Reading homework and online problem sets

Required resources

The official textbook for the course is the 11th Edition of Algebra & Trigonometry written by Sullivan and published by Prentice-Hall (Pearson).

Textbook access (DjVu instructions).

Basic trigonometry

General review:
- Reading from the textbook: Section 2.1 (pages 150–153).
- Exercises due August 29 Thursday:
  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 is the length of the hypotenuse?
- Exercises from the textbook due September 3 Tuesday on MyLab: 2.1.19, 2.1.21, 2.1.23, 2.1.25, 2.1.27, 2.1.31, 2.1.39, 2.1.41, 2.1.43.
Circles:
- Reading from the textbook: Section 2.4 (pages 185–188).
- Exercises due September 3 Tuesday:
  1. Fill in the blank: The distance from the centre (or center) of a circle to any point on the circle is the _____ of the circle.
  2. Write down an equation in the variables x and y for a circle whose centre is (h, k) and whose radius is r. (This will be an equation in which x, y, h, k, and r all appear.)
  3. If x² + y² = r² is the equation of a circle in x and y, then what are the coordinates of the centre of the circle?
- Exercises from the textbook due September 5 Thursday on MyLab: 2.4.5, 2.4.9, 2.4.11, 2.4.13, 2.4.15, 2.4.17, 2.4.21, 2.4.23, 2.4.25, 2.4.27.
Angles:
- Reading from the textbook: Section 7.1 (pages 518–526).
- Exercises due September 5 Thursday:
  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 = ___.
- Exercises from the textbook due September 10 Tuesday on MyLab: 7.1.11, 7.1.13, 7.1.15, 7.1.17, 7.1.19, 7.1.21, 7.1.23, 7.1.26, 7.1.35, 7.1.37, 7.1.71, 7.1.73.
Right triangles:
- Reading from the textbook: Section 7.2 (pages 531–538).
- Exercises due September 10 Tuesday:
  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. If the secant of an angle is 3, then what is its cosine?
- Exercises from the textbook due September 12 Thursday on MyLab: 7.2.9, 7.2.10, 7.2.11, 7.2.12, 7.2.13, 7.2.15, 7.2.17, 7.2.23, 7.2.27, 7.2.31, 7.2.33, 7.2.37, 7.2.39, 7.2.41, 7.2.43, 7.2.45, 7.2.57, 7.2.65.
Special angles:
- Reading from the textbook: Section 7.3 through Objective 2 (pages 543–546).
- Exercise due September 12 Thursday: Write down the exact values of the sine, cosine, tangent, cotangent, secant, and cosecant of π/6 (or 30°), π/4 (or 45°), and π/3 (or 60°). (This is 18 values to write down in all, which you might put into a handy table.)
- Exercises from the textbook due September 17 Tuesday on MyLab: 7.3.7, 7.3.19, 7.3.21, 7.3.32, 7.3.27, 7.3.29.
General angles:
- Reading from the textbook:
  - The rest of Section 7.3 (pages 546–549);
  - Section 7.4 through Objective 5 (pages 555–561).
- Exercises due September 17 Tuesday:
  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?
- Exercises from the textbook due September 19 Thursday on MyLab: 7.3.31, 7.3.33, 7.3.35, 7.4.11, 7.4.15, 7.4.17, 7.4.21, 7.4.31, 7.4.33, 7.4.39, 7.4.51, 7.4.53, 7.4.57, 7.4.59, 7.4.61, 7.4.75.
Circular trigonometry:
- Reading from the textbook:
  - The rest of Section 7.4 (page 562);
  - Section 7.5 through "Trigonometric Functions of Angles" (pages 566–570).
- Exercises due September 19 Thursday:
  1. Suppose that you start at the point (1, 0) in a rectangular coordinate system and move in the direction towards (0, 1) along the unit circle, for a total distance t. (This is the usual thing, not a trick question.) If you end at the point (a, b), express sin t, cos t, tan t, cot t, sec t, and csc t using only a and b.
  2. Now instead of moving along the unit circle (with radius 1), move along a circle of radius r (but still centred at the origin). That is, start at (r, 0) and move along the circle in the direction of (0, r) for a total distance of s, and let θ be s/r. (This is again the usual thing for a non-unit radius.) Now if you end at the point (x, y), express sin θ, cos θ, tan θ, cot θ, sec θ, and csc θ using only x, y, and r.
- Exercises from the textbook due September 24 Tuesday on MyLab: 7.4.83, 7.4.87, 7.4.93, 7.5.11, 7.5.13, 7.5.17, 7.5.19, 7.5.23, 7.5.31, 7.5.33, 7.5.39, 7.5.41, 7.5.47, 7.5.49.
The trigonometric functions:
- Reading from the textbook: The rest of Section 7.5 (pages 570–575).
- Exercises due September 24 Tuesday:
  1. Most of the six 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?
- Exercises from the textbook due September 26 Thursday on MyLab: 7.5.63, 7.5.65, 7.5.69, 7.5.81, 7.5.85, 7.5.87, 7.6.3.

Quiz/Exam 1, covering the material above, is on October 1 Tuesday.

Advanced trigonometry

Basic graphing:
- Reading from the textbook:
  - Section 7.6 through the box before Example 1 (pages 579&580);
  - Section 7.6 Objective 2 through the box before Example 3 (pages 581&582);
  - Section 7.7 through "The Graph of the Cotangent Function y = cot x" (pages 594–596);
  - Section 7.7 Objective 3 (pages 597&598).
- Exercises due September 26 Thursday:
  1. What are the intercepts of the graphs of the sine and tangent functions?
  2. What are the intercepts of the graph of the cotangent function?
  3. Besides the answers to (2) above, what additional intercept does the graph of the cosine function have?
  4. What is the intercept of the graph of the secant function?
  5. What are the asymptotes to the graphs of the tangent and secant functions?
  6. What are the asymptotes to the graphs of the cotangent and cosecant functions?
- Exercises from the textbook due October 3 Thursday on MyLab: 7.6.6, 7.6.8, 7.6.11, 7.6.13, 7.7.3, 7.7.6, 7.7.7, 7.7.10, 7.7.11, 7.7.12, 7.7.13, 7.7.16.
Transformations of trigonometric functions:
- Reading from the textbook:
  - Skim: Section 3.5 (pages 254–263);
  - Section 7.6 Objective 1 Examples 1&2 (pages 580&581);
  - Section 7.6 Objective 2 Example 3 (page 582);
  - Section 7.7 Objective 2 (pages 596&597);
  - Section 7.7 Objective 4 (pages 598&599).
- Exercises due October 3 Thursday:
  1. To get the graph of y = −f(x) from the graph of y = f(x), which axis do you reflect the graph across?
  2. To get the graph of y = 2f(x) from the graph of y = f(x), do you stretch the graph vertically or compress it?
  3. To get the graph of y = f(2x) from the graph of y = f(x), do you stretch the graph horizontally or compress it?
- Exercises from the textbook due October 8 Tuesday on MyLab: 3.5.41, 3.5.35, 3.5.45, 7.6.23–32, 7.7.17, 7.7.21, 7.7.23, 7.7.25, 7.7.29, 7.7.31.
Sinusoidal functions:
- Reading from (mostly) the textbook:
  - My handout on sinusoidal functions (DjVu);
  - The rest of Section 7.6 (pages 582–588);
  - Section 7.8 Objective 1 (pages 601–605).
- Exercises due October 8 Tuesday:
  1. If f(x) = A sin(ωx) for all x, with A > 0 and ω > 0, then what are the amplitude and period of f?
  2. If f(x) = A sin x + B for all x, with A > 0, then what are the maximum and minimum values of f?
  3. If f(x) = sin(ωx − φ) for all x, with 0 ≤ φ < 2π, then what is the phase shift of f?
- Exercises from the textbook due October 10 Thursday on MyLab: 7.6.35, 7.6.39, 7.6.51, 7.6.57, 7.6.61, 7.6.87, 7.8.9, 7.8.11, 7.8.19, 7.8.17.
Inverse trigonometric operations:
- Reading from (mostly) the textbook:
  - Section 8.1 (pages 622–631);
  - Section 8.2 (pages 635–638);
  - My handout on inverse trigonometric operations (DjVu).
- Exercises due October 10 Thursday: Fill in all of these blanks with algebraic expressions (most 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 either side exists).
- Exercises from the textbook due October 15 Tuesday on MyLab: 8.1.19, 8.1.21, 8.2.11, 8.2.13, 8.2.19, 8.1.39, 8.1.41, 8.1.43, 8.1.45, 8.1.51, 8.1.53, 8.1.55, 8.1.57, 8.2.33, 8.2.35, 8.2.47, 8.2.49, 8.1.59, 8.1.61, 8.2.61, 8.2.63, 8.2.65.
Sum-angle and related formulas:
- Reading from the textbook:
  - Section 8.5 Objectives 1–3 (pages 659–666);
  - Section 8.6 through Example 2 in Objective 2 (pages 672&674).
- Exercises due October 15 Tuesday: 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. sin(α − β) = ___.
  4. cos(α − β) = ___.
- Exercises from the textbook due October 17 Thursday on MyLab: 8.5.15, 8.5.17, 8.5.19, 8.5.21, 8.5.35, 8.5.37, 8.5.39, 8.5.41, 8.5.77, 8.5.83, 8.5.85.
Half-angle and related formulas:
- Reading from the textbook:
  - The rest of Section 8.6 Objective 2 through Example 3 (pages 674&675);
  - Section 8.6 Objective 3 (pages 676–678);
  - Optional: Section 8.7 (pages 683–685).
- Exercises due October 17 Thursday: Fill in the blanks with trigonometric expressions in which each trigonometric operation that appears is only applied directly to α. Make sure that each expression has at most one value for each value of α; in other words, do not use ±.
  1. sin²(α/2) = ___.
  2. cos²(α/2) = ___.
  3. tan(α/2) = ___ (notice not squared).
- Exercises from the textbook due October 24 Thursday on MyLab: 8.6.9, 8.6.11, 8.6.13, 8.6.15, 8.6.17, 8.6.19, 8.6.25, 8.6.29, 8.6.23, 8.6.27.
Simplifying trigonometric expressions:
- Reading from (mostly) the textbook:
  - My handout on simplifying trigonometric expressions (DjVu);
  - Section 8.4 (pages 651–656).
- Exercises due October 24 Thursday:
  1. Fill in the blank with an expression in which sin θ is the only trigonometric quantity: cos² θ = ___.
  2. Factor without using any trigonometric identities: sin² θ − 1 = (___)(___).
  3. If you regard a cosine as a square root, then what expression is conjugate to 1 − cos θ?
- Exercises from the textbook due October 29 Tuesday on MyLab: 8.4.1, 8.4.2, 8.4.6, 8.4.8, 8.4.11, 8.4.15, 8.4.17, 8.4.29, 8.4.55, 8.4.71, 8.4.95.
Trigonometric equations:
- Reading from the textbook:
  - Section 8.3 (pages 641–646);
  - Section 8.5 Objective 4 (pages 666–668);
  - Section 8.6 Objective 2 Examples 4&5 (pages 675&676).
- Exercises due October 29 Tuesday:
  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?
- Exercises from the textbook due October 31 Thursday on MyLab: 8.3.13, 8.3.23, 8.3.25, 8.3.27, 8.3.37, 8.3.39, 8.3.61, 8.3.73, 8.3.109, 8.3.115.

Quiz/Exam 2, covering the material above, is on November 5 Tuesday.

