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Algebra and Trigonometry

A | Proofs, Identities, and Toolkit Functions

Algebra and TrigonometryA | Proofs, Identities, and Toolkit Functions
  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Chapter Review
      1. Key Terms
      2. Key Concepts
    6. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Unit Circle: Sine and Cosine Functions
    1. Introduction to The Unit Circle: Sine and Cosine Functions
    2. 7.1 Angles
    3. 7.2 Right Triangle Trigonometry
    4. 7.3 Unit Circle
    5. 7.4 The Other Trigonometric Functions
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Periodic Functions
    1. Introduction to Periodic Functions
    2. 8.1 Graphs of the Sine and Cosine Functions
    3. 8.2 Graphs of the Other Trigonometric Functions
    4. 8.3 Inverse Trigonometric Functions
    5. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    6. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 9.1 Solving Trigonometric Equations with Identities
    3. 9.2 Sum and Difference Identities
    4. 9.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 9.4 Sum-to-Product and Product-to-Sum Formulas
    6. 9.5 Solving Trigonometric Equations
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 10.1 Non-right Triangles: Law of Sines
    3. 10.2 Non-right Triangles: Law of Cosines
    4. 10.3 Polar Coordinates
    5. 10.4 Polar Coordinates: Graphs
    6. 10.5 Polar Form of Complex Numbers
    7. 10.6 Parametric Equations
    8. 10.7 Parametric Equations: Graphs
    9. 10.8 Vectors
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 11.1 Systems of Linear Equations: Two Variables
    3. 11.2 Systems of Linear Equations: Three Variables
    4. 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 11.4 Partial Fractions
    6. 11.5 Matrices and Matrix Operations
    7. 11.6 Solving Systems with Gaussian Elimination
    8. 11.7 Solving Systems with Inverses
    9. 11.8 Solving Systems with Cramer's Rule
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 12.1 The Ellipse
    3. 12.2 The Hyperbola
    4. 12.3 The Parabola
    5. 12.4 Rotation of Axes
    6. 12.5 Conic Sections in Polar Coordinates
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. 13 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 13.1 Sequences and Their Notations
    3. 13.2 Arithmetic Sequences
    4. 13.3 Geometric Sequences
    5. 13.4 Series and Their Notations
    6. 13.5 Counting Principles
    7. 13.6 Binomial Theorem
    8. 13.7 Probability
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  15. A | Proofs, Identities, and Toolkit Functions
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index

Important Proofs and Derivations

Product Rule

log a xy= log a x+ log a y log a xy= log a x+ log a y

Proof:

Let m= log a x m= log a x and n= log a y. n= log a y.

Write in exponent form.

x= a m x= a m and y= a n . y= a n .

Multiply.

xy= a m a n = a m+n xy= a m a n = a m+n

a m+n = xy log a (xy) = m+n = log a x+ log b y a m+n = xy log a (xy) = m+n = log a x+ log b y

Change of Base Rule

log a b= log c b log c a log a b= 1 log b a log a b= log c b log c a log a b= 1 log b a

where x x and y y are positive, and a>0,a1. a>0,a1.

Proof:

Let x= log a b. x= log a b.

Write in exponent form.

a x =b a x =b

Take the log c log c of both sides.

log c a x = log c b x log c a = log c b x = log c b log c a log a b = log c b log c a log c a x = log c b x log c a = log c b x = log c b log c a log a b = log c b log c a

When c=b, c=b,

log a b= log b b log b a = 1 log b a log a b= log b b log b a = 1 log b a

Heron’s Formula

A= s( sa )( sb )( sc ) A= s( sa )( sb )( sc )

where s= a+b+c 2 s= a+b+c 2

Proof:

Let a, a, b, b, and c c be the sides of a triangle, and h h be the height.

A triangle with sides labeled: a, b and c.  A line runs through the center of the triangle, bisecting the top angle; this line is labeled: h.

So s= a+b+c 2 s= a+b+c 2 .

