Trigonometry Final Examination Review

Final Examination Review

• The final examination will take place on a specified date, and calculators are allowed.
• A reference sheet and a blank unit circle will be provided for the final examination.

Topics Covered

• Trig Unit 1
• Trig Unit 2
• Trig Unit 3
• Polars & Parametrics
• Vectors
• Sequences & Sums

Examination Format

• 20 Multiple Choice questions (40 points)
• 10 Short Answer questions (60 points)

Sample Questions

Parametric Equations

• Eliminate the parameter and write the corresponding rectangular equation: x = 2t, y = t^2

Vectors

• Determine if the vectors u = 3i + 4j and v = -2i + 3j are orthogonal
• Provide both algebraic and written explanations

Arithmetic Sequences

• Find a formula for an for the arithmetic sequence: a1 = 5, a4 = 15
• Find the 30th term in the sequence

Trigonometric Equations

• Find all solutions of the equation in the interval [0, 2π): cos(x + π/4) - cos(x - π/4) = 1

Graphing

• Sketch the graph of the function: y = -2cos(2πx) + 1
• Include at least two cycles on the graph

Triangles

• Find the angles in the triangle with vertices Chicago, St. Louis, and Atlanta

Polar Equations

• Sketch the polar curves: r = 2 + 2cosθ, r = 3
• Find the point(s) of intersection algebraically and put the angle(s) in radian measure

More Trigonometric Equations

• Solve the equation for 0 ≤ x < 2π: 4cos^2x - 3 = 0

Daylight Hours

• Model the number of daylight hours in Boston, MA with a sinusoidal graph
• Create a trigonometric equation to represent the yearly number of daylight hours

Half Angle Formula

• Use a half angle formula to find the exact value of cos(11π/12)

Vectors and Motion

• Find the plane's ground speed and true course given its heading and air speed
• Consider a wind blowing east at 20 mph

Geometric Sequences

• Find the nth term of the geometric sequence: 3, 4, 3, ... (n = 12)
• Find the 20th partial sum of the sequence

Functions

• Find the domain of the function: y = tanx

Polar Coordinates

• Find three additional polar representations of the point (5, -3π/4) on -2π < θ < 2π
• Find the corresponding rectangular coordinates for the polar coordinates (2, 7π/6)

Graphs of Polar Equations

• Provide an equation for each of the following and sketch a rough graph of the equation's graphical representation:
  • a dimpled limaçon
  • a cardioid
  • an inner-loop limaçon
  • a convex limaçon

Vector Operations

• If u = <1, -3> and v = <1, 2>, find:
  • 2u - 3v
  • u · v - ||u||
• Find the angle between vectors u = <4, -3> and v = <1, 2>
• Find the magnitude and direction angle of the vector 3i - 4j

More Vectors

• Find a unit vector in the direction of the vector w = -3i - 5j
• Find u - v if u has a magnitude of 85 and a direction of 45°, and v has a magnitude of 100 and a direction of 327°

Trigonometric Functions

• Identify the transformations given the function: y = -5sin(4x + π) - 3
• If you wanted to sketch a graph of the cosecant function, it might be helpful to first sketch which function?
• Provide a rough sketch of the secant function

Review for the final examination of a trigonometry class, covering units 1-3, polars, parametrics, vectors, and more. Calculator allowed and reference sheets provided.