Unit One Concepts
Parabolas
The equation of a vertical parabola in standard form is (x - h)^2 = 4p (y - k), where the focus is (h, k + p) and the directrix is y = k - p.
We know it is a vertical parabola because x in the equation is squared. If the p in the equation is positive it will face upwards, and if the p in the equation is negative it will face downwards.
The equation of a horizontal parabola is (y - k)^2 = 4p (x - h), where the focus is (h + p, k) and the directrix is x = h - p. The y is squared, therefore it is a horizontal parabola. Also, if the p is positive it will open up to the right and if it is negative it will open to the left.
We know it is a vertical parabola because x in the equation is squared. If the p in the equation is positive it will face upwards, and if the p in the equation is negative it will face downwards.
The equation of a horizontal parabola is (y - k)^2 = 4p (x - h), where the focus is (h + p, k) and the directrix is x = h - p. The y is squared, therefore it is a horizontal parabola. Also, if the p is positive it will open up to the right and if it is negative it will open to the left.
Ellipses
The equation for a horizontal ellipse is (x-h)^2 / a^2 + (y-k)2/b^2 =1.
For a vertical ellipse the equation is (y-k)^2/a^2+(x-h)^2/b^2=1
If the y comes first the ellipse is vertical. IF the x comes first the ellipse is horizontal.
For a vertical ellipse the equation is (y-k)^2/a^2+(x-h)^2/b^2=1
If the y comes first the ellipse is vertical. IF the x comes first the ellipse is horizontal.
Hyperbola
The equation for a vertical hyperbola is (y-k)^2/a^2-(x-h)/b^2=1.
The equation for a horizontal ellipse is (x-h)^2/a^2 - (y-k)/b^2= 1.
If the y comes first the hyperbola is vertical. IF the x comes first the hyperbola is horizontal.
The equation for a horizontal ellipse is (x-h)^2/a^2 - (y-k)/b^2= 1.
If the y comes first the hyperbola is vertical. IF the x comes first the hyperbola is horizontal.
Unit 2
Unit Circle
Sine and Cosine functions
Equation-
y=Asin(BX-C)+D
A= Amplitude
B is the coefficient of x
Period=2pi/B
(BX-C)=shift to the right (BX+C)=Shift to the left by c/b units
D-Vertical Shift
X count=(1/4)(period)
An amplitude of greater than one causes a vertical stretch
When the a is negative it causes a reflection over the x axis
The sequence of y values for sine graphs is this- 0,1,0,-1,0. It starts at 0 and then ends at 0. After you write the original sequence multiply by the amplitude and add or subtract the D if there is one.
Y=Acos(BX-C)+D
The y sequence for cosine graphs goes like this- 1,0,-1,0,1. It starts at one and ends at one. From the original y sequence you do the same as a sine graph by multiplying by the amplitude and adding or subtracting the D.
y=Asin(BX-C)+D
A= Amplitude
B is the coefficient of x
Period=2pi/B
(BX-C)=shift to the right (BX+C)=Shift to the left by c/b units
D-Vertical Shift
X count=(1/4)(period)
An amplitude of greater than one causes a vertical stretch
When the a is negative it causes a reflection over the x axis
The sequence of y values for sine graphs is this- 0,1,0,-1,0. It starts at 0 and then ends at 0. After you write the original sequence multiply by the amplitude and add or subtract the D if there is one.
Y=Acos(BX-C)+D
The y sequence for cosine graphs goes like this- 1,0,-1,0,1. It starts at one and ends at one. From the original y sequence you do the same as a sine graph by multiplying by the amplitude and adding or subtracting the D.