We can further name the parts of the base in each triangle established by the height such that p+q=c. p+q=c.

A triangle with sides labeled: a, b, and c.  A line runs through the center of the triangle bisecting the angle at the top; this line is labeled: h. The two new line segments on the base of the triangle are labeled: p and q.

Using the Pythagorean Theorem, h 2 + p 2 = a 2 h 2 + p 2 = a 2 and h 2 + q 2 = b 2 . h 2 + q 2 = b 2 .

Since q=cp, q=cp, then q 2 = ( cp ) 2 . q 2 = ( cp ) 2 . Expanding, we find that q 2 = c 2 2cp+ p 2 . q 2 = c 2 2cp+ p 2 .

We can then add h 2 h 2 to each side of the equation to get h 2 + q 2 = h 2 + c 2 2cp+ p 2 . h 2 + q 2 = h 2 + c 2 2cp+ p 2 .

Substitute this result into the equation h 2 + q 2 = b 2 h 2 + q 2 = b 2 yields b 2 = h 2 + c 2 2cp+ p 2 . b 2 = h 2 + c 2 2cp+ p 2 .

Then replacing h 2 + p 2 h 2 + p 2 with a 2 a 2 gives b 2 = a 2 2cp+ c 2 . b 2 = a 2 2cp+ c 2 .

Solve for p p to get

p= a 2 + b 2 c 2 2c p= a 2 + b 2 c 2 2c

Since h 2 = a 2 p 2 , h 2 = a 2 p 2 , we get an expression in terms of a, a, b, b, and c. c.

h 2 = a 2 p 2 = (a+p)(ap) = [ a+ ( a 2 + c 2 b 2 ) 2c ][ a ( a 2 + c 2 b 2 ) 2c ] = ( 2ac+ a 2 + c 2 b 2 )( 2ac a 2 c 2 + b 2 ) 4 c 2 = ( (a+c) 2 b 2 )( b 2 (ac) 2 ) 4 c 2 = (a+b+c)(a+cb)(b+ac)(ba+c) 4 c 2 = (a+b+c)(a+b+c)(ab+c)(a+bc) 4 c 2 = 2s(2sa)(2sb)(2sc) 4 c 2 h 2 = a 2 p 2 = (a+p)(ap) = [ a+ ( a 2 + c 2 b 2 ) 2c ][ a ( a 2 + c 2 b 2 ) 2c ] = ( 2ac+ a 2 + c 2 b 2 )( 2ac a 2 c 2 + b 2 ) 4 c 2 = ( (a+c) 2 b 2 )( b 2 (ac) 2 ) 4 c 2 = (a+b+c)(a+cb)(b+ac)(ba+c) 4 c 2 = (a+b+c)(a+b+c)(ab+c)(a+bc) 4 c 2 = 2s(2sa)(2sb)(2sc) 4 c 2

Therefore,

h 2 = 4s(sa)(sb)(sc) c 2 h = 2 s(sa)(sb)(sc) c h 2 = 4s(sa)(sb)(sc) c 2 h = 2 s(sa)(sb)(sc) c

And since A= 1 2 ch, A= 1 2 ch, then

A = 1 2 c 2 s(sa)(sb)(sc) c = s(sa)(sb)(sc) A = 1 2 c 2 s(sa)(sb)(sc) c = s(sa)(sb)(sc)

Properties of the Dot Product

u·v=v·u u·v=v·u

Proof:

u·v = u 1 , u 2 ,... u n · v 1 , v 2 ,... v n = u 1 v 1 + u 2 v 2 +...+ u n v n = v 1 u 1 + v 2 u 2 +...+ v n v n = v 1 , v 2 ,... v n · u 1 , u 2 ,... u n =v·u u·v = u 1 , u 2 ,... u n · v 1 , v 2 ,... v n = u 1 v 1 + u 2 v 2 +...+ u n v n = v 1 u 1 + v 2 u 2 +...+ v n v n = v 1 , v 2 ,... v n · u 1 , u 2 ,... u n =v·u

