# MATH-1300-ES31

Welcome to the permanent home page for Section ES31 of MATH-1300 (Precalculus) at Southeast Community College in the Fall term of 2022. I am Toby Bartels, your instructor.

## Contact information

Feel free to send a message at any time, even nights and weekends (although I'll be slower to respond then).

The official textbook for the course is the 11th Edition of Precalculus by Sullivan published by Prentice-Hall (Pearson). You automatically get an online version of this textbook through Canvas, although you can use a print version instead if you like. This comes with access to Pearson MyLab, integrated into Canvas, on which many of the assignments appear.

### Graphs and functions

1. General review:
• My online introduction;
• Skim Appendix A (except Section A.4) and review anything that you are shaky on.
• Exercises due on January 11 Tuesday (submit these here on Canvas or in class):
1. Which of the following are equations?
1. 2x + y;
2. 2x + y = 0;
3. z = 2x + y.
2. You probably don't know how to solve the equation x5 + 2x = 1, but show what numerical calculation you make to check whether x = 1 is a solution.
3. Write the set {x | x < 3} in interval notation and draw a graph of the set.
4. Suppose that ax2 + bx + c = 0 but a ≠ 0; write down a formula for x.
• Exercises from the textbook due on January 12 Wednesday (submit these through MyLab in the Next item): O.1.1, O.1.2, O.1.3, O.1.4, O.1.5, O.1.6, O.1.7, O.1.8, O.1.10, O.1.11, O.1.12, A.6.25, A.6.43, A.6.75, A.6.99, A.7.63, A.9.71, A.9.75, A.8.33, A.8.47.
2. Graphing points:
• Reading: Section 1.1 (pages 2–6) from the textbook.
• Exercises due on January 12 Wednesday (submit these here on Canvas or in class):
1. Fill in the blanks with vocabulary words: The two number lines that mark the coordinates in a rectangular coordinate system are the coordinate _____, and the point where they intersect is the _____.
2. Fill in the blank with a number: If the legs of a right triangle have lengths 3 and 4, then the length of its hypotenuse is ___.
3. Fill in the blanks with algebraic expressions: The distance between the points (x1, y1) and (x2, y2) is _____, and the midpoint between them is (___, ___).
• Exercises from the textbook due on January 13 Thursday (submit these through MyLab in the Next item): 1.1.4, 1.1.15, 1.1.17, 1.1.19, 1.1.21, 1.1.23, 1.1.27, 1.1.33, 1.1.39, 1.1.43, 1.1.47, 1.1.63, 1.1.71.
3. Graphing equations:
• Exercises due on January 13 Thursday (submit these here on Canvas or in class): Fill in the blanks with vocabulary words:
1. Given a graph in a coordinate plane, a point on the graph that lies on at least one coordinate axis is a(n) _____ of that graph.
2. If for each point (x, y) on a graph, the point (−x, −y) is also on the graph, then the graph is symmetric with respect to the _____.
3. The slope of a vertical line is _____, and the slope of a horizontal line is _____.
• Exercises from the textbook due on January 18 Tuesday (submit these through MyLab in the Next item): 1.2.1, 1.2.2, 1.2.7, 1.2.13, 1.2.17, 1.2.23, 1.2.29, 1.2.31, 1.2.33, 1.2.35, 1.2.41, 1.2.43, 1.2.45, 1.2.47, 1.2.53, 1.2.55, 1.2.61, 1.2.67, 1.2.71, 1.2.77.
4. Linear equations:
• The rest of Section 1.3 (pages 28–31);
• Section 11.1 (pages 720–730);
• My online notes and video on systems of equations.
• Exercises due on January 18 Tuesday (submit these here on Canvas or in class):
1. Fill in the blanks with numbers: Suppose that a line L has slope 2. The slope of any line parallel to L is ___, and the slope of any line perpendicular to L is ___.
2. Consider the system of equations consisting of x + 3y = 4 (equation 1) and 2x + 3y = 5 (equation 2).
1. If I solve equation (1) for x to get x = 4 − 3y and apply this to equation (2) to get 2(4 − 3y) + 3y = 5 (and continue from there), then what method am I using to solve this system?
2. If instead I multiply equation (1) by −2 to get −2x − 6y = −8 and combine this with equation (2) to get −3y = −3 (and continue from there), then what method am I using to solve this system?
