Trigonometry

SOH-CAH-TOA

an acronym helping one remember the 3 basic ratios of sides in a right triangle. Sine:Opposite/hypotenuse Cosine: Adjacent/hypotenuse Tangent: Opposite/adjacent

opposite sides

do not share a vertex (across from)

Finding Domain

finding the domain for a rational function or a function in fraction form, omit numbers that make the denominator zero. So domain can usually be all real numbers except 0. radical functions often have restrictions in their domain. When this occurs, the expression in the radicand or under the radical sign must be greater than or equal to zero, since we can't take the square root of a negative number. when there are no denominators or radicals to worry about, the domain can usually be all real numbers. As any number that you put into the function as x will yield a real number answer.

periodic function

functions with graphs that repeat the same pattern over and over. This repeated pattern, or cycle, takes place for a specific period. The period of a function is defined as the length of the smallest continuous portion of the domain over which the function completes one cycle, or more simply, the length of one cycle.

graphical vector subtraction

graph the first vector with the initial point located at the origin. draw the next vector beginning at the terminal point of the first and have it going the opposite direction. the subtracted vector is shown by the beginning at the origin and ends at the terminal point of the second vector.

the line with slope 3/4 going through the point (6, 4).

m(x-x1)=y-y1 3/4(x-6)=y-4 And we can solve this for y and simplify to get the equation of the line in point-slope form. 3/4(x-6)=y-4 y=4+3/4(x-6) y=4+3/4x=3/4(6) y=4+3/4x-9/2 y=3/4x-1/2

what are the three most common ways to represent a function?

tables, graphs, and equations.

initial side of a vector

the circle part, the origin.

absolute value function

A type of piecewise function, where the graph of f(x) = |x| looks like the letter V. The piecewise function looks like this: f(x) = -x if x < 0 and f(x) = x if x ≥ 0 Notice how you use two different functions, pieced together, to represent the graph of the absolute value of x.

step function

A type of piecewise function, where the graphed intervals look like steps.

standard position of an angle

Angles that have their initial side on the positive x-axis and their vertex at the origin (0,0) are said to be in standard position.

Cosine

Cosine of an angle= ratio of adjacent leg and the hypotenuse.

triangular definitions

Cosθ= adjacent/hypotenuse Sinθ= opposite/hypotenuse Tanθ opposite/adjacent

Equation for a function

The actual equation for a function may be viewed as the most complete representation. An equation allows you to generate all the relationships between any x and y, not just the ones you can fit into a table or onto a graph. an example of a function equation: y=2x-1

The vertical line test

This is used to determine if a graph represents a function or a relation. If a vertical line is drawn through the graph, and it goes through more than one point, it is not considered a function. If a vertical line is drawn over the through the graph, and only goes through one point at a time, it is considered a function.

Graphing vectors

When taking about the ordered pairs of a vector use arrows instead of parenthesis. EX: {0, 5} In general a vector that begins at the origin and terminates at the point (x,y) can be expressed as the vector (x,y)

Pythagorean theorem can be used to find missing side lengths of right triangles.

a and b are the legs of the triangle and c is the hypotnuse: Think of it like: (leg)^2+ (leg)^2= (Hypotenuse)^2

function vs relation

a function is a type of relation. in a function each input may be paired with only one output.

right triangle

a triangle with one right angle, and measures 90 degrees. Right triangles have two legs and a hypotenuse.

y=a sin [b(x-h)]-k

a= amplitude 2π/b= period h= phase shift k= vertical displacement d=

Graphical Vector Addition

by adding one of the vector's initial point to the other's terminal point and measure the distance between the tow combines, that will be the added vector.

