Readings and homework

Here are the assigned readings and exercises (Reading 1, Reading 2, Reading 3, Reading 4, Reading 5, Reading 6, Reading 7, Reading 8, Reading 9, Reading 10, Reading 11, Reading 12, Reading 13, Reading 14, Reading 15, Reading 16, Reading 17); but anything whose assigned date is in the future is subject to change!

1. Graphing review:
   • Date assigned: October 4 Thursday.
   • Date due: October 9 Tuesday.
   • Reading: Pages 74–83 from the textbook (§2.1).
   • Exercises due:
     1. In which number quadrant are both coordinates positive?
     2. Write down a formula for the distance between the points (x₁, y₁) and (x₂, y₂) in a rectangular coordinate system.
     3. If the two short sides of a right triangle have lengths 3 and 4, then what it the length of the hypotenuse?
2. Angles:
   • Date assigned: October 9 Tuesday.
   • Date due: October 11 Thursday.
   • Reading: Pages 576–590 (§7.1).
   • Exercises due:
     1. If a central angle in a circle subtends an arc whose length equals the circle's radius, then what is the measure of that angle?
     2. How many radians is 360°?
     3. Fill in the blank with an algebraic expression: In a circle of radius r, a central angle whose measure is θ radians subtends an arc whose length is s = ___.
3. Acute angles:
   • Date assigned: October 11 Thursday.
   • Date due: October 16 Tuesday.
   • Reading: Pages 593–600 (§7.2).
   • Exercises due:
     1. Fill in the blank: The sine of the complement of θ is the ___ of θ.
     2. If θ is the measure of an acute angle in a right triangle, then what is the cotangent of θ as a ratio of the lengths of the adjacent leg, the opposite leg, and/or the hypotenuse?
     3. What is the sine of π/6 radians?
4. The trigonometric operations:
   • Date assigned: October 16 Tuesday.
   • Date due: October 18 Thursday.
   • Reading:
     • Pages 604–616 (§7.3);
     • Pages 620–630 (§7.4).
   • Exercises due:
     1. Two angles that differ by one or more full turns are called ___ angles.
     2. If the six fundamental trigonometric functions all have the same absolute values at θ as at α and α is an acute angle, then α is the ___ angle of θ.
     3. Which of the six fundamental trigonometric functions of θ are positive when θ terminates in Quadrant III?
5. Graphs of the trigonometric functions:
   • Date assigned: October 18 Thursday.
   • Date due: October 23 Tuesday.
   • Reading:
     • Pages 641–644 (§8.1.1).
     • Page 659 and the first half of page 660 (§8.2.1).
     • The bottom of page 664 and page 665. (§8.2.3).
     • The middle of page 670 (§8.2.5).
   • Exercises due:
     1. Most trigonometric functions have a period of 2π; which two have a period of π instead?
     2. Consider the numbers 2 and 1/2. Which is in the range of the sine function, and which is in the range of the cosecant function?
     3. Out of 0, π/2, π, and 2π, which is not in the domain of the tangent function?
6. Transformations of trigonometric functions:
   • Date assigned: October 23 Tuesday.
   • Date due: October 25 Thursday.
   • Reading:
     • Optional: Review pages 222–242 (§3.5);
     • The bottom of page 644 through page 645 (the rest of §8.1);
     • My handout on sinusoidal functions.
     • The bottom half of page 660 through the top half of page 664 (§8.2.2);
     • The bottom of page 665 through the top half of page 670 (§8.2.4);
     • The box at the bottom of page 670 and page 671–673 (§§8.2.6&8.2.7).
   • Exercises due:
     1. To get the graph of y = −f(x) from the graph of y = f(x), what do you reflect the graph across?
     2. If f(x) = A cos(ωx) for all x, with A > 0 and ω > 0, then what are the amplitude and period of f?
     3. If f(x) = sin(x + ψ) for all x, with 0 < ψ < 2π, then what is the phase shift of f?
7. None (continue previous topic).
8. Inverse trigonometric operations:
   • Date assigned: October 30 Tuesday.
   • Date due: November 1 Thursday.
   • Reading:
     • Pages 677–685 (§8.3);
     • My handout on inverse trigonometric operations.
   • Exercises due: Fill in all of these blanks with algebraic expressions (some of which will be constants):
     1. That y = sin⁻¹ x means that x = ___ and ___ ≤ y ≤ ___.
     2. cos⁻¹ x exists if and only if ___ ≤ x ≤ ___.
     3. cos⁻¹ cos θ = θ if and only if ___ ≤ θ ≤ ___.
     4. cos sin⁻¹ x = ___ (if it exists).
9. Expanding trigonometric expressions:
   • Date assigned: November 1 Thursday.
   • Date due: November 6 Tuesday.
   • Reading:
     • Pages 706–715 (§9.2 through §9.2.4);
     • Pages 720–722 (§9.3.1);
     • Pages 724–729 (§§9.3.3&9.3.4).