• Exercises from the textbook due on January 19 Wednesday (submit these through MyLab in the Next item): 1.3.2, 1.3.7, 1.3.8, 1.3.13, 1.3.15, 1.3.17, 1.3.19, 1.3.21, 1.3.23, 1.3.25, 1.3.27, 1.3.29, 1.3.31, 1.3.45, 1.3.51, 1.3.53, 1.3.57, 1.3.63, 1.3.67, 1.3.73, 1.3.75, 1.3.79, 1.3.85, 1.3.91, 1.3.93, 1.3.111, 1.3.113, 11.1.3, 11.1.4, 11.1.6, 11.1.11, 11.1.19, 11.1.21, 11.1.27, 11.1.31, 11.1.45, 11.1.47, 11.1.65, 11.1.73.
5. Functions:
• Section 2.1 (pages 47–59);
• Most of Section 2.2 (pages 63–67), but you may skip parts D and E of Example 4;
• My online notes on functions.
• Exercises due on January 19 Wednesday (submit these here on Canvas or in class):
1. Fill in the blank with a mathematical expression: If g(x) = 2x + 3 for all x, then g(___) = 2(5) + 3 = 13.
2. Fill in the blanks with mathematical expressions: If (3, 5) is a point on the graph of a function f, then f(___) = ___.
3. Fill in the blank with a geometric word: The graph of a relation is the graph of a function if and only if every _____ line goes through the graph at most once.
4. True or false: The graph of a function can have any number of x-intercepts.
5. True or false: The graph of a function can have any number of y-intercepts.
• Exercises from the textbook due on January 20 Thursday (submit these through MyLab in the Next item): 2.1.1, 2.1.2, 2.1.3, 2.1.31, 2.1.33, 2.1.35, 2.1.43, 2.1.49, 2.1.103, 2.2.7, 2.2.9, 2.2.11, 2.2.13, 2.2.15, 2.2.17, 2.2.19, 2.2.21, 2.2.27, 2.2.29, 2.2.31, 2.2.33, 2.2.39, 2.2.45, 2.2.47.
6. Properties of functions:
• Exercises due on January 20 Thursday (submit these here on Canvas or in class): Fill in the blanks with vocabulary words:
1. If f(3) = 5, then 3 belongs to the _____ of the function, and 5 belongs to its _____.
2. Suppose that f is a function and, whenever f(x) exists, then f(−x) also exists and equals f(x). Then f is _____.
3. If c is a number and f is a function, and if f(c) = 0, then c is a(n) _____ of f.
4. Suppose that a function f is defined on (at least) a nontrivial interval I and that, whenever a ∈ I and b ∈ I, if a < b, then f(a) < f(b). Then f is (strictly) _____ on I.
• Exercises from the textbook due on January 24 Monday (submit these through MyLab in the Next item): 2.1.10, 2.1.51, 2.1.53, 2.1.55, 2.1.59, 2.1.63, 2.1.71, 2.1.79, 2.1.81, 2.3.2, 2.3.3, 2.3.5, 2.3.13, 2.3.15, 2.3.17, 2.3.19, 2.3.21, 2.3.23, 2.3.26, 2.3.27, 2.3.29, 2.3.31, 2.3.37, 2.3.39, 2.3.41, 2.3.43, 2.3.45, 2.3.49, 2.3.51.
7. Word problems with functions:
• Exercise due on January 24 Monday (submit this here on Canvas or in class): Suppose that you have a problem with three quantities, A, B, and C; and suppose that you have two equations, equation (1) involving A and B, and equation (2) involving B and C. If you wish to find A as a function of C, then which equation should you solve first, and which variable should you solve it for? (Although there is a single best answer in my opinion, there is more than one answer that will progress the solution, and I'll accept either of them.)
• Exercises from the textbook due on January 25 Tuesday (submit these through MyLab in the Next item): 2.6.5, 2.6.13, 2.6.15, 2.6.21, 2.6.23.
8. Examples of functions:
• Section 2.4 through Objective 1 (pages 86–90);
• My online notes and video on partially-defined functions;
• The rest of Section 2.4 (pages 91–93).
• Exercises due on January 25 Tuesday (submit these here on Canvas or in class): Fill in the blanks with vocabulary words:
1. In the _____ function, the output is always defined and equal to the input.
2. A _____-defined function is defined by a formula together with a condition restricting its inputs.
3. A _____-defined function is defined by more than one formula, each with a condition restricting its inputs.
• Exercises from the textbook due on January 26 Wednesday (submit these through MyLab in the Next item): 2.4.9, 2.4.10, 2.4.11–18, 2.4.19, 2.4.20, 2.4.21, 2.4.22, 2.4.23, 2.4.24, 2.4.25, 2.4.26, 2.4.27, 2.4.29, 2.4.31, 2.4.33, 2.4.35, 2.4.43, 2.4.45, 2.4.51.
Quiz 1, covering the material in Problem Sets 1–8, is on January 31 Monday.

### Polynomial functions

1. Linear functions:
• Reading: Section 3.1 (pages 125–131).
• Exercises due on January 26 Wednesday (submit these here on Canvas or in class):
1. Suppose that y is a linear function of x. If the rate of change of the function is m and the initial value of the function is b, then write an equation relating x and y.
2. Suppose that f is a linear function. If you know f(a) and f(b) for two distinct real numbers a and b, then give a formula for the slope of the graph of f using a, b, f(a), and f(b).
• Exercises from the textbook due on January 27 Thursday (submit these through MyLab in the Next item): 3.1.2, 3.1.13, 3.1.15, 3.1.17, 3.1.19, 3.1.21, 3.1.23, 3.1.25, 3.1.27, 3.1.37, 3.1.43, 3.1.45, 3.1.47, 3.1.49.
2. Composite and inverse functions:
• Exercises due on February 1 Tuesday (submit these here on Canvas or in class):
1. Fill in the blanks with a vocabulary word and a mathematical expression: If f and g are functions, then their _____ function, denoted f ∘ g, is defined by (f ∘ g)(x) = _____.
2. Fill in the blanks with mathematical expressions: A number x is in the domain of f ∘ g if and only if ___ belongs to the domain of g and ___ belongs to the domain of f.
3. Fill in the blank with a geometric word: A function is one-to-one if and only if every _____ line goes through its graph at most once.
4. Fill in the blank with a vocabulary word: If f is a one-to-one function, then its _____ function, denoted f−1, exists.
5. Fill in the blank with an ordered pair: If f is one-to-one and (2, −3) is on the graph of f, then ___ is on the graph of f−1.
• Exercises from the textbook due on February 2 Wednesday (submit these through MyLab in the Next item): 5.1.2, 5.1.9, 5.1.11, 5.1.15, 5.1.19, 5.1.25, 5.1.27, 5.1.29, 5.1.33, 5.1.55, 5.2.4, 5.2.5, 5.2.7, 5.2.8, 5.2.9, 5.2.12, 5.2.21, 5.2.23, 5.2.25, 5.2.27, 5.2.29, 5.2.31, 5.2.35, 5.2.37, 5.2.41, 5.2.43, 5.2.45, 5.2.55, 5.2.57, 5.2.59, 5.2.61, 5.2.75, 5.2.77, 5.2.79, 5.2.87.
3. Coordinate transformations:
• Exercises due on February 2 Wednesday (submit these here on Canvas or in class): Assume that the axes are oriented in the usual way (positive x-axis to the right, positive y-axis upwards).
1. Fill in the blank with a direction: To change the graph of y = f(x) into the graph of y = f(x − 1), shift the graph to the ___ by 1 unit.
2. To change the graph of y = f(x) into the graph of y = −f(x), do you reflect the graph left and right or up and down?
3. To change the graph of y = f(x) into the graph of y = f(2x), do you compress or stretch the graph left and right?
• Exercises from the textbook due on February 3 Thursday (submit these through MyLab in the Next item): 2.5.5, 2.5.6, 2.5.7–10, 2.5.11–14, 2.5.15–18, 2.5.19, 2.5.21, 2.5.23, 2.5.25, 2.5.29, 2.5.30, 2.5.33, 2.5.35, 2.5.37, 2.5.41, 2.5.43, 2.5.45, 2.5.47, 2.5.53, 2.5.61, 2.5.63, 2.5.73, 2.5.89.
• Exercises due on February 3 Thursday (submit these here on Canvas or in class):
1. Fill in the blank with a vocabulary word: The shape of the graph of a nonlinear quadratic function is a(n) _____.
2. Fill in the blanks with algebraic expressions: Given a ≠ 0 and f(x) = ax2 + bx + c for all x, the vertex of the graph of f is (___, ___).
3. Given a ≠ 0, b2 − 4ac > 0, and f(x) = ax2 + bx + c for all x, how many x-intercepts does the graph of y = f(x) have?
• Exercises from the textbook due on February 7 Monday (submit these through MyLab in the Next item): 3.3.1, 3.3.2, 3.3.3, 3.3.4, 3.3.15–22, 3.3.31, 3.3.33, 3.3.43, 3.3.49, 3.3.53, 3.3.57, 3.3.61, 3.3.63, 3.3.67, 3.3.70.
• Exercises due on February 7 Monday (submit these here on Canvas or in class):
1. If you make and sell x items per year at a price of p dollars per item, then what is your revenue (in dollars per year)?
2. If a business's revenue is R dollars per year and its costs are C dollars per year, then what is its profit (in dollars per year)?
3. If the width of a rectangle is w metres and its length is l metres, then what is its area (in square metres)?
• Exercises from the textbook due on February 8 Tuesday (submit these through MyLab in the Next item): 3.3.87, 3.3.89, 3.3.93, 3.3.95, 3.4.3, 3.4.5, 3.4.7, 3.4.9, 3.4.11, 3.4.13, 3.4.15.
6. Polynomial functions:
• Section 4.1 (pages 175–186);
• My online notes on graphing polynomials (but the last paragraph is optional);
• Section 4.2 through Objective 1 (pages 190–192).
• Exercises due on February 8 Tuesday (submit these here on Canvas or in class):
1. Give the coordinates of a point on the graph of every power function, another point (different from the previous point) on the graph of every power function with a positive exponent, another point on the graph of every power function with an even exponent, and another point on the graph of every power function with an odd exponent.
2. If a root (zero) of a polynomial function has odd multiplicity, does the graph cross (go through) or only touch (bounce off) the horizontal axis at the intercept given by that root? Which does the graph do if the root has even multiplicity?
• Exercises from the textbook due on February 9 Wednesday (submit these through MyLab in the Next item): 4.1.1, 4.1.2, 4.1.11, 4.1.15, 4.1.17, 4.1.19, 4.1.21, 4.1.27, 4.1.29, 4.1.33, 4.1.41, 4.1.43, 4.1.47, 4.1.49, 4.1.59, 4.1.61, 4.1.69, 4.1.71, 4.1.73, 4.1.75, 4.2.1, 4.2.2, 4.2.5, 4.2.11.
• Section A.4 (pages A31–A34);
• Section 4.6 through Objective 1 (pages 231–234);
• Section 4.6 Objectives 3–5 (pages 235–239);
• Section 4.7 (pages 245–250).
• Exercises due on February 9 Wednesday (submit these here on Canvas or in class):
1. Suppose that f is a polynomial function and c is a number. If you divide f(x) by x − c, then what will the remainder be?
2. Suppose that f is a polynomial function with rational coefficients and c is an integer. If x − c is a factor of f(x), then what is f(c)?
3. Suppose that f is a polynomial function with real coefficients and a and b are real numbers with b ≠ 0. If the imaginary complex number a + bi is a root (or zero) of f, then what other number must be a root of f?
4. What polynomial in x (with real coefficients) must be a factor of f(x)?
• Exercises from the textbook due on February 10 Thursday (submit these through MyLab in the Next item): 4.6.2, 4.6.3, 4.6.4, 4.6.11, 4.6.15, 4.6.19, 4.6.33, 4.6.35, 4.6.37, 4.6.45, 4.6.51, 4.6.53, 4.6.57, 4.6.59, 4.6.65, 4.6.67, 4.6.93, 4.6.99, 4.6.101, 4.7.1, 4.7.2, 4.7.9, 4.7.11, 4.7.13, 4.7.15, 4.7.17, 4.7.19, 4.7.21, 4.7.23, 4.7.25, 4.7.29, 4.7.35, 4.7.39.
8. Rational functions:
• Section 4.3 (pages 198–205);
• Section 4.4 through Objective 1 (pages 209–219);
• My online notes on rational functions.
• Exercises due on February 10 Thursday (submit these here on Canvas or in class):
1. If a graph gets arbitrarily close to a line (without necessarily reaching it) in some direction, then the line is a(n) _____ of the graph.
2. If the reduced form of a rational function is defined somewhere where the original (unreduced) form is not, then the graph of the original function has a(n) _____ there.
• Exercises from the textbook due on February 14 Monday (submit these through MyLab in the Next item): 4.3.2, 4.3.3, 4.3.4, 4.3.15, 4.3.17, 4.3.19, 4.3.23, 4.3.27, 4.3.29, 4.3.31, 4.3.35, 4.3.45, 4.3.47, 4.3.49, 4.3.51, 4.4.1, 4.4.5, 4.4.7, 4.4.9, 4.4.11, 4.4.17, 4.4.19, 4.4.21, 4.4.23, 4.4.31, 4.4.33, 4.4.35, 4.4.51, 4.4.53.
9. Inequalities:
• Exercise due on February 14 Monday (submit this here on Canvas or in class): Suppose that you have a rational inequality in one variable that you wish to solve. You investigate the inequality and discover the following facts about it:
• the left-hand side is always defined;
• the right-hand side is undefined when x is 2 but is otherwise defined;
• the left-hand side and right-hand side are equal when x is −3/2 and only then;
• the original inequality is true when x is −3/2 or 3 but false when x is −2, 0, or 2.
What are the solutions to the inequality?
• Exercises from the textbook due on February 15 Tuesday (submit these through MyLab in the Next item): 4.5.1, 4.5.5, 4.5.7, 4.5.9, 4.5.13, 4.5.15, 4.5.19, 4.5.21, 4.5.23, 4.5.27, 4.5.29, 4.5.35, 4.5.39, 4.5.41, 4.5.43, 4.5.47.
Quiz 2, covering the material in Problem Sets 9–17, is on February 21 Monday.

