Download exam review - ICHSPre-Calc

Final Exam Review:
Part 1: Vocab
1. trigonometric identity
11. asymptotes
2. circle
12. parabola
3. center
13. directrix
4. radius
14. sequence
5. conic section
15. terms
6. ellipse
16. common difference
7. foci
17. common ratio
8. minor axis
18. infinite sequence
9. major axis
19. limit
10. hyperbola
20. infinite series
Chapter Seven: Section Two Verifying Trigonometric Identities
Key Concepts/ Suggestions for Verifying Trigonometric Identities:
- Transform the more complicated side of the equation into the simpler side.
-
Substitute one or more basic trigonometric identities to simplify expressions.
-
Factor or multiply to simplify expressions.
-
Multiply expressions by an expression equal to 1.
-
Express all trigonometric functions in terms of sine and cosine.
1. Tan B= sec B/ csc B
2. Cos n= sin n cot n
3. Sec y csc y= tan y + cot y
4. Sin x + cos x= 2 sin^2 x – 1/ sin x- cos x
5. Cos y/ 1- sin y= 1 + sin y/ cos y
7.3 Sum and Difference Identities
VOCABULARY / FORMULAS
·
COSINE
o Sum Identity
 Cos(A+B) = cosAcosB - sinAsinB
o Difference Identity
 cos(A-B) = cosAcosB + sinAsinB
·
SINE
o Sum Identity
 sin(A+B) = sinAcosB + cosA+sinB
o Difference Identity
 sin(A-B) = sinAcosB – cosAsinB
·
TAN
o Sum Identity
tan(A-B) = tanA – tanB
1+ tanAtanB
o Difference Identity
tan(A+B) = tanA + tanB
1 – tanAtanB
PROBLEMS
Use sum or difference identities to find the exact value of each trigonometric
function.
1. (sin 165°)
2. (sin pi/12)
Find each exact value if 0<x<pi/2 and 0<y<pi/2
3. cos (x-y) if cosx = 3/5 and cosy =4/5
4. sec (x-y) if cscx = 5/3 and tany = 12/5
Verify that each equation is an identity.
5. cos (180°+ x) = -cosx
7.4 Double Angle & Half Angle Identities
Important Formulas and Identities
Double Angle Identities
Half Angle Identities
Sin2x = 2SinxCosx
Cos2x = Cos^2x – Sin^2x
= 2Cos^2x – 1
= 1 -2Sin^2x
Tan2x = 2tanx
1 – Tan^2x
The Pythagorean Identities is not in this section but is necessary to know.
Sin^2x + Cos^2x = 1
Tan^2x + 1 = Sec^2x
1 + Cot^2x = Csc^2x
The chart containing the value of trigonometric functions in page 305 is also important.
Solving Double Angles
· First identify the information that you have, such as given values (i.e. Cos = 2/3)
as well as the quadrant it is in.
· Next use the Pythagorean identities to find any missing values for trigonometric
functions.
· Once you have the values of both trigonometric functions necessary to solve the
double angle identity you can substitute those values.
Solving Half Angle Identities
· The value of the function is multiplied by 2 to find the value of a
· If the value of the function is in radians then divide only its denominator by two,
the resulting fraction is the value of a
· Substitute a into the appropriate half angle identity formula and refer to the value
of trigonometric functions chart to determine the value, remember to take the
quadrant into consideration when choosing a formula and value
· Finally, solve.
5 Simple Mathematical Problems
1. Use the half angle identity to find the exactly value of the function:
Cos 7π/12
2. If Sin x = 2/3 and is in the 1st quadrant, find the exact value of each of its double
angles.
3. Verify that the following equation is an identity
csc 2x = ½ sec x csc x
4. Verify that the following equation is an identity
CosA – SinA = Cos2A CosA + SinA
5. Use the information provided to find the value of Sin2x, Cos2x, and Tan2x
Cos x = 4/5 0 < x < 90
Sec. 7.5 Solving Trig. Equations
Principal Values – the solutions of a trigonometric equation in which the variable is
restricted to two adjacent quadrants.
5) Solve equation for principal values of x.
2 sinx + 1 = 0
9) Solve equation for 0 < x < 2π
3 tan2 x-1 = 0
27) Solve equation for 0° < x < 360°
sin 2x = -sin x
31) Solve equation for 0 < x < 2π
4 sin2x + 1 = -4 sinx
51) Solve inequality for 0 < θ < 2π
√2 sinθ - 1 < 0
Chapter 10.1
Definitions
•
Analytic geometry-the study of coordinate geometry from an algebraic
perspective.