   • Exercises due: Fill in the blanks with trigonometric expressions in which each trigonometric operation that appears is only applied directly to α or β.
     1. sin(α + β) = ___.
     2. cos(α + β) = ___.
     3. sin²(α/2) = ___.
     4. cos²(α/2) = ___.
     5. tan(α/2) = ___.
10. Simplifying trigonometric expressions:
    • Date assigned: November 6 Tuesday.
    • Date due: November 8 Thursday.
    • Reading:
      • My handout on simplifying trigonometric expressions;
      • Pages 696–703 (§9.1);
      • Pages 715–717 (§9.2.5);
      • The very botom of page 722 and page 723 (§9.3.2).
    • Exercises due:
      1. Fill in the blank with an expression in which the only trigonometric quantity is sin θ: cos² θ = ___.
      2. Factor: sin² θ − 1 = (___)(___).
      3. If you regard a cosine as a square root, then what expression is conjugate to 1 − cos θ? (If you were to multiply the two expressions together, then cos θ should appear only as cos² θ.)
11. Trigonometric equations:
    • Date assigned: November 8 Thursday.
    • Date due: November 13 Tuesday.
    • Reading: Page 739 through the top of page 748 (§9.5 through §9.5.6).
    • Exercises due:
      1. Write a general form for the solutions of tan x = b using tan⁻¹ b and an arbitrary integer k.
      2. Similarly, give the general solution of sin x = b. (Hint: This one is more complicated than the last one.)
      3. To obtain θ ∈ [0, 2π) (that is, 0 ≤ θ < 2π), what interval should 3θ  belong to?
12. Solving right triangles:
    • Date assigned: November 13 Tuesday.
    • Date due: November 15 Thursday.
    • Reading: Page 748&749 (§9.5.7).
    • Exercises due:
      1. Answer this in degrees, and also answer it in radians: If A and B are the two acute angles in a right triangle, then A + B = ___.
      2. True or false: Knowing any two of the three sides of a right triangle is enough information to solve the triangle completely.
      3. True or false: Knowing any two of the three angles of a right triangle is enough information to solve the triangle completely.
13. The Law of Sines:
    • Date assigned: November 15 Thursday.
    • Date due: November 20 Tuesday.
    • Reading: Pages 762–769 (§10.1).
    • Exercises due: In each of the following forms of the Law of Sines, fill in the blank to get a true theorem (where a, b and c are the lengths of the three sides of a triangle and α, β, and γ are the measures of the respective opposite angles).
      1. a ÷ sin α = b ÷ ___.
      2. b ÷ c = sin β ÷ ___.
      3. sin α ÷ a = sin β ÷ ___.
14. The Law of Cosines:
    • Date assigned: November 20 Tuesday.
    • Date due: November 27 Tuesday.
    • Reading:
      • Pages 776–782 (§10.2);
      • My handout on solving triangles.
    • Exercises due:
      1. Which law do you use to solve a triangle, if you are given two angles and one of the sides?
      2. Which law do you use if you are given the three sides?
      3. What do you do if you are given only the angles?
15. Polar coordinates:
    • Date assigned: November 27 Tuesday.
    • Date due: November 29 Thursday.
    • Reading:
      • Pages 788–796 (§10.3);
      • Pages 802–812 (§§10.4.2&10.4.3).
    • Exercises due:
      1. Fill in the blanks with expressions: Given a point with polar coordinates (r, θ), its rectangular coordinates are (x, y) = (___, ___).
      2. True or false: For each point P in the coordinate plane, for every pair (r, θ) of real numbers that gives P in polar coordinates, r ≥ 0 and 0 ≤ θ < 2π.
      3. True or false: For each point P in the coordinate plane, for some pair (r, θ) of real numbers that gives P in polar coordinates, r ≥ 0 and 0 ≤ θ < 2π.
16. Vectors:
    • Date assigned: November 29 Thursday.
    • Date due: December 4 Tuesday.
    • Reading: Pages 847–854 (§10.8 through §10.8.5).
    • Exercises due:
      1. Give a formula for the vector from the initial point (x₁, y₁) to the terminal point (x₂, y₂).
      2. Give a formula for the magnitude (or norm, or length) of the vector ⟨a, b⟩.
17. Vectors and angles:
    • Date assigned: December 13 Thursday.
    • Date due: December 18 Tuesday.
    • Reading: The bottom of page 854 through page 860 (§§10.8.6–10.8.10).
    • Exercises due:
      1. State a formula for the dot product u ⋅ v of two vectors using only their lengths |u| and |v|, the angle ∠(u, v) between them, and real-number operations.
      2. State a formula for the dot product of ⟨a, b⟩ and ⟨c, d⟩ using only real-number operations and the rectangular components a, b, c, and d.

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