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Solving Systems of Equations
This unit introduces how to systematically solve a system of equations, namely linear equations. Examples of non-linear systems, including systems of 3 unknowns will be of emphasis.
Graphs of Trigonometric Functions
The unit focuses primarily on how to graph periodic sinusoidal functions, and how to identify features of a waveform to produce an equation by inspection.
Polar Coordinate Functions
An introduction to the polar coordinate system.
Exponents and Radicals
This unit is an extension of what was introduced in Math 1131. To learn how to work with radicals, knowing your exponent laws in crucial. Hence, this unit begins with a thorough review.
This chapter introduces you to exponential functions, and how they can be solved using logarithms.
Trigonometric Identities and Equations
Multiply and Divide Complex Numbers
Multiplying Complex Numbers
Imaginary and complex numbers are multiplied the same way you multiply polynomials, with the addition of what you learned in the previous lesson about i when raised to varying exponents. Examples are shown below:
Similarly, this idea can be expanded to imaginary numbers found within larger algebraic expressions found in polynomials. For example:
Starting with (1), you’d expand the factor 3 into the binomial (5 + 2i), giving you: 15 + 6i. In question (2), 3i×2 equals 6i and 3i×(–4i) equals –12i². Together, 6i – 12i² = 6i + 12 ⇒ 12 + 6i. For question (3), you’d have to use the foil method (linked), giving you: –12 + 15i + 8i – 20i² ⇒ –12 + 23i + 20 ⇒ 8 + 23i.
As discussed briefly in the previous lesson, the conjugate of a complex number is one that changes the sign of the imaginary portion. In case you’re asked to multiply a complex number by its conjugate, follow the steps outlined in the video below:
Beware of this common mistake
If you’re asked to multiply, for example:
Always convert radicals to imaginary numbers first, otherwise contradictions may result:
This rule only applies to be the radicand is positive. Rather, do this:
Dividing Complex Numbers
Dividing complex numbers is relatively easy if the denominator is a single term. However, if you’re given an expression where it’s in the form of two binomials, such as ( a ± bi ) ÷ ( a ∓ bi ), this can be reduced by multiplying and dividing the expression by the conjugate of the denominator. This demonstrated is illustrated below: