How do trigonometric functions work? Please explain in detail. 25 POINTS!!!

The sign of a trigonometric function is dependent on the signs of the coordinates of the points on the terminal side of the angle. By knowing in which quadrant the terminal side of an angle lies, you also know the signs of all the trigonometric functions.

There are eight regions in which the terminal side of an angle may lie: in any of the four quadrants, or along the axes in either the positive or negative direction (the quadrantal angles). Each situation means something different for the signs of the trigonometric functions.

Terms

Domain

The set of all possible inputs of a function.  (x)

Function

An operation that assigns a correspondence from elements of one set to elements of another set.  (y)

Period

The repeating interval of a periodic function; the period of a function is a real number.

Periodic Function

A function that repeats itself in regular intervals; it follows this equation: f (x + c) = f (x), where c is a constant.

Range

The set of all possible outputs of a function.

Reference Angle

The positive acute angle formed between the terminal side of an angle and the x-axis.

Rule

The part of a function that dictates the exact correspondence between the elements of one set and the elements of another set.

Trigonometric Functions

There are six trigonometric functions: Sine, cosine, tangent, cosecant, secant, and cotangent.

Unit Circle

The circle whose center is at the origin and whose radius is one.

Formulas

CosecantGiven a point P(x, y) on the terminal side of an angle θ in standard position, distance d from the origin, cosecant(θ) = csc(θ) = .

CosineGiven a point P(x, y) on the terminal side of an angle θ in standard position, distance d from the origin, cosine(θ) = cos(θ) = .

CotangentGiven a point P(x, y) on the terminal side of an angle θ in standard position, cotangent(θ) = cot(θ) = .

SecantGiven a point P(x, y) on the terminal side of an angle θ in standard position, distance d from the origin, secant(θ) = sec(θ) = .

SineGiven a point P(x, y) on the terminal side of an angle θ in standard position, distance d from the origin, sine(θ) = sin(θ) = .

TangentGiven a point P(x, y) on the terminal side of an angle θ in standard position, tangent(θ) = tan(θ) = .

4.0 ft

step-by-step explanation:

4.0 ft

for the friend request.

when a figure, such as triangle (xyz) is reflected over the line y = x to create x'y'z' the x-coordinates and y-coordinates of the original image (pre-image) will be reversed for the image. (e.g., (x, y) > (y, x)) this means that the pre-image position of the x- and y-coordinates switch places which in turn provides a mirror like image of the original xyz.  

i notice that when i draw a line segment from x to x', that the points are parallel from each other -- they are exactly across from each other and do not intersect with any other points.  

yes, think i would see the same characteristic if you drew the line segment connecting y with the reflecting line and then y' with the reflecting line because of the essence of reflections. because it is a mirror image, each point was effected the same way which leads me to conclude that they same would apply to y and y'.