u·( v+w )=u·v+u·w u·( v+w )=u·v+u·w

Proof:

u·(v+w) = u 1 , u 2 ,... u n ·( v 1 , v 2 ,... v n + w 1 , w 2 ,... w n ) = u 1 , u 2 ,... u n · v 1 + w 1 , v 2 + w 2 ,... v n + w n = u 1 ( v 1 + w 1 ), u 2 ( v 2 + w 2 ),... u n ( v n + w n ) = u 1 v 1 + u 1 w 1 , u 2 v 2 + u 2 w 2 ,... u n v n + u n w n = u 1 v 1 , u 2 v 2 ,..., u n v n + u 1 w 1 , u 2 w 2 ,..., u n w n = u 1 , u 2 ,... u n · v 1 , v 2 ,... v n + u 1 , u 2 ,... u n · w 1 , w 2 ,... w n =u·v+u·w u·(v+w) = u 1 , u 2 ,... u n ·( v 1 , v 2 ,... v n + w 1 , w 2 ,... w n ) = u 1 , u 2 ,... u n · v 1 + w 1 , v 2 + w 2 ,... v n + w n = u 1 ( v 1 + w 1 ), u 2 ( v 2 + w 2 ),... u n ( v n + w n ) = u 1 v 1 + u 1 w 1 , u 2 v 2 + u 2 w 2 ,... u n v n + u n w n = u 1 v 1 , u 2 v 2 ,..., u n v n + u 1 w 1 , u 2 w 2 ,..., u n w n = u 1 , u 2 ,... u n · v 1 , v 2 ,... v n + u 1 , u 2 ,... u n · w 1 , w 2 ,... w n =u·v+u·w

u·u= | u | 2 u·u= | u | 2

Proof:

u·u = u 1 , u 2 ,... u n · u 1 , u 2 ,... u n = u 1 u 1 + u 2 u 2 +...+ u n u n = u 1 2 + u 2 2 +...+ u n 2 =| u 1 , u 2 ,... u n | 2 =v·u u·u = u 1 , u 2 ,... u n · u 1 , u 2 ,... u n = u 1 u 1 + u 2 u 2 +...+ u n u n = u 1 2 + u 2 2 +...+ u n 2 =| u 1 , u 2 ,... u n | 2 =v·u

Standard Form of the Ellipse centered at the Origin

1= x 2 a 2 + y 2 b 2 1= x 2 a 2 + y 2 b 2

Derivation

An ellipse consists of all the points for which the sum of distances from two foci is constant:

( x( c ) ) 2 + ( y0 ) 2 + ( xc ) 2 + ( y0 ) 2 =constant ( x( c ) ) 2 + ( y0 ) 2 + ( xc ) 2 + ( y0 ) 2 =constant

An ellipse centered at the origin on an x, y-coordinate plane.  Points C1 and C2 are plotted at the points (0, b) and (0, -b) respectively; these points appear on the ellipse.  Points V1 and V2 are plotted at the points (-a, 0) and (a, 0) respectively; these points appear on the ellipse.  Points F1 and F2 are plotted at the points (-c, 0) and (c, 0) respectively; these points appear on the x-axis, but not the ellipse. The point (x, y) appears on the ellipse in the first quadrant.  Dotted lines extend from F1 and F2 to the point (x, y).

Consider a vertex.

An ellipse centered at the origin.  The points C1 and C2 are plotted at the points (0, b) and (0, -b) respectively; these points are on the ellipse.  The points V1 and V2 are plotted at the points (-a, 0) and (a, 0) respectively; these points are on the ellipse.  The points F1 and F2 are plotted at the points (-c, 0) and (c, 0) respectively; these points are on the x-axis and not on the ellipse.  A line extends from the point F1 to a point (x, y) which is at the point (a, 0).  A line extends from the point F2 to the point (x, y) as well.