Secant and Cosecant Functions
Equation- Y=Asec(BX-C)+D
All of the variable values mean the same thing as they do in the previous equations. You set up the x values exactly the same as you would for the functions before. Secant is the inverse of cosine, therefore when you take the original Y values of cosine you find the inverse of them. Instead of 1,0,-1,-0,1 it becomes 1,undefined,-1,undefined,1. Once you have done this, multiply them by the amplitude and add or subtract the d just like you would for sin and cosine functions. They become undefined because you cannot divide a number by zero. Where you have the undefined values on the chart is where you will set up the asymptotes. Once you have done this you graph the points to look like this-
All of the variable values mean the same thing as they do in the previous equations. You set up the x values exactly the same as you would for the functions before. Secant is the inverse of cosine, therefore when you take the original Y values of cosine you find the inverse of them. Instead of 1,0,-1,-0,1 it becomes 1,undefined,-1,undefined,1. Once you have done this, multiply them by the amplitude and add or subtract the d just like you would for sin and cosine functions. They become undefined because you cannot divide a number by zero. Where you have the undefined values on the chart is where you will set up the asymptotes. Once you have done this you graph the points to look like this-
Equation- Y=Acsc(BX-C)+D This equation goes almost identical to the secant graph. The only difference is the y values. When you take the y values remember that co-secant is the inverse of sine. Therefore you would use the parent function of sine to find the inverses to graph cosecant. Instead of 0,1,0,-1,0, it becomes, undefined,1,undefined,-1,undefined. You set up the asymptotes just like with secant, and your graphing should look like this.
Tangent and co-tangent Graphs.
Equation-Y=Atan(BX-C)+ D
For tangent you will still apply the amplitude,phase shift, and vertical shift the same. However to calculate the period the equation is Pi/B instead of 2Pi/B. Then multiply by 1/4 to get the x count just like the others. The y sequence for tangent graphs are as follows 0,1,undefined,-1,0. Just like with the previous graphs, where there is an undefined value you will set up an asymptote. After you set up the asymptote and plot your points, your graph should look like this.
For tangent you will still apply the amplitude,phase shift, and vertical shift the same. However to calculate the period the equation is Pi/B instead of 2Pi/B. Then multiply by 1/4 to get the x count just like the others. The y sequence for tangent graphs are as follows 0,1,undefined,-1,0. Just like with the previous graphs, where there is an undefined value you will set up an asymptote. After you set up the asymptote and plot your points, your graph should look like this.
Equation - Y=Acotan(BX-C)+D
Everything is the same except when you find the y values you will find the inverse of the tangent graph because the value of cotan is the inverse of tan. The y sequence will go like this, Undefined,1,0,-1,undefined. Set up the asymptotes like before and plot the points. Just like always once you have found the inverse y values multiply by the amplitude and add or subtract the D. When you plot your points it should look like this-
Everything is the same except when you find the y values you will find the inverse of the tangent graph because the value of cotan is the inverse of tan. The y sequence will go like this, Undefined,1,0,-1,undefined. Set up the asymptotes like before and plot the points. Just like always once you have found the inverse y values multiply by the amplitude and add or subtract the D. When you plot your points it should look like this-
Unit 3
Trigonometric Values and their Inverses
Rules For the values of sine,cosine, and tangent for Pi-X,Pi+X, and 2Pi-X in terms of X-
- Sine pi-x=sinx, Sinepi+x= -sineX, Sine 2pi-x= -sine x
-Cos pi-x= -cosx, Cospi+x= -cosX, Cos2pi-x=cosx
-Tanpi-x= -tanX, Tan pi+x=tanx, Tan2pi-x= -tanx
- Sine pi-x=sinx, Sinepi+x= -sineX, Sine 2pi-x= -sine x
-Cos pi-x= -cosx, Cospi+x= -cosX, Cos2pi-x=cosx
-Tanpi-x= -tanX, Tan pi+x=tanx, Tan2pi-x= -tanx
Inverse Functions
Check The Videos Tab under Unit 3 for detailed explanation on inverse trigonometric functions.
Law of Sine and Cosine
These two laws are formulas used to find unknown information about triangles that do not contain a 90 degree right angle, so long that the information needed to complete the equation is given in the triangle the formula is being applied to.
In these triangles AB & C represents the various angles of the triangle.
ab and c represent the sides of the triangle. The letter of each side is applied to the side opposite of the given angle.
Ex the side on a triangle opposite of angle A would be named side a.
Law of Sine- a/sin A = b/sin B = c/sin C
Law of Cosine- c2 = a2 + b2 − 2ab cos(C)
The lettering of the law of cosine can be switched around to find unknown information about a triangle depending on what information is already given on the triangle.
