trigonometry

1. Introduction to Trigonometry
2. Basic Trigonometric Ratios
3. Law of Sines
4. Law of Cosines
5. Radians
6. Unit Circle
7. Sine Function
8. Cosine Function
9. Tangent Function
10. Trigonometric Identities
11. Solving Trigonometric Functions

KiteModeler was developed in an effort to foster hands-on, inquiry-based learning in science and math. KiteModeler is a simulator that models the design, trimming, and flight of a kite. The program works in three modes: Design Mode, Trim Mode, or Flight Mode. In the Design Mode (shown below), you pick from five basic types of kite designs. You can then change design variables including the length and width of various sections of the kite. You can also select different materials for each component. When you have a design that you like, you switch to the Trim Mode where you set the length of the bridle string and tail and the location of the knot attaching the bridle to the control line. Based on your inputs, the program computes the center of gravity and pressure, the magnitude of the aerodynamic forces and the weight, and determines the stability of your kite. With a stable kite design, you are ready for Flight Mode. In Flight Mode you set the wind speed and the length of control line. The program then computes the sag of the line caused by the weight of the string and the height and distance that your kite would fly. Using all three modes, you can investigate how a kite flies, and the factors that affect its performance.

This general math site offers reference material on a host of math topics, plus a math message board and links to relevant material online. The tables cover a range of math skills, from basic fraction-decimal conversion to the more advanced calculus and discrete math. The information is presented in notation form, with diagrams, graphs, and tables. The site is available in English, Spanish, and French. This resource is part of the Teaching Quantitative Skills in the Geosciences collection. http://serc.carleton.edu/quantskills/

This is an assessment framework, not a curriculum framework. In broad terms, this framework attempts to answer the question: What mathematics should be assessed in 2009 on NAEP at grades 4, 8 and 12? The answer to this question must necessarily take into account the constraints of a large-scale assessment such as NAEP, with its limitations on time and resources. Of critical importance is the fact that this document does not attempt to answer the question: What mathematics should be taught (or how)? The framework was developed with the understanding that some concepts, skills, and activities in school mathematics are not suitable to be assessed on NAEP, though they may well be important components of a school curriculum. An example would be an extended project that involves gathering data, or a group project.

In this lesson, students will learn that math is important in navigation and engineering. Ancient land and sea navigators started with the most basic of navigation equations (Speed x Time = Distance). Today, navigational satellites use equations that take into account the relative effects of space and time. However, even these high-tech wonders cannot be built without pure and simple math concepts basic geometry and trigonometry that have been used for thousands of years. In this lesson, these basic concepts are discussed and illustrated in the associated activities.