Then, ( x( c ) ) 2 + ( y0 ) 2 + ( xc ) 2 + ( y0 ) 2 =2a ( x( c ) ) 2 + ( y0 ) 2 + ( xc ) 2 + ( y0 ) 2 =2a

Consider a covertex.

An ellipse centered at the origin.  The points C1 and C2 are plotted at the points (0, b) and (0, -b) respectively; these points are on the ellipse.  The points V1 and V2 are plotted at the points (-a, 0) and (a, 0) respectively; these points are on the ellipse.  The points F1 and F2 are plotted at the points (-c, 0) and (c, 0) respectively; these points are on the x-axis and not on the ellipse.  There is a point (x, y) which is plotted at (0, b). A line extends from the origin to the point (c, 0), this line is labeled: c.  A line extends from the origin to the point (x, y), this line is labeled: b.  A line extends from the point (c, 0) to the point (x, y); this line is labeled: (1/2)(2a)=a.  A dotted line extends from the point (-c, 0) to the point (x, y); this line is labeled: (1/2)(2a)=a.

Then b 2 + c 2 = a 2 . b 2 + c 2 = a 2 .

(x(c)) 2 + (y0) 2 + (xc) 2 + (y0) 2 = 2a (x+c) 2 + y 2 = 2a (xc) 2 + y 2 (x+c) 2 + y 2 = ( 2a (xc) 2 + y 2 ) 2 x 2 +2cx+ c 2 + y 2 = 4 a 2 4a (xc) 2 + y 2 + (xc) 2 + y 2 x 2 +2cx+ c 2 + y 2 = 4 a 2 4a (xc) 2 + y 2 + x 2 2cx+ y 2 2cx = 4 a 2 4a (xc) 2 + y 2 2cx 4cx4 a 2 = 4a (xc) 2 + y 2 1 4a ( 4cx4 a 2 ) = (xc) 2 + y 2 a c a x = (xc) 2 + y 2 a 2 2xc+ c 2 a 2 x 2 = (xc) 2 + y 2 a 2 2xc+ c 2 a 2 x 2 = x 2 2xc+ c 2 + y 2 a 2 + c 2 a 2 x 2 = x 2 + c 2 + y 2 a 2 + c 2 a 2 x 2 = x 2 + c 2 + y 2 a 2 c 2 = x 2 c 2 a 2 x 2 + y 2 a 2 c 2 = x 2 ( 1 c 2 a 2 )+ y 2 (x(c)) 2 + (y0) 2 + (xc) 2 + (y0) 2 = 2a (x+c) 2 + y 2 = 2a (xc) 2 + y 2 (x+c) 2 + y 2 = ( 2a (xc) 2 + y 2 ) 2 x 2 +2cx+ c 2 + y 2 = 4 a 2 4a (xc) 2 + y 2 + (xc) 2 + y 2 x 2 +2cx+ c 2 + y 2 = 4 a 2 4a (xc) 2 + y 2 + x 2 2cx+ y 2 2cx = 4 a 2 4a (xc) 2 + y 2 2cx 4cx4 a 2 = 4a (xc) 2 + y 2 1 4a ( 4cx4 a 2 ) = (xc) 2 + y 2 a c a x = (xc) 2 + y 2 a 2 2xc+ c 2 a 2 x 2 = (xc) 2 + y 2 a 2 2xc+ c 2 a 2 x 2 = x 2 2xc+ c 2 + y 2 a 2 + c 2 a 2 x 2 = x 2 + c 2 + y 2 a 2 + c 2 a 2 x 2 = x 2 + c 2 + y 2 a 2 c 2 = x 2 c 2 a 2 x 2 + y 2 a 2 c 2 = x 2 ( 1 c 2 a 2 )+ y 2

Let 1= a 2 a 2 . 1= a 2 a 2 .

a 2 c 2 = x 2 ( a 2 c 2 a 2 )+ y 2 1 = x 2 a 2 + y 2 a 2 c 2 a 2 c 2 = x 2 ( a 2 c 2 a 2 )+ y 2 1 = x 2 a 2 + y 2 a 2 c 2

Because b 2 + c 2 = a 2 , b 2 + c 2 = a 2 , then b 2 = a 2 c 2 . b 2 = a 2 c 2 .