### Transcendental functions

1. Exponential functions:
• Exercises due on February 15 Tuesday (submit these here on Canvas or in class): Let f(x) be Cbx for all x.
1. What is f(x + 1)/f(x)?
2. What are f(−1), f(0), and f(1)?
(Write your answers using b and C, and simplify them as much as possible.)
• Exercises from the textbook due on February 16 Wednesday (submit these through MyLab in the Next item): 5.3.1, 5.3.15, 5.3.16, 5.3.21, 5.3.23, 5.3.25, 5.3.27, 5.3.29, 5.3.31, 5.3.33, 5.3.35, 5.3.37–44, 5.3.45, 5.3.47, 5.3.51, 5.3.53, 5.3.57, 5.3.59, 5.3.61, 5.3.65, 5.3.67, 5.3.71, 5.3.73, 5.3.76, 5.3.77, 5.3.79, 5.3.83, 5.3.85, 5.3.91, 5.3.93.
2. Logarithmic functions:
• Exercises due on February 16 Wednesday (submit these here on Canvas or in class): Suppose that b > 0 and b ≠ 1.
1. Rewrite logb(M) = r as an equation involving exponentiation.
2. What are logb(b), logb(1), and logb(1/b)?
• Exercises from the textbook due on February 17 Thursday (submit these through MyLab in the Next item): 5.4.11, 5.4.13, 5.4.15, 5.4.17, 5.4.19, 5.4.21, 5.4.23, 5.4.25, 5.4.27, 5.4.29, 5.4.31, 5.4.33, 5.4.35, 5.4.37, 5.4.39, 5.4.43, 5.4.51, 5.4.53, 5.4.55, 5.4.57, 5.4.65–72, 5.4.73, 5.4.79, 5.4.83, 5.4.85, 5.4.89, 5.4.91, 5.4.93, 5.4.95, 5.4.97, 5.4.99, 5.4.101, 5.4.103, 5.4.105, 5.4.107, 5.4.109, 5.4.111, 5.4.119, 5.4.129, 5.4.131.
3. Properties of logarithms:
• Section 5.5 (pages 309–315);
• My online notes on laws of logarithms;
• Section 5.6 through Objective 2 (pages 318–321).
• Exercises due on February 22 Tuesday (submit these here on Canvas or in class):
1. Fill in the blanks to break down these expressions using properties of logarithms. (Assume that b, u, and v are all positive and that b ≠ 1.)
1. logb (uv) = ___;
2. logb (u/v) = ___;
3. logb (ux) = ___.
2. In solving which of the following equations would it be useful to have a step in which you take logarithms of both sides of the equation? (Say Yes or No for each one.)
1. log2 (x + 3) = 5;
2. (x + 3)2 = 5;
3. 2x + 3 = 5.
• Exercises from the textbook due on February 23 Wednesday (submit these through MyLab in the Next item): 5.5.7, 5.5.11, 5.5.13, 5.5.15, 5.5.17, 5.5.19, 5.5.21, 5.5.23, 5.5.25, 5.5.27, 5.5.37, 5.5.39, 5.5.41, 5.5.43, 5.5.45, 5.5.47, 5.5.49, 5.5.51, 5.5.53, 5.5.55, 5.5.57, 5.5.61, 5.5.63, 5.5.65, 5.5.67, 5.5.69, 5.5.71, 5.5.73, 5.5.75, 5.5.78, 5.5.87, 5.5.91, 5.5.97, 5.6.1, 5.6.2, 5.6.5, 5.6.7, 5.6.9, 5.6.15, 5.6.19, 5.6.21, 5.6.23, 5.6.25, 5.6.27, 5.6.29, 5.6.31, 5.6.39, 5.6.43, 5.6.45, 5.6.49, 5.6.57, 5.6.61.
4. Compound interest:
• Exercises due on February 23 Wednesday (submit these here on Canvas or in class):
1. The original amount of money that earns interest is the _____.
2. If you borrow P dollars at 100r% annual interest compounded n times per year, then how much will you owe after t years (if you make no payments)?
• Exercises from the textbook due on February 24 Thursday (submit these through MyLab in the Next item): 5.7.1, 5.7.2, 5.7.7, 5.7.11, 5.7.13, 5.7.15, 5.7.21, 5.7.31, 5.7.33, 5.7.41, 5.7.43.
5. Applications of logarithms:
• Exercise due on February 24 Thursday (submit this here on Canvas or in class): Suppose that a quantity A undergoes exponential growth with a relative growth rate of k and an initial value of A0 at time t = 0. Write down a formula for the value of A as a function of the time t.
• Exercises from the textbook due on February 28 Monday (submit these through MyLab in the Next item): 5.8.1, 5.8.3, 5.8.5, 5.8.7, 5.8.9, 5.8.11, 5.8.13, 5.8.15, 5.8.17, 5.8.19, 5.8.21, 5.8.23.
6. Circles:
• Reading from the textbook: Section 1.4 (pages 35–39).
• Exercises due on February 28 Monday (submit these here on Canvas or in class):
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 on March 1 Tuesday (submit these through MyLab in the Next item): 1.4.5, 1.4.9, 1.4.11, 1.4.13, 1.4.15, 1.4.17, 1.4.21, 1.4.23, 1.4.25, 1.4.27.
7. Angles:
• Reading from the textbook: Section 6.1 (pages 362–370).
• Exercises due on March 1 Tuesday (submit these here on Canvas or in class):
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 blanks with algebraic expressions:
1. In a circle of radius r, a central angle whose measure is θ radians subtends an arc whose length is s = ___.
2. In a circle of radius r, a central angle whose measure is θ forms a sector whose area is A = ___.
3. Around a circle of radius r, an object with an angular speed of ω has a linear speed of v = ___.
• Exercises from the textbook due on March 2 Wednesday (submit these through MyLab in the Next item): 6.1.11, 6.1.13, 6.1.15, 6.1.17, 6.1.19, 6.1.21, 6.1.23, 6.1.26, 6.1.35, 6.1.37, 6.1.71, 6.1.73, 6.1.79, 6.1.81, 6.1.87, 6.1.91, 6.1.95, 6.1.99.
8. The trigonometric operations:
• Section 6.2 through Objective 2 (pages 375–380);
• Section 6.2 Objectives 6&7 (pages 385–387).
• Exercises due on March 2 Wednesday (submit these here on Canvas or in class):
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 (x, y), express sin t, cos t, tan t, cot t, sec t, and csc t using only x and y.
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.
3. If you want to calculate the secant of 50 degrees on a calculator with buttons only for sine, cosine, and tangent, then what do you enter on the calculator?
• Exercises from the textbook due March 3 Thursday (submit these through MyLab in the Next item): 6.2.13, 6.2.15, 6.2.17, 6.2.19, 6.2.65, 6.2.67, 6.2.69, 6.2.71, 6.2.77, 6.2.79.
9. Right triangles:
• Reading from the textbook: Section 8.1 through Objective 3 (pages 522–525).
• Exercises due on March 3 Thursday (submit these here on Canvas or in class):
1. If θ is the measure of an acute angle in a right triangle, then express the six basic trigonometric functions of θ as ratios of the lengths of the adjacent leg, the opposite leg, and the hypotenuse.
2. Fill in the blank: The sine of the complement of θ is the _____ of θ.
3. If the secant of an angle is 3, then what is its cosine?
• Exercises from the textbook due on March 7 Monday (submit these through MyLab in the Next item): 8.1.9, 8.1.11, 8.1.13, 8.1.19, 8.1.21, 8.1.23.
10. Special angles:
• Reading from the textbook: The rest of Section 6.2 (pages 380–385).
• Exercises due on March 7 Monday (submit these here on Canvas or in class):
1. Write down the exact values of the sine, cosine, tangent, cotangent, secant, and cosecant of 0, π/6, π/4, π/3, and π/2. (This is 30 values to write down in all, which you might put into a handy table. One way or another, be sure to label which value is which.)
2. Two angles that differ by one or more full turns are called _____ angles.
3. 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.
4. For each of Quadrants I, II, III, and IV, Which of the six fundamental trigonometric functions of θ are positive and which are negative when θ terminates in that quadrant? (This is 24 positive/negative answers in all, which you might also put into a table.)
• Exercises from the textbook due March 8 Tuesday (submit these through MyLab in the Next item): 6.2.31, 6.2.33, 6.2.35, 6.2.41, 6.2.43, 6.2.45, 6.2.47, 6.2.49, 6.2.51, 6.2.53, 6.2.55, 6.3.11, 6.3.19, 6.3.31.
Quiz 3, covering the material in Problem Sets 18–27, is on March 10 Thursday.