Applications

Solving right triangles:
- Reading from the textbook: Section 9.1 (pages 694–696).
- Exercises due October 31 Thursday:
  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.
- Exercises from the textbook due November 7 Thursday on MyLab: 9.1.4, 9.1.11, 9.1.13, 9.1.15, 9.1.17, 9.1.19, 9.1.21, 9.1.23, 9.1.37.
The Law of Sines:
- Reading from the textbook: Section 9.2 (pages 700–706).
- Exercises due November 7 Thursday: 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 A, B, and C are the measures of the respective opposite angles).
  1. a ÷ sin A = b ÷ ___.
  2. b ÷ c = sin B ÷ ___.
  3. sin A ÷ a = sin C ÷ ___.
- Exercises from the textbook due November 12 Tuesday on MyLab: 9.2.9, 9.2.11, 9.2.13, 9.2.15, 9.2.18, 9.2.27, 9.2.29, 9.2.33, 9.2.35, 9.2.37.
The Law of Cosines:
- Reading from (mostly) the textbook:
  - Section 9.3 (pages 711–714);
  - My handout on solving triangles (DjVu).
- Exercises due November 12 Tuesday:
  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?
- Exercises from the textbook due November 14 Thursday on MyLab: 9.3.9, 9.3.11, 9.3.13, 9.3.15, 9.2.39, 9.3.45.
Area of triangles:
- Reading from the textbook: Section 9.4 (pages 718–720).
- Exercises due November 14 Thursday:
  1. If two sides of a triangle have lengths a and b and the angle between them has measure C, then what is the area of the triangle?
  2. If a triangle's sides have lengths a, b, and c, then what is the area of the triangle? (Express this using only a, b, c, and non-trigonometric operations. You may use the perimeter or semiperimeter as well, if you find it convenient, but then you must state what it is using only a, b, and c.)
- Exercises from the textbook due November 19 Tuesday on MyLab: 9.4.9, 9.4.11, 9.4.13, 9.4.15, 9.4.17, 9.4.19, 9.4.21, 9.4.25, 9.4.27, 9.4.37, 9.4.43, 9.4.53.
Polar coordinates:
- Reading from the textbook: Section 10.1 (pages 740–747).
- Exercises due November 19 Tuesday:
  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 each 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.
- Exercises from the textbook due November 21 Thursday on MyLab: 10.1.13–20, 10.1.21, 10.1.23, 10.1.25, 10.1.27, 10.1.31, 10.1.33, 10.1.35, 10.1.45, 10.1.47, 10.1.49, 10.1.51, 10.1.53, 10.1.59, 10.1.63.
Graphing in polar coordinates:
- Reading from the textbook:
  - Section 10.2 (pages 749–761);
  - Optional: Section 10.3 (pages 764–771).
- Exercises due November 21 Thursday:
  1. Let a be a positive number, and consider the circle given in polar coordinates by r = 2a sin θ. The radius of this circle is ___, and its centre is (___, ___) in rectangular coordinates.
  2. Let n be a positive integer, and consider the rose curve given in polar coordinates by r = sin(nθ). If n is even, then this rose has ___ petals; if n is odd, then it has ___ petals.
- Exercises from the textbook due November 26 Tuesday on MyLab: 10.2.15, 10.2.17, 10.2.19, 10.2.21, 10.2.23, 10.2.31–38, 10.2.39, 10.2.43, 10.2.47, 10.2.51, 10.2.55, 10.2.59.
Vectors:
- Reading from the textbook: Section 10.4 (pages 773–783).
- Exercises due November 26 Tuesday:
  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⟩.
- Exercises from the textbook due December 3 Tuesday on MyLab: 10.4.11, 10.4.13, 10.4.15, 10.4.17, 10.4.27, 10.4.29, 10.4.37, 10.4.39, 10.4.43, 10.4.45, 10.4.49, 10.4.51, 10.4.61, 10.4.63, 10.4.65, 10.4.67, 10.4.69.
Vectors and angles:
- Reading from the textbook: Section 10.5 (pages 788–793).
- Exercises due December 3 Tuesday:
  1. State a formula for the dot product u ⋅ v of two vectors using only their lengths |u| and |v|, the angle θ = ∠(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.
- Exercises from the textbook due December 5 Thursday on MyLab: 10.5.9, 10.5.11, 10.5.13, 10.5.15, 10.5.17, 10.5.19, 10.5.21, 10.5.23, 10.5.25.

Quiz/Exam 3, covering the material above, is on December 5 Thursday.

Final

A comprehensive final exam is on December 18 Wednesday from 1:00 PM to 3:00 PM. The exam will consist of questions similar in style and content to those in the practice final exam (DjVu).

Go back to the the course homepage.

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