Ok so this week we did more stuff in the study guides for trig exam and yea..anyway im going to show you the half and double angle formulas for yall...
Half-Angle and Double Angle Formulas:
sin(2alpha) = 2sin(alpha)cos(alpha)
cos(2alpha) = cos^2(alpha)-sin^2(alpha)=1-2sin^2(alpha)=2cos^2
(alpha)-1
tan(2alpha) = 2tan(alpha)/1-tan^2(alpha)
sin(alpha/2)= +- sqrt(1-cos(alpha)/2)
cos(alpha/2)= +- sqrt(1+cos(alpha)/2)
tan(alpha/2)= +- sqrt(1-cos(alpha)/1+cos(alpha))=sin(alpha)/1+cos
(alpha)=1-cos(alpha)/sin(alpha)
Ok so like i said in class, everything with graphs is gonnneee. I need help working on conics and graphing them so if anyone can help i would be grateful
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Conics
ReplyDeleteEllipses
Steps:
1. find the center
2. determine the major axis
3. find the vertex (± √big denom)
4. find the other intercept ( ± √small denom)
5. find the focus (c^2 = a^2 + b^2)
6. determine the length of the major axis (2√big denom)
7. find the length of the minor axis (2√small denom)
8. finally graph
Example 1: Graph the following ellipse. Find its major intercepts, length of the major axis, minor intercepts, length of the minor axis, and foci.
x^2/4 + y^2/9 = 1
This ellipse is centered at (0, 0). Since the larger denominator is with the y variable, the major axis lies along the y-axis.
Since a^2 = 9 then a = 3 & Since b^2 = 4 then b = 2
Major intercepts: (0, 3), (0, –3)
Length of major axis: 2 √9 = 6
Minor intercepts: (2, 0), (–2, 0)
Length of minor axis: 2√4 = 4
c^2 = a^2 + b^2
= 9 - 4
= 5
= √5
Foci: (0, √5) , (0, -√5)
then you graph your points..
Parabolas:
how to find the axis of symmetry, vertex, focus, & directrx??
1.) to find the axis of symmetry: x = -b/2a
2.) for the vertex: (-b/2a, f(-b/2a)) or use complete the square:
y = (x+a)^2 + b.....a & b are numbers and (-a,b) = vertex
3.) to find the focus: 1/4p= the coefficient of x^2 and then add p
Note:
*If opens up, add to y value from vertex, if opens down, subtract
*If opens right, add to x value to vertex, if opens left, subtract)
4.) directrix: is p units behind the vertex
Note:
*If opens up, subtract; if opens down, add from y-value of vertex.
*If opens right, subtract x-value
*If opens left, add x-value
Example: x^2 + 1
~vertex:
x = -b/2a
x = 0/2(1) = 0
0^2 + 1 = 1
(0,1)
~Focus:
1/4p = 1
4p = 1
p = 1/4
(0, 1 + 1/4)
(0, 5/4)
~directrix:
y = 1 - 1/4
y = 3/4
CIRCLES
The standard equation of a circle is (x-h)^2+(y-k)^2 .....the center is (h,k)
If the equation is not in standard form, you must complete the square to put it in standard form.
If you are given a center and a point, you can use the distance formula to find the radius.
To find the intersection of a line and a circle:
1. solve the linear eqn for y.
2. substitute in the circle eqn.
3. solve for x.
4. plug the x value in to get the y value.
***Reminder. If your x value is imaginary, then there is no point of intersection.
EX: find the center and radius.
(x-3)^2+(y+7)^2=19 c:(h,k)
center: (3,-7) radius: square root of 19
EX: find the eqn of the circle with the center (1,4) through (3,7)
in the problem you are given a center and a point so you would plug into the distance formula.
square root of (3-1)^2+(7-4)^2= square root of 4+9=square root of 13. **13 has no root.
Your answer should be (x-1)^2+(y-4)^2=13
Hyperbola
(x^2-h/a^2)+(y^2-k/b^2)=1
* Your center is (h,k)
* your major axis has the larger denominator
Graphing Conics
ReplyDeleteCIRCLES
-the equation of a circle in standard form
(x-h)^2+(y-k)^2=r^2
where the center is (h,k) and radius=r
-if not in standard form, you must complete the square for both x and y to put in standard form
-give nthe center and point, you can use the distance formula to find the radius
TO FIND THE INTERSECTION OF A LINE AND CIRCLE
-solve the linear equation of a line
-substitute in circle equation
-solve for x
-plug x value in to get y value
*if your x value is imaginary, there is no point of intersection
TO GRAPH A CIRCLE
-find the center and radius
-draw your circle
ELLIPSES
((x-H)^2/(length of X/2)^2)+((Y-K)^2/(length of Y/2)^2)=1
-(h,k)=center
-major axis is the larger denominator
-vertex is on major axis
-focus> smaller#^2=larger#^2-focus^2
focus is on major axis
1. center
2. major axis- bigger denominator (x or y)
3. vertex +/- squareroot larger denominator
4. other intercepts +/- squareroot smaller denominator
5. focus
6. length of major axis 2squareroot larger denominator
7. length of minor axis 2squareroot smaller denominator
8. graph
HYPERBOLAS
((x-h)^2/(length/2)^2)-((y-k)^2/(length/2)^2)=1
-center (h/k)
-major axis is the non-negative denominator
-vertex> +/- squareroot non-negative denominator
-asymptotes> y=+/- (squareroot y/ squareroot x)X
-focus^2= x denom+ y denom
or focus^2= vertex^2+ other denom
TO SKETCH
1. shape
2. center
3. major
4. minor
5. other intercepts (none)
6. focus
7. asymptotes
8. vertex
9. sketch
PARABOLAS
- x=(-b/2a)
-two ways for find the vertex
((-b/2a), f(-b/2a))or complete the square to get in vertex form y=(x+a)^2 +/- b >(-a,b)
-focus (1/4p)=coefficient of x^2
-directrix is P units behind vertex
if opens up, add y value to vertex, if down, subtract y value from vertex
if upens < add x value to vertex, if > subtract x value from the vertex