1 = x 2 a 2 + y 2 a 2 c 2 1 = x 2 a 2 + y 2 b 2 1 = x 2 a 2 + y 2 a 2 c 2 1 = x 2 a 2 + y 2 b 2

Standard Form of the Hyperbola

1= x 2 a 2 y 2 b 2 1= x 2 a 2 y 2 b 2

Derivation

A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances between two fixed points is constant.

Side-by-side graphs of hyperbole.  In Diagram 1: The foci F’ and F are labeled and can be found a little in front of the opening of the hyperbola.  A point P at (x,y) on the right curve is labeled.  A line extends from the F’ focus to the point P labeled: D1.  A line extends from the F focus to the point P labeled: D2.  In Diagram 2:  The foci F’ and F are labeled and can be found a little in front of the opening of the hyperbola.  A point V is labeled at the vertex of the right hyperbola.  A line extends from the F’ focus to the point V labeled: D1.  A line extends from the F focus to the point V labeled: D2.

Diagram 1: The difference of the distances from Point P to the foci is constant:

(x(c)) 2 + (y0) 2 (xc) 2 + (y0) 2 =constant (x(c)) 2 + (y0) 2 (xc) 2 + (y0) 2 =constant

Diagram 2: When the point is a vertex, the difference is 2a. 2a.

( x( c ) ) 2 + ( y0 ) 2 ( xc ) 2 + ( y0 ) 2 =2a ( x( c ) ) 2 + ( y0 ) 2 ( xc ) 2 + ( y0 ) 2 =2a

(x(c)) 2 + (y0) 2 (xc) 2 + (y0) 2 = 2a (x+c) 2 + y 2 (xc) 2 + y 2 = 2a (x+c) 2 + y 2 = 2a+ (xc) 2 + y 2 (x+c) 2 + y 2 = ( 2a+ (xc) 2 + y 2 ) x 2 +2cx+ c 2 + y 2 = 4 a 2 +4a (xc) 2 + y 2 x 2 +2cx+ c 2 + y 2 = 4 a 2 +4a (xc) 2 + y 2 + x 2 2cx+ y 2 2cx = 4 a 2 +4a (xc) 2 + y 2 2cx 4cx4 a 2 = 4a (xc) 2 + y 2 cx a 2 = a (xc) 2 + y 2 ( cx a 2 ) 2 = a 2 ( (xc) 2 + y 2 ) c 2 x 2 2 a 2 c 2 x 2 + a 4 = a 2 x 2 2 a 2 c 2 x 2 + a 2 c 2 + a 2 y 2 c 2 x 2 + a 4 = a 2 x 2 + a 2 c 2 + a 2 y 2 a 4 a 2 c 2 = a 2 x 2 c 2 x 2 + a 2 y 2 a 2 ( a 2 c 2 ) = ( a 2 c 2 ) x 2 + a 2 y 2 a 2 ( a 2 c 2 ) = ( c 2 a 2 ) x 2 a 2 y 2 (x(c)) 2 + (y0) 2 (xc) 2 + (y0) 2 = 2a (x+c) 2 + y 2 (xc) 2 + y 2 = 2a (x+c) 2 + y 2 = 2a+ (xc) 2 + y 2 (x+c) 2 + y 2 = ( 2a+ (xc) 2 + y 2 ) x 2 +2cx+ c 2 + y 2 = 4 a 2 +4a (xc) 2 + y 2 x 2 +2cx+ c 2 + y 2 = 4 a 2 +4a (xc) 2 + y 2 + x 2 2cx+ y 2 2cx = 4 a 2 +4a (xc) 2 + y 2 2cx 4cx4 a 2 = 4a (xc) 2 + y 2 cx a 2 = a (xc) 2 + y 2 ( cx a 2 ) 2 = a 2 ( (xc) 2 + y 2 ) c 2 x 2 2 a 2 c 2 x 2 + a 4 = a 2 x 2 2 a 2 c 2 x 2 + a 2 c 2 + a 2 y 2 c 2 x 2 + a 4 = a 2 x 2 + a 2 c 2 + a 2 y 2 a 4 a 2 c 2 = a 2 x 2 c 2 x 2 + a 2 y 2 a 2 ( a 2 c 2 ) = ( a 2 c 2 ) x 2 + a 2 y 2 a 2 ( a 2 c 2 ) = ( c 2 a 2 ) x 2 a 2 y 2

