7.3.1: Resources and Key Concepts

Prerequisite Topics

This page lists the prerequisite skills (all of which can be reviewed in CRC's Corequisite Codex) needed for this section that have not already been mentioned in any previous section. If you are enrolled in a course with a Support section, some (but definitely not all) of these topics might be covered or reviewed in the Support section of your course.

• Solving Equations
  • Solving Quadratic Equations: This section requires solving equations that are quadratic-in-form. The "Skills Refresher" also includes basic quadratic equations.
  • Solving Equations Quadratic-in-Form: This section uses this skill frequently.
  • Solving Rational Equations: This skill is needed for the "Skills Refresher" exercises.

Videos

• Solving Trigonometric Equations
  • Solving Trigonometric Equations - The Concepts Behind Solving Trigonometric Equations
  • Solving Trigonometric Equations - Basic Equations

Definitions

• Trigonometric Equation: An equation that involves one or more trigonometric functions of a variable.
• Quadratic-in-Form: An equation that can be written in the form \(a[f(x)]^2 + b[f(x)] + c = 0\), where \(f(x)\) is some function.

Common Mistakes

• Forgetting to Find All Solutions: Trigonometric equations typically have an infinite number of solutions due to the periodic nature of the functions. Forgetting to add multiples of the period (\(+2\pi k\) or \(\pi k\)) will result in an incomplete solution set.
• Stopping After the Inverse Function: Using an inverse trigonometric function on a calculator provides only one solution (the principal value). You must use that angle as a reference angle to find all solutions in the required quadrants.
• Incorrectly Handling Non-Standard Arguments: When solving an equation like \(\sin(2x) = k\), the solutions found are for the argument \(2x\). A common mistake is forgetting the final step of dividing by 2 to solve for \(x\).
• Not Checking for Extraneous or Invalid Solutions: It is important to check for realism. For example, an equation that simplifies to \(\sin(\beta) = 20\) has no solution, since the range of the sine function is \([-1, 1]\).
• Algebraic Errors: Common algebraic mistakes, such as incorrectly applying the zero-factor principle or dividing by a variable that could be zero, can lead to incorrect or incomplete solutions.