Trigonometric+Functions+-+B3

= __//**Trigonometric Functions**//__ =

**Radians versus Degrees**
 * Radians and degrees are both are a standard unit of measurement when expressing angles. There are some key differences that can help you to identify whether an angle measurement is being expressed in degrees or radians. First of all, degrees are written out in the form //degrees-minutes-seconds.// Degrees are expressed using the ° symbol, minutes using the ’ symbol, and seconds using the ” symbol. Radians on the other hand are expressed using terms of pi.**
 * To convert from radians to degrees use the following formula:**


 * Radian x 180 °/pi = equivalent degree measurement **


 * To convert from degrees to radians use the following formula:**


 * Degree x pi/180 ° = equivalent radian measurement **


 * Khan explains this in completion with examples: **

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**Sine, Cosine, and Tangent in relation to right triangles**

On a right triangle, the longest side of the triangle that is opposite of the 90 degree angle is call the hypotenuse. For the other two sides, in context with angle A, the two sides are labeled opposite and adjacent. The opposite side is the side the angle is across from and not touching. The adjacent side is the side angle A is touching that is not the hypotenuse. The ratios of sine, cosine and tangent come from the sides of the right triangle. The ratios come from the lengths of the 2 sides that are being used on the triangle. Put easily, the ratios are:

**Domain and Range: Sine and Cosine**

The domain and range of sine, cosine, and tangent come from when they are graphed.

Sine and Cosine graphs both have a domain of any real number and have a range of -1 to 1 exclusively.

The tangent graph has a domain of all real numbers except ± π/2+πk where k is any integer, and has a range of all real numbers.

**Basic Trig Identities**

As a refresher of the basic trigonometric identities, here is a list: (Identities are proven for all values A)

Reciprocal Identity: Quotient Identity:
 * sinA= || 1/cscA ||  || cosA= || 1/secA ||   || tanA= || 1/cotA ||
 * cscA= || 1/sinA ||  || secA= || 1/cosA ||   || cotA= || 1/tanA ||

tanA=sinA/cosA or sinA=tanAcosA cotA=cosA/sinA or cosA=cotAsinA

Pythagorean Identity:

sin^2 A + cos^2 A = 1 sec^2 A - tan^2 A = 1 csc^2 A - cot^2 A = 1

**Cosecant, Secant, and Cotangent**

Cosecant(csc), Secant(sec) and Cotangent(cot) are reciprocals of the Sine, Cosine, and Tangent functions. (For all real numbers A)
 * =  ||= Is equal to: ||=   ||
 * = Cosecant ||=  ||= 1/sinA ||
 * = Secant ||=  ||= 1/cosA ||
 * = Cotangent ||=  ||= 1/tanA ||

**The Unit Circle** The unit circle is a circle with the radius of one, it is frequently used in trigonometry. The trigonometric functions cosine and sine may be defined on the unit circle as follows.If (x, y) is a point of the unit circle, and if the ray from the origin (0, 0) to (x, y) makes an angle t from the positive x-axis,(where counterclockwise turning is positive), then

The equation x2 + y2 = 1 gives the relation ** The unit circle also demonstrates that sine and cosine are periodic functions, with the identities ** for any integer k. **Reference Angles and Determining the Values of Trig operations**

If you know that sin(pi/4) = root 2/2 then finding the value of sin(3pi/4) and sin(5pi/4) is simple. Each value is still equal to root 2/2 because each unit is divided by 4. However, depending on which quadrant those units fall in the value could be negative or positive. For example, sin(3pi/4) = root 2/2 but sin(5pi/4) = negative root2/2. This pattern works for every trig value. If you know that the value of cos(pi/6) = root 3/2, then you know the value of any unit divided by 6, depending on whether you know if its negative or positive. There's a pattern to determine which values are negative and which are positive. For sin operations, all values less than 180 degrees are positive and those greater are negative. For cos operations, all values on the right side of the y axis are positive and those on the left are negative. Lastly, for tan operations, all values in the first and third quadrant are positive and those in the second and fourth are negative. The same pattern works for the cot, csc and sec.

**Ways to Remember Trig values**


 * To memorize all the trig values for common angles, you need only remember what is in the first quadrant. After that, the X and Y values are translated across the X and Y axises.



To remember the Y values, simply remember your 1 2 3's. All of the sine and cosine values in this quadrant are divisible by two. The numerator of the sine values are just counting by radicals increasing by one each time (the square root of 1 is 1). Remembering your Trig values is as easy as 1, 2, 3! (~Taz)
 * Another trick to remember is that in the parentheses, the X values are the cos values and the Y values are the sin values. ~Ethan


 * ** An easy way to remember what values are positive and which ones are negative on the unit circle is to know the acronym **** ASTC or All Students Take Calculus. **** In the first quadrant, All of the trigonometric values are positive (sin, cos, tan, csc, sec, cot). In the second quadrant only the Sine values are positive (sin, csc) and the everything else is negative. In the the third quadrant only the Tangent values are positive, and in the fourth quadrant only the Cosine values are positive. So, All Sine Tangent Cosine = ASTC = All Students Take Calculus.~Dan **

[|__http://etc.usf.edu/clipart/43200/43215/unit-circle7_43215_lg.gif__] [|__http://www.algebralab.org/lessons/lesson.aspx?file=trigonometry_trigsincostan.xml__] [|__http://ssepkowitz.pbworks.com/w/page/9000806/Geometry%20-%20Sine,%20Cosine,%20and%20Tangent__] [|__http://literacy.purduecal.edu/student/mrrieste/quadrant1.png__] [|__http://www.dummies.com/how-to/content/how-to-graph-sine-cosine-and-tangent.html__]
 * Bibliography: **

http://www.khanacademy.org/math/trigonometry/v/radians-and-degrees