circular definition

cosθ= x/1=x sinθ= y/1=y tanθ= y/x Since secant, cosecant, and cotangent are reciprocals of cosine, sine, and tangent, their circular definitions follow directly from the above. sec θ= 1/cos θ=1/x csc θ= 1/sin θ=1/y cot θ 1/tan θ=x/y

product of functions

(f*g)(x)=f(x)*g(x) Example f(x)=x^2 g(x)=x f(x)*g(x) x^2*x x^3

sum of functions

(f+g)(x)=f(x)+g(x) example: f(x)=2x+6 g(x)=x-10 f(x)+g(x) 2x+6+x-10 3x-4

Difference of functions

(f-g)(x) = f(x) - g(x) Example f(x)=2x+6 g(x)=x-10 f(x)-g(x) 2x+6-(x-10) 2x+6-x+10 x+16

quotient of functions

(f/g)(x) is the same as f(x)/g(x) g(x) cannot equal 0. if g(x0 does equal 0, the 0 is the denominator. Example: f(x)=x^2 g(x)=x f(x)/g(x) x^2/x x where x doesn't equal 0

coterminal angles practice

1.Find one positive and one negative angle that are each coterminal with 65 degrees. To find the positive coterminal angle add 360: 65+360=425 degrees to find the negative coterminal angle subtract 360: 65-360=-295 degrees. 2. Find one positive and one negative coterminal angle to 475 degrees. since 475 is greater than 360. you can find a positive coterminal angle by subtracting 360: 475-360=115 degrees to find the negative coterminal angle, subtract 360 again: 115-360=-245 degrees. 3. what quadrant does the terminal side of 475 degrees fall in since 475 is coterminal 115, and 90<115<180, the terminal side falls in quadrant two.

Practice degrees minutes and seconds.

1.write 13 degrees, 19 degrees, 24 degrees using decimal degrees. 13 degrees, 19 minutes, 13 seconds= 13+19/60+24/3,600 13+0.3167+0.0067≈13.323 2. write 54.726 in degrees, minutes and seconds. 54.726=54+(0.726x60)' =54=43'+(0.56x60)" ≈54+43'+34 =54 degrees, 43', 34"

determine the domain and range of the following ordered pairs: {(1,2) (2,3) (3,4)}

Domain: {1, 2, 3} Range= {2, 3, 4}

Adding vectors

EX: vector d= {4,6} vector g= {-2,5} d+g {4,6}+{-2,5} {4+(-2), 6+5} {2,11}

subtracting vectors

EX: vector d= {4,6} vector g= {-2,5} d-g {4,6}-{-2,5} {4-(-2), 6-5} {6,1}

Abscicca

Each element from the left side of the table or the first number of each fired pair, is an abscissa, or x-value. The set of all abscissa is the domain.

ordinate

Each element from the right side of the table is an ordinate, or y-value.The set of all ordinates is the range.

is the following equation a function? what are the domain and range? y=x+2

Graph the equation and use the vertical line test. according to the test this equation is a function, and the domain and range or all real numbers, since there are no radicals or fractions to make restrictions.

clockwise rotation

If an angle is rotated and measured in the clockwise direction, then the angle is negative. how many degrees are in 4.4 clockwise rotation? 4.4x-360=-1,584 the angle measure is -1,54 degrees.

Parallel lines

Parallel lines have the same slope and their graphs will not intersect. Are these two lines parallel? y=-2/3x+5 and 2x+3y=12 2x+3y=12 3y=-2x+12 y=-2/3x+4 so yes these lines are parallel because the slopes are the same.

quadrantal angles

Quadrantal angles are angles with terminal sides that fall on one of the axes. Using the unit circle, we can find the sine and cosine of these angles. 0°, 90°, 180°, 270°, 360°

relation/function

These sets of information can be displayed in many ways: Ordered pairs, mappings, or graphs. A relation is any set of ordered pairs that relates an item (sometimes these are called elements) from one set ( called the domain) to an item from a second set ( called the range.) Sometimes a relation is also called a function. If a relation relates each element from the domain to only one element of the range, then the reaction is Laos a function.