Define b b as a positive number such that b 2 = c 2 a 2 . b 2 = c 2 a 2 .

a 2 b 2 = b 2 x 2 a 2 y 2 a 2 b 2 a 2 b 2 = b 2 x 2 a 2 b 2 a 2 y 2 a 2 b 2 1 = x 2 a 2 y 2 b 2 a 2 b 2 = b 2 x 2 a 2 y 2 a 2 b 2 a 2 b 2 = b 2 x 2 a 2 b 2 a 2 y 2 a 2 b 2 1 = x 2 a 2 y 2 b 2

Trigonometric Identities

Pythagorean Identities cos 2 θ+ sin 2 θ=1 1+ tan 2 θ= sec 2 θ 1+ cot 2 θ= csc 2 θ cos 2 θ+ sin 2 θ=1 1+ tan 2 θ= sec 2 θ 1+ cot 2 θ= csc 2 θ
Even-Odd Identities cos(−θ)=cosθ sec(−θ)=secθ sin(−θ)=sinθ tan(−θ)=tanθ csc(−θ)=cscθ cot(−θ)=cotθ cos(−θ)=cosθ sec(−θ)=secθ sin(−θ)=sinθ tan(−θ)=tanθ csc(−θ)=cscθ cot(−θ)=cotθ
Cofunction Identities cosθ=sin( π 2 θ ) sinθ=cos( π 2 θ ) tanθ=cot( π 2 θ ) cotθ=tan( π 2 θ ) secθ=csc( π 2 θ ) cscθ=sec( π 2 θ ) cosθ=sin( π 2 θ ) sinθ=cos( π 2 θ ) tanθ=cot( π 2 θ ) cotθ=tan( π 2 θ ) secθ=csc( π 2 θ ) cscθ=sec( π 2 θ )
Fundamental Identities tanθ= sinθ cosθ secθ= 1 cosθ cscθ= 1 sinθ cotθ= 1 tanθ = cosθ sinθ tanθ= sinθ cosθ secθ= 1 cosθ cscθ= 1 sinθ cotθ= 1 tanθ = cosθ sinθ
Sum and Difference Identities cos(α+β)=cosαcosβsinαsinβ cos(αβ)=cosαcosβ+sinαsinβ sin(α+β)=sinαcosβ+cosαsinβ sin(αβ)=sinαcosβcosαsinβ tan(α+β)= tanα+tanβ 1tanαtanβ tan(αβ)= tanαtanβ 1+tanαtanβ cos(α+β)=cosαcosβsinαsinβ cos(αβ)=cosαcosβ+sinαsinβ sin(α+β)=sinαcosβ+cosαsinβ sin(αβ)=sinαcosβcosαsinβ tan(α+β)= tanα+tanβ 1tanαtanβ tan(αβ)= tanαtanβ 1+tanαtanβ
Double-Angle Formulas sin(2θ)=2sinθcosθ cos(2θ)= cos 2 θ sin 2 θ cos(2θ)=12 sin 2 θ cos(2θ)=2 cos 2 θ1 tan(2θ)= 2tanθ 1 tan 2 θ sin(2θ)=2sinθcosθ cos(2θ)= cos 2 θ sin 2 θ cos(2θ)=12 sin 2 θ cos(2θ)=2 cos 2 θ1 tan(2θ)= 2tanθ 1 tan 2 θ
Half-Angle Formulas sin α 2 =± 1cosα 2 cos α 2 =± 1+cosα 2 tan α 2 =± 1cosα 1+cosα tan α 2 = sinα 1+cosα tan α 2 = 1cosα sinα sin α 2 =± 1cosα 2 cos α 2 =± 1+cosα 2 tan α 2 =± 1cosα 1+cosα tan α 2 = sinα 1+cosα tan α 2 = 1cosα sinα
Reduction Formulas sin 2 θ= 1cos( 2θ ) 2 cos 2 θ= 1+cos( 2θ ) 2 tan 2 θ= 1cos( 2θ ) 1+cos( 2θ ) sin 2 θ= 1cos( 2θ ) 2 cos 2 θ= 1+cos( 2θ ) 2 tan 2 θ= 1cos( 2θ ) 1+cos( 2θ )
Product-to-Sum Formulas cosαcosβ= 1 2 [ cos(αβ)+cos(α+β) ] sinαcosβ= 1 2 [ sin(α+β)+sin(αβ) ] sinαsinβ= 1 2 [ cos(αβ)cos(α+β) ] cosαsinβ= 1 2 [ sin(α+β)sin(αβ) ] cosαcosβ= 1 2 [ cos(αβ)+cos(α+β) ] sinαcosβ= 1 2 [ sin(α+β)+sin(αβ) ] sinαsinβ= 1 2 [ cos(αβ)cos(α+β) ] cosαsinβ= 1 2 [ sin(α+β)sin(αβ) ]
Sum-to-Product Formulas sinα+sinβ=2sin( α+β 2 )cos( αβ 2 ) sinαsinβ=2sin( αβ 2 )cos( α+β 2 ) cosαcosβ=2sin( α+β 2 )sin( αβ 2 ) cosα+cosβ=2cos( α+β 2 )cos( αβ 2 ) sinα+sinβ=2sin( α+β 2 )cos( αβ 2 ) sinαsinβ=2sin( αβ 2 )cos( α+β 2 ) cosαcosβ=2sin( α+β 2 )sin( αβ 2 ) cosα+cosβ=2cos( α+β 2 )cos( αβ 2 )
Law of Sines sinα a = sinβ b = sinγ c a sinα = b sinβ = c sinγ sinα a = sinβ b = sinγ c a sinα = b sinβ = c sinγ
Law of Cosines a 2 = b 2 + c 2 2bccosα b 2 = a 2 + c 2 2accosβ c 2 = a 2 + b 2 2abcosγ a 2 = b 2 + c 2 2bccosα b 2 = a 2 + c 2 2accosβ c 2 = a 2 + b 2 2abcosγ
Table A1

