The broad goals of the course include the following:

To gain an analytical and conceptual understanding of the nature and applications of trigonometric functions, conic sections, matrices, sequences and binomial probability.

Understanding of the nature of the mathematical modeling process, its uses and its limitations.

Proficiency in mathematical modeling using trigonometric functions, matrices, sequences and binary probability.

Specific objectives are listed by module in the table below.  These objectives appear in the Assignments Page for the course, where the appropriate objectives are listed for each individual assignment.

Module 1 Objectives
Objectives:

Relate {theta, x, y} on the unit circle.

Relate {omega, t, theta(t), x(t), y(t)} for a reference point initially at (1, 0) on the circle, moving with angular velocity omega.

Relate for a reference point initially at coordinates x = cos(phi), y = sin(phi)) on the circle, moving with angular velocity omega.

  • period
  • omega
  • t
  • theta
  • phi
  • x
  • y
  • h
  • k
  • v_arc (the speed of the reference point as it moves around the arc of the circle)
  • `dt
  • `ds_arc (the distance traveled by the reference point along the arc during time interval `dt)
Objectives:

1.  Construct a unit circle showing all standard angular positions which are multiples of pi/6 or pi/4.

2.  Given starting point and angular velocity model motion on the unit circle.

3.  Relate angular displacement on the unit circle to arc distance and vice versa.

4.  Relate

  • reference circle angles which are multiples of pi / 6 or pi / 4
  • x and y coordinates on the unit circle
  • x and y coordinates on a reference circle of radius A, centered at the origin
  • table of y vs. x

5.  Using triangles and the Pythagorean Theorem determine the exact values of the sines and cosines of pi/4, pi/6 and pi/3 and use these values to construct a table of the sines and cosines of all angles which are multiples of pi/4 and pi/6.

6.  Relate for a reference point initially at the point (cos(phi), sin(phi)) on the circle, moving with angular velocity omega.

  • period
  • omega
  • t
  • theta
  • phi
  • x
  • y
  • h
  • k
    •
  • graph of sin(t), cos(t)
  • v_arc (the speed of the reference point as it moves around the arc of the circle)
  • `dt
  • `ds_arc (the distance traveled by the reference point along the arc during time interval `dt)

7.  Apply all the above to applications and problems.

 

 Objectives:

Estimate the x and y coordinates and angle theta associated with a given point on a picture of the unit circle.

Hand-sketch a unit circle and the points corresponding to angles which are multiples of pi/4 and pi/6:

  • Using the sketch estimate the values of the sine and cosine of each angle
  • graph the sine and cosine functions based on your estimates
  • use periodicity properties to extend the graph.
  • For motion of the reference point at constant angular velocity omega, determine the change in t associated with a complete cycle and use along with estimates from the hand sketch to graph y = sin(omega * t) and y = cos(omega * t)

Apply the periodicity property of trigonometric functions to evaluation at given points, to construction of graphs and to tables.

Explain and apply the even-odd behavior of given trigonometric functions.

Relate for a reference point initially at the point (cos(phi), sin(phi)) on the circle, moving with angular velocity omega.

  • period
  • omega
  • t
  • theta
  • phi
  • x
  • y
  • h
  • k
  • A cos(omega t + phi) + k
  • A sin(omega t + phi) + k
  • A cos(omega ( t – h) + k
  • A sin(omega (t – h) ) + k
  • graph of y = A cos(omega t + phi) + k
  • graph of y = A A sin(omega t + phi) + k
  • graph of y = A A cos(omega ( t – h) + k
  • graph of y = A A sin(omega (t – h) ) + k
  • v_arc (the speed of the reference point as it moves around the arc of the circle)
  • `dt
  • `ds_arc (the distance traveled by the reference point along the arc during time interval `dt)

 

 Objectives:

1.   Construct a table of the values of y = A sin(x) for a given value of A, extending for a complete cycle of this function, with x equal to multiples of pi/6 or pi/4, and using the table construct a graph of one cycle of y = A sin(x ). 

2.  Given a function y = A sin(theta) with theta given as a function of x, construct a table of the values of y = A sin(theta) for a complete cycle of this function with theta equal to multiples of pi/6 or pi/4, then determine the x value corresponding to each value of theta.  Using a table of y vs. x construct a graph of one cycle of y = A sin(theta) in terms of the given function theta of x, clearly labeling the x axis for each quarter-cycle of the function.   

3.  Interpret the function and graph corresponding to Goal 2 in terms of angular motion on a unit circle.

4.  Use transformations to construct graphs of A cos(omega t + theta_0) + k, and similarly translated graphs for the other trigonometric functions.

  

Objectives:

1.   Construct a table of the values of y = A sin(x) for a given value of A, construct a graph of a single cycle of this function, then extending the graph forward and/or backward for any specified number of complete cycles. 

