List of trigonometric identities
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity. Notation[edit] Angles[edit] 1 full circle = 360 degrees = 2 radians = 400 grads. The following table shows the conversions for some common angles: Unless otherwise specified, all angles in this article are assumed to be in radians, but angles ending in a degree symbol (°) are in degrees. Trigonometric functions[edit] The primary trigonometric functions are the sine and cosine of an angle. The Sine of an angle is defined in the context of a Right Triangle, as the ratio of the length of the side that is opposite to the angle, divided by the length of the longest side of the triangle (the Hypotenuse ). Inverse functions[edit] and Related identities[edit]

Amplitude and Period
Amplitude and Period Learning Objective(s) · Understand amplitude and period. · Graph the sine function with changes in amplitude and period. · Graph the cosine function with changes in amplitude and period. · Match a sine or cosine function to its graph and vice versa. You know how to graph the functions and . or , where a and b are constants. We used the variable previously to show an angle in standard position, and we also referred to the sine and cosine functions as . for the input (as well as to label the horizontal axis). . You know that the graphs of the sine and cosine functions have a pattern of hills and valleys that repeat. . (or ) on the interval looks like the graph on the interval . The graph below shows four repetitions of a pattern of length . is on the interval is one cycle. You know from graphing quadratic functions of the form that as you changed the value of a you changed the “width” of the graph. and see how changes to b will affect the graph. periodic, and if so, what is the period?

Function Visualizer Model Documents
written by Wolfgang Christian and Francisco Esquembre The Function Visualizer Model displays the graph of a function f(x) with arbitrary parameters. The function can contain polynomial, trigonometric, and exponential expressions as well a parameters. This applet was created using the Easy Java Simulations (EJS) modeling tool. Download ejs_math_FunctionVisualizer.jar - 1108kb Java Archive File
Solving Trigonometric Equations
Solving Trigonometric Equations (page 1 of 2) Solving trig equations use both the reference angles you've memorized and a lot of the algebra you've learned. Be prepared to need to think! Solve sin(x) + 2 = 3 for 0° < x < 360° Just as with linear equations, I'll first isolate the variable-containing term: sin(x) + 2 = 3 sin(x) = 1 Now I'll use the reference angles I've memorized: x = 90° Solve tan2(x) + 3 = 0 for 0° < x < 360° There's the temptation to quickly recall that the tangent of 60° involves the square root of 3 and slap down an answer, but this equation doesn't actually have a solution: tan2(x) = –3 How can the square of a trig function evaluate to a negative number? no solution Solve on 0° < x < 360° To solve this, I need to do some simple factoring: Now that I've done the algebra, I can do the trig. x = 30°, 90°, 270°, 330° Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved Solve sin2(x) – sin(x) = 2 on 0° < x < 360° Only one of the factor solutions is sensible. x = 270° Hmm...

Exploring Multivariable Calculus
And then relaunch Chrome. Instructors: If you are an instructor using this project in any way, please send me an email to let me know of your interest. I would love to see more people using the materials from this project, and it is important that I be able to report how the project is doing to the NSF. I also encourage anyone who is interested in this project to take the time to write something on the discussion board. The following PDF documents give more information about the goals and current state of the project. 9-27-2011 Graph implicit equations in spherical or cylindrical coordinates like rho^2 = -sec(2ϕ) or r^2*theta^2 = r +theta.Implicit surfaces/equations can now be graphed in spherical and cylindrical coordinates as well as in cartesian coordinates. Future Presentations: Jan. 2014: I will present a 2-hour poster session on CalcPlot3D and several 15-minute sessions on the project at the 2014 JMM conference in Baltimore, MD. Past Presentations: June 29, 2011 March 17-19, 2011

Graphing Trigonometric Functions
Graphing Trigonometric Functions (page 1 of 3) Sections: Introduction, Examples with amplitude and vertical shift, Example with phase shift You've already learned the basic trig graphs. But just as you could make the basic quadratic x2, more complicated, such as –(x + 5)2 – 3, so also trig graphs can be made more complicated. Let's start with the basic sine function, f(t) = sin(t). Now let's look at g(t) = 3sin(t): Do you see that the graph is three times as tall? Now let's look at h(t) = sin(2t): Copyright © Elizabeth Stapel 2010 All Rights Reserved Do you see that the graph is squished in from the sides? For sines and cosines (and their reciprocals), the "regular" period is 2π, so the formula is: For tangents and cotangents, the "regular" period is π, so the formula is: In the sine wave graphed above, the value of B was 2. (Note: Different books use different letters to stand for the period formula. Now let's looks at j(t) = sin(t – π/3): Now let's look at k(t) = sin(t) + 3:

Math, Physics, and Engineering Applets
Oscillations and Waves Acoustics Signal Processing Electricity and Magnetism: Statics Electrodynamics Quantum Mechanics Linear Algebra Vector Calculus Thermodynamics Miscellaneous Licensing info. Links to other educational sites with math/physics-related information or java applets useful for teaching: And when you get tired of learning, here is some fun stuff: Java Pong Applet a cute little pong game I wrote a while ago.
4. Graphs of tan, cot, sec and csc
by M. Bourne The graphs of tanx, cotx, secx and cscx are not as common as the sine and cosine curves that we met earlier in this chapter. However, they do occur in engineering and science problems. They are interesting curves because they have discontinuities. [For more on this topic, go to Continuous and Discontinuous Functions in an earlier chapter.] Recall from Trigonometric Functions, that tanx is defined as: tanx=cos xsin x For some values of x, the function cosx has value 0. When this happens, we have 0 in the denominator of the fraction and this means the fraction is undefined. The same thing happens with cotx, secx and cscx for different values of x. The Graph of y = tan x Sketch y = tan x. Solution As we saw above, tanx=cosxsinx This means the function will have a discontinuity where cos x = 0. x=…,−25π,−23π,−2π,2π,23π,25π,… It is very important to keep these values in mind when sketching this graph. Graph of y = tan x: Interactive Tangent Curve π 2π t

Math-Blog: Mathematics is wonderful!
The Graphs of Sine and Cosine
A sine wave, or sinusoid, is the graph of the sine function in trigonometry. A sinusoid is the name given to any curve that can be written in the form (A and B are positive). Sinusoids are considered to be the general form of the sine function. In addition to mathematics, sinusoidal functions occur in other fields of study such as science and engineering. This function also occurs in nature as seen in ocean waves, sound waves and light waves. Let's start with an investigation of the simpler graphs of y = A sin(Bx) and y = A cos(Bx). The value A affects the amplitude. The value B is the number of cycles it completes in an interval of from 0 to or 360º. . and the graph will be a horizontal stretching. and the graph will be a horizontal shrinking. This problem is a combination of dealing with the values of A and B. (there will be a horizontal stretch). (or 720º). This problem is also a combination of dealing with the values of A and B. (there will be a horizontal shrink). (or 120º).