ToolKit Functions

Three graphs side-by-side. From left to right, graph of the identify function, square function, and square root function. All three graphs extend from -4 to 4 on each axis.
Figure A1
Three graphs side-by-side. From left to right, graph of the cubic function, cube root function, and reciprocal function. All three graphs extend from -4 to 4 on each axis.
Figure A2
Three graphs side-by-side. From left to right, graph of the absolute value function, exponential function, and natural logarithm function. All three graphs extend from -4 to 4 on each axis.
Figure A3

Trigonometric Functions

Unit Circle

Graph of unit circle with angles in degrees, angles in radians, and points along the circle inscribed.
Figure A4
Angle 0 0 π 6 ,or 30° π 6 ,or 30° π 4 ,or 45° π 4 ,or 45° π 3 ,or 60° π 3 ,or 60° π 2 ,or 90° π 2 ,or 90°
Cosine 1 3 2 3 2 2 2 2 2 1 2 1 2 0
Sine 0 1 2 1 2 2 2 2 2 3 2 3 2 1
Tangent 0 3 3 3 3 1 3 3 Undefined
Secant 1 2 3 3 2 3 3 2 2 2 Undefined
Cosecant Undefined 2 2 2 2 3 3 2 3 3 1
Cotangent Undefined 3 3 1 3 3 3 3 0
Table A2
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