### Analytic trigonometry

1. The trigonometric functions:
• Reading from the textbook: Section 6.3 (pages 392–403).
• Exercises due on March 8 Tuesday (submit these here on Canvas or in class):
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 on March 9 Wednesday (submit these through MyLab in the Next item): 6.4.3, 6.3.35, 6.3.37, 6.3.43, 6.3.45, 6.3.53, 6.3.55, 6.3.89, 6.3.115.
2. Graphs of the trigonometric functions:
• Section 6.4 through the box before Example 1 (pages 407&408);
• Section 6.4 Objective 2 through the box before Example 3 (pages 409&410).
• Section 6.5 through "The Graph of the Cotangent Function y = cot x" (pages 422–424);
• Section 6.5 Objective 3 (pages 425&426).
• Exercises due on March 21 Monday (submit these here on Canvas or in class):
1. List at least five consecutive intercepts of the graphs of the sine and tangent functions.
2. List at least five consecutive intercepts of the graphs of both the cosine and cotangent functions.
3. Give the one point that's an intercept of the graphs of both the cosine and secant functions.
4. List at least five consecutive turning points of the graphs of the sine and cosecant functions.
5. List at least five consecutive turning points of the graphs of the cosine and secant functions.
6. List at least five consecutive linear asymptotes to the graphs of the tangent and secant functions.
7. List at least five consecutive linear asymptotes to the graphs of the cotangent and cosecant functions.
• Exercises from the textbook due on March 22 Tuesday (submit these through MyLab in the Next item): 6.4.6, 6.4.8, 6.5.3, 6.5.6, 6.5.7, 6.5.10, 6.5.11, 6.5.12, 6.5.13, 6.5.16.
3. Sinusoidal functions:
• Reading from (mostly) the textbook:
• My handout on sinusoidal functions (DjVu);
• Section 6.4 Objective 1 Examples 1&2 (pages 408&409);
• Section 6.4 Objective 2 from Example 3 (pages 410–416);
• Section 6.5 Objective 2 (pages 424&425);
• Section 6.5 Objective 4 (pages 426&427);
• Section 6.6 Objective 1 (pages 429–433).
• Exercises due March 22 Tuesday (submit these here on Canvas or in class):
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 ω > 0 and 0 ≤ φ < 2π, then what is the phase shift of f?
• Exercises from the textbook due on on March 23 Wednesday (submit these through MyLab in the Next item): 6.4.11, 6.4.13, 6.4.23–32, 6.4.35, 6.4.39, 6.4.51, 6.4.57, 6.4.61, 6.4.87, 6.5.17, 6.5.21, 6.5.23, 6.5.25, 6.5.29, 6.5.31, 6.6.9, 6.6.11, 6.6.17, 6.6.19.
4. Inverse trigonometric operations:
• Exercises due on March 23 Wednesday (submit these here on Canvas or in class): Fill in all of these blanks with algebraic expressions (or constants). Work only in the real number system.
1. That y = sin−1x means that x = ___ and ___ ≤ y ≤ ___.
2. cos−1x exists if and only if ___ ≤ x ≤ ___.
3. cos−1 cos θ = θ if and only if ___ ≤ θ ≤ ___.
4. cos sin−1x = ___ (if either side exists).
5. If f is the function given by f(x) = sin−1x, then what is its inverse function f −1? (Write down a formula that involves one or more of the six basic trigonometric operations and that includes all necessary conditions.)
• Exercises from the textbook due on March 24 Thursday (submit these through MyLab in the Next item): 7.1.19, 7.1.21, 7.1.39, 7.1.41, 7.1.43, 7.1.45, 7.1.51, 7.1.53, 7.1.55, 7.1.57, 7.1.59, 7.1.61, 7.2.11, 7.2.13, 7.2.19, 7.2.33, 7.2.35, 7.2.47, 7.2.49, 7.2.61, 7.2.63, 7.2.65.
5. Sum-angle formulas:
• Section 7.5 through Objective 3 (pages 487–494);
• Section 7.6 through Example 3 in Objective 2 (pages 500&503).
• Exercises due on March 24 Thursday (submit these here on Canvas or in class): 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. tan(α + β) = ___.
5. sin(2α) = ___.
6. Express sin2α using sin(2α) and/or cos(2α).
• Exercises from the textbook due on March 28 Monday (submit these through MyLab in the Next item): 7.5.15, 7.5.17, 7.5.19, 7.5.21, 7.5.35, 7.5.37, 7.5.39, 7.5.41, 7.5.77, 7.6.83, 7.6.85, 7.6.87.
• Section 7.7 (pages 511–513);
• Section 7.6 Objective 3 (pages 504–506).
• Exercises due on March 28 Monday (submit these here on Canvas or in class):
1. Express sin2(α/2) and cos2(α/2) using sin α and/or cos α.
2. tan(α/2) = ___ (notice not squared).
3. Express sin α sin β using sin(α + β), sin(α − β), cos(α + β), and/or cos(α − β).
4. Factor sin α + sin β so that each factor has at most one trigonometric operation.
• Exercises from the textbook due on March 29 Tuesday (submit these through MyLab in the Next item): 7.6.25, 7.6.29, 7.6.23, 7.6.27, 7.6.9, 7.6.11, 7.6.13, 7.6.15, 7.6.17, 7.6.19, 7.7.7, 7.7.9, 7.7.11, 7.7.13, 7.7.15, 7.7.17, 7.7.19, 7.7.21, 7.7.23.
7. Simplifying trigonometric expressions:
• Reading from (mostly) the textbook:
• Exercises due on March 29 Tuesday (submit these here on Canvas or in class):
1. Fill in the blank with an expression in which sin θ is the only trigonometric quantity: cos2θ = ___.
2. Factor without using any trigonometric identities: sin2θ − 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 cos2θ.)
• Exercises from the textbook due on March 30 Wednesday (submit these through MyLab in the Next item): 7.4.1, 7.4.2, 7.4.6, 7.4.8, 7.4.11, 7.4.15, 7.4.17, 7.4.29, 7.4.55, 7.4.71, 7.4.95.
8. Trigonometric equations:
• Section 7.3 (pages 469–474);
• Section 7.5 Objective 4 (pages 494–496);
• Section 7.6 Objective 2 Examples 4&5 (pages 503&504).
• Exercises due on March 30 Wednesday (submit these here on Canvas or in class):
1. Write a general form for the solutions of tan x = b using tan−1b and an arbitrary integer k.
2. Similarly, give the general solution of sin x = b. (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?
4. Since you can factor x + xy as x(1 + y), how can you factor cos θ + sin θ cos θ?
5. To solve a sin θ + b cos θ = c with the help of a sum-angle formula, what should you multiply both sides of the equation by?
6. To solve sin(aθ) + sin(bθ) = 0, how can you factor the left-hand side?
• Exercises from the textbook due on March 31 Thursday (submit these through MyLab in the Next item): 7.3.13, 7.3.23, 7.3.25, 7.3.27, 7.3.37, 7.3.39, 7.3.61, 7.3.73, 7.3.115, 7.5.93, 7.5.97, 7.6.75, 7.6.77, 7.7.47.
Quiz 4, covering the material in Problem Sets 28–35, is on April 4 Monday.

