Problem sets and exams

Here are the exams and their associated problem sets (Exam 1, Exam 2, Exam 3, Exam 4, extra material):

1. Exam 1:
   • Date taken: October 23 Tuesday.
   • Knewton modules:
     • Cartesian Coordinates and Distances;
     • Angles as Rotations and Radian Measures;
     • Arc Length and Area of a Sector;
     • The Six Trigonometric Ratios;
     • Use Right Triangle Trigonometry in Solving Problems;
     • Sine and Cosine Values in the First Quadrant;
     • Sine and Cosine Values with Reference Angles;
     • The Other Trigonometric Ratios on the Unit Circle;
     • Use Given Trigonometric Ratios to Find Other Ratios.
   • Extra credit: In the diagram below, the radius of the circle is 1, so the sine and cosine of the indicated angle θ are the lengths of the line segments labelled ‘sin’ and ‘cos’. That is, given that AO = 1 and ∠AOD = θ, you know that AC = sin θ and CO = cos θ. Explain why the line segments labelled ‘tan’ and ‘sec’ are the tangent and secant, respectively; that is, why AE = tan θ and EO = sec θ. Hint: Use similar triangles: If two triangles have the same angles, then the lengths of their corresponding sides will be proportional.

     

     (This picture shows where the terms ‘tangent’ and ‘secant’ came from, as the tangent is a segment from a line that is touching the circle, while the secant is a segment from a line that is cutting the circle; ‘tangent’ and ‘secant’ are Latin for touching and cutting respectively. The origin of ‘sine’ is much more complicated, involving a mistranslation among other things. This diagram also includes a few of the obsolete trigonometric operations that were used to help make precise calculations in the days before handheld calculators.)
2. Exam 2:
   • Date taken: November 6 Tuesday.
   • Knewton modules:
     • Characteristics of Sine and Cosine Graphs;
     • Characteristics of Tangent and Cotangent Graphs;
     • Characteristics of Secant and Cosecant Graphs;
     • Transformations of Functions;
     • Transformations of Sine and Cosine Graphs;
     • Graph Sine and Cosine Functions;
     • Introduction to Inverse Trigonometric Functions;
     • Compose Functions with Inverse Trigonometric Functions.
   • Extra credit: In some computer programming languages, an inverse tangent function is provided but not the inverse sine or any other inverse trigonometric function. Write down formulas for the other five inverse trigonometric functions using only the inverse tangent operation (in addition to rational and radical operations), similar to the list of formulas in the handout from October 25 Thursday (which uses the inverse sine instead of the inverse tangent).
3. Exam 3:
   • Date taken: November 20 Tuesday.
   • Knewton modules:
     • Sum and Difference Formulas;
     • Double-Angle Formulas;
     • Half-Angle and Power-Reduction Formulas;
     • Simplify Expressions with Basic Trigonometric Identities;
     • Use Pythagorean and Cofunction Identities;
     • Verify Trigonometric Identities;
     • Trigonometric Equations in Sine and Cosine;
     • Trigonometric Equations Involving a Single Trigonometric Function;
     • Solve Triangles with Inverse Trigonometric Functions.
   • Extra credit: Exercise 9.4.34 on page 737 of the textbook. (Although this can be done using the methods of my handout, that will take a very long time. Use sum-to-product and/or product-to-sum identities from Section 9.4 instead. Show at least the steps just before and just after applying these identities.)
4. Exam 4:
   • Date taken: December 6 Thursday.
   • Knewton modules:
     • Law of Sines;
     • Law of Cosines;
     • Area of Oblique Triangles;
     • Convert Coordinates Between Rectangular and Polar Forms;
     • Convert Equations Between Rectangular and Polar Forms;
     • Introduction to Graphing Polar Equations;
     • Graph Classic Polar Curves;
     • Properties of Vectors;
     • Vector Addition and Scalar Multiplication.
   • Extra credit: (For brevity, I will write cis θ to abbreviate cos θ + i sin θ. See Section 10.5 of the textbook for why this comes up so much.) In the 1740s, Leonhard Euler defined a general rule for raising any nonzero complex number to the power of any other complex number, called the principal power. If you write the base in polar coordinates as r cis θ, with r > 0 and 0 ≤ θ < 2π, and the exponent in rectangular coordinates as x + yi, then the principal power is written in polar coordinates as

     (r cis θ)^x+yi = (r^x e^−yθ) cis (y ln r + xθ),

     where e ≈ 2.718 and ln is the natural logarithm (so ln means log_e). Show at least one intermediate step for each of these:
     1. If the base is positive (θ = 0) and the exponent is real (y = 0), then what is the principal power r^x? (The answer to this one is so simple that it might be confusing; it's just r^x in the usual sense for real numbers, but you should check to see how that comes out of the general formula. The other answers will be more complicated.)
     2. If the base is negative (θ = π) and the exponent is 1/2 (x = 1/2 and y = 0), then what is the principal power (−r)^1/2?
     3. If the base is negative and the exponent is 1/n for an odd natural number n, then what is the principal power (−r)^1/n? (This is called the principal nth root of −r, in contrast to the real nth root of −r that you should know from Algebra, which is −r^1/n.)
     4. If the base is e (r = e and θ = 0), then what is the principal power e^x+yi? (For this reason, ln a + yi is called the principal natural logarithm of a cis y.)
     5. If the base is e and the exponent is purely imaginary (x = 0), then what is the principal power e^iy? (This result is called Euler's formula.)
     6. What is e^iπ + 1? (This result is called Euler's identity.)
5. No exam:
   • Date taken: Never.
   • Knewton modules (optional):
     • The Unit Vector;
     • The Dot Product and Vector Applications.

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