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|Name: __________________________________________________ Date: ___________ |
|CP10 Trigonometry |
|DO NOW: |
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|Solve the equation. Leave your answer in simplest radical form. |
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|4.1 Complex Numbers |
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|Objectives: I will be able to EXPLAIN how to SOLVE quadratic equations with complex solutions and perform operations with complex numbers. |
|Imaginary Unit: |Mathematicians created an expanded system of numbers using the imaginary unit i, defined as i = [pic] , because . . . there is no |
| |real number x that can be squared to produce − 1. |
|Properties: |1. If r is a positive real number, then [pic] |Example: |
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| |2. By Property (1), it follows that [pic] |Example: |
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|Prove: |[pic] |[pic] |
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|Principal Square Root: |If a is a positive number, the principal square root of the negative |
| |Number –a is defined as |
|Example 1: Solving Quadratic |a) [pic] |b) [pic] |
|Equation | | |
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|Now you try: |c)[pic] |d) [pic] |
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|Real part |[pic] | |
|and | | |
|Imaginary part | | |
|Example 2: | |Imaginary | |
|Plotting Complex numbers |[pic] |[pic] | |
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| |[pic] | |real |
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|Add Complex Numbers: |To add two complex numbers, . . . add the real parts and the |
| |imaginary parts of the numbers separately. |
|Subtract Complex Numbers: |To subtract two complex numbers, . . . subtract the real parts |
| |and the imaginary parts of the numbers separately. |
|NOTE: |The additive identity in the complex number system is 0. |
| |The additive inverse of the complex number a + bi is : −(a+bi)=−a−bi . |
|Example 3: |a) [pic] |b) [pic] |
|Adding and Subtracting | | |
|Complex Numbers | | |
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|Now you try: |c) [pic] |d) [pic] |
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|Multiply Complex Numbers: |To multiply two complex numbers a + bi and c + di, . . . use the multiplication rule (ac − bd) + (ad + bc)i or use the Distributive |
| |Property to multiply the two complex numbers, similar to using the FOIL method for multiplying two binomials. |
|Example 4: |a) [pic] |b) [pic] |
|Multiplying Complex Numbers | | |
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| |c) [pic] |d) [pic] |
| |(Complex Conjugates) | |
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|Now you try: |c) [pic] |d) [pic] |
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| | |e) [pic] |
| |d) [pic] | |
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|Complex Conjugate: |The product of a pair of complex conjugates is a real number. |
| |To find the quotient of the complex numbers a + bi and c + di, where c and d are not both zero, multiply the numerator and denominator|
| |by the conjugate of the denominator. |
|Example 5: |Write the quotient in standard form: [pic] |
|Dividing Complex Numbers | |
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|Now you try: |b) [pic] |c) [pic] |
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| |d) [pic] |e) [pic] |
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|Absolute Value: |The absolute value of a complex number [pic] , denoted with [pic], is a nonnegative real number defined as follows [pic] |
| |Geometrically, the absolute value of a complex number is the number’s distance from the origin in the complex plane. |
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|Example 6: |Which number is closest to the origin in the complex plane? |
|Finding absolute Value |a) [pic] b) [pic] c) [pic] |
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|NOTE: |To avoid problems with multiplying square roots of negative numbers, be sure to convert to standard form before multiplying. |
|Example 7: |Perform the operation and write the result in standard form: |
| |a) [pic] b) [pic] |
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| |c) [pic] d) [pic] |
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|Now You Try: |e) [pic] f) [pic] |
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| |g) [pic] h) [pic] |
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|Example 8: |a) [pic] |
|Find the solutions of the | |
|quadratic equation: | |
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| |b) [pic] |
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|Now you try: |c) [pic] |
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|Example 9: |a) [pic] b) [pic] |
|Simplify the complex number | |
|and write the solution in | |
|standard form. | |
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| |c) [pic] d) [pic] |
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|Example 10: |[pic] |
|Finding Powers of i. | |
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|Now You Try: |a) [pic] b) [pic] c) [pic] |
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|SUMMARY: | |
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|Homework: |Pg. 333 – 334: # 3 – 9 Odd, 19 – 23 Odd, 27 – 39 Odd, 49 – 59 Odd, 63, 67, 73 – 81 Odd |
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