### Applications

1. The Law of Sines:
• Reading from the textbook: Section 8.2 through Objective 2 (pages 535–539).
• Exercises due on April 5 Tuesday (submit these here on Canvas or in class): 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 on April 6 Wednesday (submit these through MyLab in the Next item): 8.1.2, 8.1.29, 8.1.31, 8.1.33, 8.1.35, 8.1.37, 8.1.39, 8.2.9, 8.2.11, 8.2.13, 8.2.15, 8.2.18, 8.2.27, 8.2.29, 8.2.33, 8.2.35, 8.2.37.
2. The Law of Cosines:
• Reading from (mostly) the textbook:
• Section 8.3 through Objective 2 (pages 546–548);
• My handout on solving triangles (DjVu).
• Exercises due on April 6 Wednesday (submit these here on Canvas or in class):
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 on April 7 Thursday (submit these through MyLab in the Next item): 8.1.41, 8.3.9, 8.3.11, 8.3.13, 8.3.15, 8.3.35, 8.3.37, 8.3.39, 8.3.41.
3. Area of triangles:
• Reading from the textbook: Section 8.4 (pages 553–555).
• Exercises due on April 7 Thursday (submit these here on Canvas or in class):
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 on April 11 Monday (submit these through MyLab in the Next item): 8.4.9, 8.4.11, 8.4.13, 8.4.15, 8.4.17, 8.4.19, 8.4.21, 8.4.25, 8.4.27, 8.4.37.
4. Applications of solving triangles:
• Section 8.1 Objective 4 (pages 524–529);
• Section 8.2 Objective 3 (pages 539–541);
• Section 8.3 Objective 3 (pages 548&549).
• Exercises due on April 11 Monday (submit these here on Canvas or in class):
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.
4. If you know the horizontal distance to the base of an object and the angle of elevation to the top of the object and you want to find the height of the object, then would you use the sine, the cosine, or the tangent of the angle of elevation?
5. If a bearing is N30°E, then what is the angle that this direction makes with due north, and what angle does it make with due east?
6. If you divide a polygon with n sides into triangles, then how many triangles will you need?
• Exercises from the textbook due on April 12 Tuesday (submit these through MyLab in the Next item): 8.1.43, 8.1.45, 8.1.47, 8.1.51, 8.1.63, 8.2.39, 8.2.49, 8.3.45, 8.3.57, 8.4.46, 8.4.53.
5. Harmonic motion:
• Reading from the textbook: Section 8.5 (pages 559–565).
• Exercises due on April 12 Tuesday (submit these here on Canvas or in class):
1. Fill in the blank with more than one word: If the position of an object is a sinusoidal function of time, then the object is undergoing _________ motion.
2. If the sinusoidal function is modified so that the amplitude is an exponential function with a negative growth rate (instead of a constant), then the object is undergoing _________ motion.
• Exercises from the textbook due on April 13 Wednesday (submit these through MyLab in the Next item): 8.5.7, 8.5.9, 8.5.11, 8.5.13, 8.5.15, 8.5.17, 8.5.19, 8.5.21, 8.5.23, 8.5.25, 7.3.109.
6. Polar coordinates:
• Reading from the textbook: Section 9.1 through Objective 3 (pages 576–582).
• Exercises due on April 13 Wednesday (submit these here on Canvas or in class):
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 on April 14 Thursday (submit these through MyLab in the Next item): 9.1.13–20, 9.1.21, 9.1.23, 9.1.25, 9.1.27, 9.1.31, 9.1.33, 9.1.35, 9.1.45, 9.1.47, 9.1.49, 9.1.51, 9.1.53, 9.1.59, 9.1.63, 9.1.77, 9.1.79, 9.1.83, 9.1.85.
7. Equations in polar coordinates:
• Section 9.1 Objective 4 (pages 582&583);
• Section 9.2 through Objective 1 (pages 585–589).
• Exercises due on November 19 Friday (submit these here on Canvas or in class): Let x and y be rectangular coordinates, and let r and θ be the corresponding polar coordinates.
• Express the following quantities using only x and y:
1. r2,
2. tan θ;
• Express the following quantities using x, y, and/or r:
1. sin θ,
2. cos θ.
• Exercises from the textbook due on November 22 Monday (submit these through MyLab in the Next item): 9.1.77, 9.1.79, 9.1.83, 9.1.85, 9.2.15, 9.2.17, 9.2.19, 9.2.21, 9.2.23.
8. Graphing in polar coordinates:
• Reading from the textbook: The rest of Section 9.2 (pages 589–597).
• Exercises due on April 14 Thursday (submit these here on Canvas or in class):