Cotangent, secant, cosecant

Three other trigonometric ratios. they are reciprocals of Sine, cosine and tangent.. cosecant= opposite of sin secant= opposite of cos cotangent= opposite of tan

Finding the components of a vector

To find the horizontal (x) and vertical (y) components of a vector, subtract the coordinates of the vector's initial point from its terminal point. (x, y) = (xt - xi, yt - yi) xt = the x-coordinate of the terminal point xi = the x-coordinate of the initial point yt = the y-coordinate of the terminal point yi = the y-coordinate of the initial point EX: {x,y}={4-1, 4-2}= {3,2}

perpendicular lines

Two lines that have slopes that are negative reciprocals of each other are said to be perpendicular lines. Perpendicular lines create right angles when they cross each other.

quadrantal angle

an angle in standard position whose terminal side coincides with one of the axes

angle

an angle is formed by two rays that meet at a point called a vertex. The vertex is always a fixed point. One of the rays does not rotate and is called the initial side of the angle; the ray that rotates is called the terminal side of the angle.

central angle

an angle whose vertex is the center of the circle. can have the sine, cosine, and tangent values defined not only in terms of the legs of the triangle, but also in terms of the x- and y-coordinates of the point.

X intercepts and zeros

an x intercept is where the graph crosses the x-axis. Zero is another name for an x intercept. You find zeros of an equation by plugging in 0 for y and solving for x.

function notation

another way to write the equation is using function notation. When you use function notation, you replace the y with f(x). heres y-2x-1 written in function notation: f(x)-2x-1 With function notation, input values can be evaluated in the equation to generate a corresponding function value. Example: Evaluate f(2) for the function, f(x)=2x+4 so x=2 f(2)=2(2)+4 F(2)=8

adjacent sides

two sides sharing a common vertex (next to)

90º

π/2 rad (0,1) Cosine90º= 0 Sine90º=1

coinciding lines

same slope, same y-intercept

the trigonometric ratios

sine, cosine, tangent: Quantifies relationship between measure of each angle and lengths of sides.

Unit Cirlce

A unit circle is a tool to help find values of trigonometric functions. A unit circle has a radius, r, of 1, and the center of the circle is at the origin. Any angle in standard position intersects the circumference of the unit circle at a point. This point can then be used to draw a right triangle with a hypotenuse always equal to 1 (because the hypotenuse will always be a radius of the unit circle). The legs of the triangle are equal to the x, the adjacent side, and y, the opposite side, which also correspond to the coordinates of the point.

vector quantity

A vector gives both magnitude and direction. EX: 70 miles per hour north. Vectors can be represented by arrows or rays. The length of the arrow represents the magnitude of the vector. Therefore, the longer the arrow the larger the magnitude of the vector.

degrees

Degrees are one way to measure angles. A degree measure is based on revolutions around a circle. One revolution around a circle is equal to 360 degrees or 360°. One degree is equal to 60 minutes and one minute is equal to 60 seconds. Another way of writing this is: 1 degree=60' ('=minutes and "=seconds) 1 minute is 1/60 of a degree 1 second is 1/3,600 of a degree. Example: what is the degrees-minute-seconds equivalent of 60.425 degrees? 1. Multiply the decimal portion of the degree measure by 60 to find the number of minutes. 60.425° = 60° + (0.425 • 60)' = 60° 25.5' 2. Multiply the decimal portion of the resulting minute measure by 60 to find the number of seconds. Round answer to the nearest whole number. 60° 25.5' = 60° 25'+ (0.5 • 60)" = 60° 25' 30" The degrees-minutes-seconds equivalent of 60.425° is 60° 25' 30".

composition of functions

Functions are like big machines. We can input x-values to generate an output or function value. These inputs and outputs can then be displayed in a variety of ways: a table, ordered pairs, a mapping, or a graph. When you do a composition of two functions, you take the output from one function and use it as an input for the next function. The composition of a function is defined as: (f o g)(x)=f(g(x)) Composition of functions is often confused with multiplication of functions. Notice the notation for multiplication uses a closed dot while composition uses an open dot. Example: Find (f o g)(x)=f(g(x)) f(x)=4x+2 g(x)=x-5 use function,f, and substitute, the function,g, for each x in f. f(g(x))=f(x-5)=4(x-5)+2 =4x-20+2 =4x-18

zero slope

Horizontal lines have a slope of 0 because there is no rise.

counterclockwise rotation

If an angle is rotated and measured in the counterclockwise direction, then the angle is positive. how many degrees are in a 2.3 counterclockwise rotation?

Sine

Sine of an angle= ratio of opposite leg and hypotnuse.

Tangent

Tangent of an angle= Ratio of opposite leg and adjacent leg.

correlation coefficient

The correlation coefficient, r, is a number that represents the correlation of the data. The value of r can range from -1 up to 1. This is not the slope value. The closer the r-value is to 1, the more closely the correlation is to being a perfect, positive correlation, y = x. The closer the r-value is to -1, the more closely the correlation is to being a perfect, negative correlation, y = -x. An r-value of 0 shows no correlation. Example: a correlation coefficient of -0.3 would have a weak negative correlation.

terminal point of a vector

The endpoint of a vector, the arrow side.

Tables

a table is quick way to examine what is just a small part of a function. it's usually organized with the domain in the left column (x) and the range in the right column (y or f(x)). Tables are limited in the amount of information they can provide. They only represent a few of the data points from the function, and it does not clearly show the values of other data points within the function.

Pythagorean Theorem

a²+b²=c²

slope formula

(y₂- y₁) / (x₂- x₁) To find the slope (m) between two points. rise/run=change in y/change in x Example: (2,3) and (4,0) 0-3/4-2 -3/2 =3/2 is the slope.

cofunction

A function of an angle is equal to the cofunction of its complement. Additionally, recall the sum of two complementary angles is 90°. There are three pairs of cofunctions: sine and cosine secant and cosecant tangent and cotangent

graph

A graph is a visual representation of a function. It takes all the ordered pairs of the functions and plots them on a coordinate plane. usually there are so many ordered pairs that they form a solid line or curve. Graphs are useful because they give a "big picture" image of what the function looks like.

scalar quantity

A physical measurement that does not contain directional information. represents a situation by giving only its magnitude. EX: 70 miles per hour

piecewise functions

A piecewise function is one that has different equations for different intervals of the domain. There are different types of piecewise functions.

scatter plots

A scatter plot will show if there is a relationship between two variables. If there is a correlation, a relationship between two variables, then we can draw a line. The line that is drawn is called the line of best fit, or regression line. To find a line of best fit, you first draw a line to represent the general trend of the data, typically with the same number of data points above and below the line. Then follow the same procedure you used to determine the equation of a line in the form y = mx + b. Choose two points on the line you drew and label them as (x1, y1) and (x2, y2). You can then calculate the slope, or m, of the line. Next, find the point on the y-axis where the line would intersect the axis. This is your b-value. Substitute the slope and y-intercept into your slope-intercept formula and you have the equation of the line of best fit.

function iteration

Iteration means "repetition." Function iteration is when you repeatedly evaluate a function. You start with some initial value that is given and then the answer to each step tells you what to evaluate for the next step. For example, if f(x)=x^2+3 , find the first three iterations of f(x) starting with an initial value of x0=0 . x1=f(x0)=f(0)=0^2+3=3 x2=f(x1)=f(3)=3^2+3=12 x3=f(x2)=f(12)=12^2+3=147

linear function

Linear functions are a set of ordered pairs (x, y) with a constant rate of change. For each unit increase or decrease in the domain (x), there is always the same increase or decrease in the range (y). As a result of the constant rate of change, the graph of a linear function forms a straight line.

reciprocals and opposite reciprocals

One of the expression's denominator is the other expression's numerator (except with a sign switch) and vice versa. Fractions like these are reciprocals. we call the slopes of perpendicular lines opposite reciprocals. if one slope is a, the perpendicular slope is -1/a Here are some more examples: 1/2 and -2 3 and -1/3 -2/3 and 3/2

calculating magnitude

Recall that the magnitude of a vector is equivalent to its length. To calculate the magnitude of a vector graphed in the coordinate plane, use the distance formula. D=√(x2-x1)^2 + (y2-y1)^2 EX: {3,4} D=√(4-0)^2+(3-0)^2 D=√4^2+3^2 D=√16+9 D=√25 D=5 therefore the magnitude of this vector is 5 units.

how would you graph the function, f(x)=1/2|x|+1

Remember that absolute value is the positive distance a number or value is from 0. Therefore, any number that is substituted in for x will be positive. Multiply the absolute value of the number you substitute in for x by 1/2 ; then add 1 to that value, to determine the value for f(x).

is the following equation a function? what are the domain and range? y=x^2+2

The equations passed the vertical line test, making it a function. The graph uses all x-values so the domain is all real numbers. also the equation does not contain a fraction or radical, therefore there are no restrictions on the domain. the graph has y values larger than 2 (the lowest point of the parabola is at y=2) so range is y≥2.

Slope

The rate of change of a linear function is called its slope. The slope of a line is calculated by finding the change in i-values (rise) divided by the change in x-values (run). To find slope from a graph, pick two points and determine how far up or down divided by how far left or right. In the graph below, the slope is found by counting two units up divided by two units to the right. The graph has a slope of 1.

y-intercept

The y-intercept is where the line crosses (or intercepts) the y-axis. This is usually easy to find visually from a graph, if the intercept is a whole number. When given two pints lets use (2,3) and (4,0) again, find the y intercept. To find the y-intercept, b, choose one of the two points. For this example, choose (4, 0) because it is easier to work with an ordered pair where either the x- or y-value is 0. substitute x=4 and y=0 into the equation y=-3/2 and solve for b. *-3/2 is the slope we found earlier* 0=-3/2(4)+b 0=-6+b b=6 so the finished equation: y=-3/2x+6

The equation for the graph of a linear function

To write the equation of a line, two pieces of information are required. Slope (m) A point on the line (x, y) Linear equations are most often written in slope-intercept form. The equation can be written after identifying the slope and the y-intercept. The y-intercept, or b-value, is written as an ordered pair in the form (0, b). These dimensions can be found by looking at the graph, or using the slope formula and the slope-intercept of an equation. Again, after determining the slope and y-intercept, a linear equation can be written. slope intercept form: y=mx+b, where m-slope and b=y-intercept.

coterminal angles

two angles in standard position that have the same terminal side. How do you identify coterminal angles? Determine the number of complete rotations (k) by dividing the given angle A by 360° and then rounding down to the nearest whole number. Find the number of remaining degrees (α) by substituting k into α + 360°k = A and solving for α. This will give a positive angle (α) less than 360° that is coterminal with the given angle. Find a coterminal angle of a negative measure by subtracting 360° from α. Determine the quadrant in which the terminal side lies using the ranges for each quadrant. If 0° < α < 90°, then the terminal side lies in the first quadrant. If 90° < α < 180°, then the terminal side lies in the second quadrant. If 180° < α < 270°, then the terminal side lies in the third quadrant. If 270° < α < 360°, then the terminal side lies in the fourth quadrant.

undefined slope

vertical lines have undefined slope since there is no run.

point slope form

y-y1=m(x-x1) Another equation that is useful in generating linear equations is called the point-slope form. It is used when you know one point on the line and the slope. Given slope -2 and point (5, 2), find the linear equation of this line in point-slope form. y-2=-2(x-5) y-2=-2x+10 y=-2x+12

find the x-intercept of the following function: f(x)=3x-12

y=0/f(x)_=0 so... 0=3x-12 12=3x x=4 so the zero of the function is x=4

ordered pairs

yo usan represent the same information in a table using a set of ordered pairs. When we are defining a "set" we always use a curly bracket "{}" Each row of the table then becomes an ordered pair (x and y). Each element from the left side of the table or the first number of each fired pair, is an abscissa, or x-value. Each element from the right side of the table is an ordinate, or y-value. The set of all abscissa is the domain. The set of all ordinates is the range. since they are sets, we can describe them using a list inside of curly brackets: Domain: {-1, 0, 1, 2} Range: {-3,-1,1,3}

y intercepts

you can find y intercepts or initial value of an equation by plugging in zero for x and then solving for y.

180º

π rad (-1,0) Cosine180º=-1 Sine180º=0