2.  Given a function y = A sin(theta) with theta given as a function of x, construct a y vs. x graph of a single cycle of this function, then extending the graph forward and/or backward for any specified number of complete cycles. 

3.  Interpret the function and graph corresponding to Goal 2 in terms of angular motion on a circle of appropriate radius.

 

Module 2 Objectives
Objectives:

1.  Construct the basic triangles corresponding to angles pi/6, pi/4, pi/3, as appropriate, in order to find the sines and cosines of these angles.

2.  Using exact values construct tables and graphs for the basic trigonometric functions. 

3.  Using exact values construct y vs. x graphs of y = A sin(theta) or y = A cos(theta), where theta is given as a function of x. 

4 .  Use the technique of reversing columns and restricting domain to construct tables and graphs for the arcsin and arccos functions. 

5.  Given an equation in which the argument of the sine or cosines function is a function of x, solve the equation for all values of x within a given interval.

 

Objectives:

1.   Understand and explain how the vertical asymptotes of the graph of the tangent function occur.

2.  Using exact values construct y vs. x graphs of y = A sin(theta) or y = A cos(theta), where theta is given as a function of x. 

 

Objectives:

1.   Given an identity involving sine, cosine, tangent, cosecant, secant and cotangent functions prove or disprove it using the Pythagorean identities and the definitions of these functions.

2.  Find the trigonometric functions of arbitrary angles by first finding the reference angle, then applying the sign appropriate to the quadrant.

3.  Given a side of a right triangle and another side or angle, determine all of its angles and sides.

4.  Given a side of a triangle, another side or and angle, and still another side or angle of a triangle, find all of its sides and angles.

Module 3 Objectives
Objectives:

1.  Given the magnitude and direction of a vector determine its components.

2.  Given the components of a vector determine its magnitude and direction.

3.  Given two or more vectors determine the magnitude and angle of their sum.

4.  Model simple harmonic motion using sine or cosine functions and appropriate transformations.

 Objectives:

Relate

  • v, a vector in 2 or 3 dimensions
  • w, a vector in 2 or 3 dimensions
  • theta, the angle between v and w
  • v dot w, the dot product of v and w
  • || v ||
  • || w ||
  • scalar projection of v on w
  • v / || v ||
  • w / || w ||
  • vector projection of v on w
 

Objectives:

1.  Express the distance from general point (x, y) from a specific point (x_0, y_0), or from a line y = c, or from a line x = c (where c is constant).

2.  Express the definitions of the conic sections as equations in terms of distances from a point (x, y) on the conic section to appropriate points and/or lines, and simplify the equations.

 Objectives:

1.  Know the equations of parabolas with vertex at the origin, and of ellipses and hyperbolas centered at the origin.

2.  Sketch the graph of the equation of an ellipse or a hyperbola centered at the origin, using the 'basic rectangle'.

3.  Find, as appropriate, the vertices, foci, eccentricities and asymptotes of the above.

4.  Use shifting and stretching transformations and the process of completing the square to find the equation of and graph a general conic section of form A x^2 + B x + C y^2 + D y + E = 0.

5.  Relate the equation of an ellipse to its properties (center, vertices, foci, semimajor and semiminor axes) and to its geometric definition.

 Objectives:

1.  Perform the operations of addition or multiplication of two compatible matrices, and multiplication of a matrix by a scalar.

2.  Relate the equation of a hyperbola to its properties (center, vertices, foci, semimajor and semiminor axes) and to its geometric definition.

 Objectives:

1.  Determine whether a system of two or three simultaneous linear equations in an equal number of variables has a solution and if so find it using substitution or elimination, as specified. 

2.  Write a system of simultaneous linear equations an augmented matrix and solve by matrix reduction.

 

Module 4 Objectives
Objectives: 

1.  Apply Cramer's Rule to solve a system of linear equations

2.  Know and apply the properties of determinants

3.  Write a system of linear equations as a matrix equation.

4.  Apply the properties of matrices to find sums of multiples and products of compatible matrices.

5.  Find the inverse of a matrix.

6.  Solve matrix equations by applying the inverse matrix.

Objectives:

1.  Given an explicit or recursive rule for a sequence, write its first several terms.

2.  Use summation notation to represent the sum of the members of a sequence.

3.  Apply the additive and multiplicative properties of sequences.

4.  Recognize, write, sum and apply arithmetic sequences.

Objectives:

1.  Recognize, write, sum and apply geometric sequences.

2.  Determine convergence of infinite geometric sequences. 

3.  Use the binomial theorem to expand binomials.

Objectives:

1.  Apply permutations, combinations and the properties of probability to calculate properties of specified events.