1. Let r and θ be polar coordinates, and let x and y be the corresponding rectangular coordinates. Express the following quantities using only x and y: r2, tan θ.
2. Again let r and θ be polar coordinates, and let x and y be the corresponding rectangular coordinates. Express the following quantities using x, y, and/or r: sin θ, cos θ.
3. 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.
4. 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 on April 18 Monday (submit these through MyLab in the Next item): 9.2.15, 9.2.17, 9.2.19, 9.2.21, 9.2.23, 9.2.31–38, 9.2.39, 9.2.43, 9.2.47, 9.2.51, 9.2.55, 9.2.59.
9. Complex numbers:
• Reading from (mostly) the textbook:
• Exercises due on April 18 Monday (submit these here on Canvas or in class):
1. What is the magnitude (absolute value) of the complex number x + iy?
2. Write the complex number with magnitude r and argument θ.
3. What is the product of r1(cos θ1 + i sin θ1) and r2(cos θ2 + i sin θ2)?
• Exercises from the textbook due on April 19 Tuesday (submit these through MyLab in the Next item): 9.3.13, 9.3.15, 9.3.17, 9.3.19, 9.3.21, 9.3.23, 9.3.25, 9.3.29, 9.3.33, 9.3.35, 9.3.37, 9.3.41, 9.3.43, 9.3.45, 9.3.47, 9.3.49, 9.3.53, 9.3.55, 9.3.57, 9.3.59, 9.3.61, 9.3.63.
10. Vectors:
• Reading from the textbook: Section 9.4 (pages 609–619).
• Exercises due on April 19 Tuesday (submit these here on Canvas or in class):
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 on April 20 Wednesday (submit these through MyLab in the Next item): 9.4.11, 9.4.13, 9.4.15, 9.4.17, 9.4.27, 9.4.29, 9.4.37, 9.4.39, 9.4.43, 9.4.45, 9.4.49, 9.4.51.
11. Vectors and angles:
• Reading from the textbook: Section 9.5 (pages 624–629).
• Exercises due on April 20 Wednesday (submit these here on Canvas or in class):
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 on April 21 Thursday (submit these through MyLab in the Next item): 9.4.61, 9.4.63, 9.4.65, 9.4.67, 9.4.69, 9.5.9, 9.5.11, 9.5.13, 9.5.15, 9.5.17, 9.5.19, 9.5.21, 9.5.23, 9.5.25.
Quiz 5, covering the material in Problem Sets 36–45, is on April 25 Monday.

## Quizzes

1. Graphs and functions:
• Review date: January 27 Thursday (in class).
• Date taken: January 31 Monday (in class).
• Corresponding problem sets: 1–8.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.
2. Polynomial functions:
• Review date: February 17 Thursday (in class).
• Date taken: February 21 Monday (in class).
• Corresponding problem sets: 9–17.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.
3. Transcendental functions:
• Review date: March 9 Wednesday (in class).
• Date taken: March 10 Thursday (in class).
• Corresponding problem sets: 18–27.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.
4. Analytic trigonometry:
• Review date: March 31 Thursday (in class).
• Date taken: April 4 Monday (in class).
• Corresponding problem sets: 28–35.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.
5. Applications:
• Review date: April 21 Thursday (in class).
• Date taken: April 25 Monday (in class).
• Corresponding problem sets: 36–45.
• Help allowed: Your notes, calculator.
• NOT allowed: Textbook, my notes, other people, websites, etc.

## Final exam

There is a comprehensive final exam on May 4 Wednesday, in our normal classroom at the normal time but lasting until 2:10 PM. (You can also arrange to take it at a different time from April 29 to May 5.) To speed up grading at the end of the term, the exam is multiple choice, with no partial credit.

For the exam, you may use one sheet of notes that you wrote yourself. However, you may not use your book or anything else not written by you. You certainly should not talk to other people! Calculators are allowed, although you shouldn't really need one, but not communication devices (like cell phones).

The exam consists of questions similar in style and content to those in the practice final exam (DjVu).

This web page and the files linked from it (except for the official syllabus) were written by Toby Bartels, last edited on 2022 December 12. Toby reserves no legal rights to them.

The permanent URI of this web page is `http://tobybartels.name/MATH-1300/2022FA/`.