Formulas
•
Distance Formula for Two Points- The distance, of units, between two points with
the coordinates[x1,y1] and [x2,y2] is given by d=(radical)[x2-x1]2 +[y2-y1]2
•
Midpoint of a Line Segment- If the coordinates of P1 and P2 are [x1,y1] and
[x2,y2], respectively, then the midpoint of (line)P1P2 has the coordinates
[x1+x2/2,y1+y2/2]
Problems
•
D=(radical) (x2-x1)2 + (y2-y1)2
•
•
D=(radical) (x2-x1)2 + (y2_y1)2
Determine whether quadrilateral ABCD with vertices A(3,2), B(2,-4), C(-2,-3), and
D(-1,3) is a parallelogram.
•
Find the coordinates of the midpoint of the segment that has endpoints at (-2,4)
and (6,-5)
Prove that the measure of the median of a trapezoid is equal to one half of
•
the sum of the measures of the two bases.
Section 10-2 Circles
Key Facts
Standard Form of a circle – (x-h)^2 + (y-k)^2 = r^2
General form of circle – x^2+y^2+Dx+Ey+F= 0
***Remember if given the coordinates of diameter use the distance formula to
find radius
10.2 Ellipse
1. An ellipse is the set of all points in the plane, the sum of whose distances from
two fixed points, called foci, is constant.
2. has two axes of symmetry
a. the x-axis and the y-axis
3. Longer axis, or major axis, has the foci.
4. Other axis is called minor axis.
Graphing
h)2
(y -
(x k)2
+ 2
b
a2
where c2 = a2 -
k)2
(y -
a2
b2
(x h)2
+ b2
where c2 = a2 -
= 1,
b2
-
= 1,
Center: (h, k)
Foci: (h +/- c, k)
Major axis: y = k
Major axis vertices: (h +/- a, k)
Minor axis: x = h
Minor axis vertices: (h, k +/- b)
Center: (h, k)
Foci: (h, k +/- c)
Major axis: x =h
Major axis vertices: (h, k +/- a)
Minor axis: y = k
Minor axis vertices: (h +/- b, k)
Which is major/Which is minor ?
In all ellipses, a2 > b2. You can use this information to determine the orientation of
the major axis from the equation. If a2 is the denominator of the x2 term, the major axis
is parallel to the x-axis. If a2 is the denominator of the y2 term, the major axis is parallel
to the y-axis.
10-4 Hyperbolas
Formulas:
Standard Form of the Equation of a Hyperbola:
(x-h)² - (y-k)² =1
a²
b²
where b²=c²-a²
Description
center: (h, k)
foci: (h ± c, k)
vertices: (h ± a, k)
equation of
transverse axis: y =k
(Parallel to x-axis)
(y-k)² - (x-h)² = 1
a²
b²
Description
center: (h, k)
foci: (h, k ± c)
vertices: (h, k ± a)
equation of
transverse axis: x = h
(parallel to y-axis)
Equations of the Asymptotes of the Hyperbola:
y - k =± b/a (x-h),
for a hyperbola with standard form
(x-h)² - (y-k)² =1
a²
b²
y-k=± a/b (x-h),
for the hyperbola with standard form
(y-k)² - (x-h)² = 1
a²
b²
Rectangular Hyperbola: xy _ c
Rectangular Hyperbola: xy = c
Value of c
Location of branches of hyperbola
Positive
Quadrants I and III
Negative
Quadrants II and IV
Problems: pgs.’ 649 &650 #5,9,10,18 & 32
For the equation of each hyperbola, find the coordinates of the center, the foci,
and the vertices and the equations of the asymptotes of its graph. Then graph
the equation.
5. X² - Y² =1
25 4
18. (y-7)² - (x+1)² =1
64
4
9. XY= -9
Write an equation of the hyperbola that meets each set of conditions.
10. The center is at (1, _4), a = 5, b = 2, and it has a horizontal transverse axis.
32. The vertices are at (0, 3) and (0, -3), and a focus is at (0, -9).
Section 10-5 Parabolas
Formulas
Standard form (y - k)² = 4p(x - h)
Vertex: (h,k)
focus: (h+p,k) axis of symmetry: y = k
opening: right if p>0, left if p<0
(x – h)² = 4p(y – k)
Vertex: (h , k) focus: (h , k + p) axis of symmetry: x = h
opening: upward if p>0, downward if p<0
Practice problems
Pg. 658 6-8 (example 1 pg 655), 10-11.
directrix: x = h - p
directrix: y = k – p