In these triangles AB & C represents the various angles of the triangle.
ab and c represent the sides of the triangle. The letter of each side is applied to the side opposite of the given angle.
Ex the side on a triangle opposite of angle A would be named side a.
Law of Sine- a/sin A = b/sin B = c/sin C
Law of Cosine- c2 = a2 + b2 − 2ab cos(C)
The lettering of the law of cosine can be switched around to find unknown information about a triangle depending on what information is already given on the triangle.
Area of a Non Right Triangle Using Trig Formula
The formula for a non-right triangle is 1/2(side one)(side two)sine(included angle)
To use the formula the triangle must possess 2 sides and an angle in the order of side angle side; sort of like the congruency statements we learned in trigonometry. One you get these three things, plug them in in the right places using the formula.
To use the formula the triangle must possess 2 sides and an angle in the order of side angle side; sort of like the congruency statements we learned in trigonometry. One you get these three things, plug them in in the right places using the formula.
Unit 4
Sine Addition Formula- sin(α + β) = sinα cos β + cos α sin β.
Sine Subtraction Formula- sin(x-y)=sinxcosy-cosxsiny
Cosine Addition formula- cos(x+y)=cosxcosy-sinxsiny
Cosine Subtraction Formula- cos(x-y)=cosxcosy+sinxsiny
Tangent Addition- tan(x+y)= tanx+tany/1-tanxtany
Tangent Subtraction- tan(x-y)=tanx-tany/1+tanxtany
Double Angle Trig ID
x= angle value or theta
Sine(2x)= 2sinxcosx
cos(2x)= cos^2x-sin^2x
tan(2x)=2tanx/1-tan^2x
Half Angle Formulas
Tangent x/2=+/- √1-cosx/1+cosx
Sine x/2 =+/-√1-cosx/2
Cosinex/2= +/-√1+cosx/2
Sine Subtraction Formula- sin(x-y)=sinxcosy-cosxsiny
Cosine Addition formula- cos(x+y)=cosxcosy-sinxsiny
Cosine Subtraction Formula- cos(x-y)=cosxcosy+sinxsiny
Tangent Addition- tan(x+y)= tanx+tany/1-tanxtany
Tangent Subtraction- tan(x-y)=tanx-tany/1+tanxtany
Double Angle Trig ID
x= angle value or theta
Sine(2x)= 2sinxcosx
cos(2x)= cos^2x-sin^2x
tan(2x)=2tanx/1-tan^2x
Half Angle Formulas
Tangent x/2=+/- √1-cosx/1+cosx
Sine x/2 =+/-√1-cosx/2
Cosinex/2= +/-√1+cosx/2
Proving Trigonometric Identities
With trig identities that are set equal to each other, all you are trying to do is verify and prove that both sides are equal to each other. To do this you will have to use all of the trig identities that you have learned, and a lot of the algebra knowledge that you already have. These identities are a good way of pushing yourself to use all of the knowledge you have to solve an equation. Since it is hard to explain how to solve one of these equations, watch this video for help-
Matrices
Vocabulary-
Matrix: a rectangular arrangement of numbers into rows and columns.
Dimensions or Order of a Matrix: the number of rows by the number of columns.
Scalar: In matrix algebra, a real number is called a scalar.
Square Matrix: a matrix with the same number of rows and columns
Zero Matrix: a matrix whose all entries are zero.
Identity Matrix: the matrix that has 1's on the main diagonal and 0's elsewhere.
Inverse Matrices: Matrices whose product(in both orders) is the identity matrix.
Determinant: the product of the elements on the main diagonal minus the product of the elements off the main diagonal.
Matrix: a rectangular arrangement of numbers into rows and columns.
Dimensions or Order of a Matrix: the number of rows by the number of columns.
Scalar: In matrix algebra, a real number is called a scalar.
Square Matrix: a matrix with the same number of rows and columns
Zero Matrix: a matrix whose all entries are zero.
Identity Matrix: the matrix that has 1's on the main diagonal and 0's elsewhere.
Inverse Matrices: Matrices whose product(in both orders) is the identity matrix.
Determinant: the product of the elements on the main diagonal minus the product of the elements off the main diagonal.