Additional Topics in math Lessons 5-7 Degree, Radians, and Arc Length The Unit Circle and Trigonometric Functions with Radian Measure Circle Definitions and Equations Simplification of Imaginary Monomial Expressions Arithmetic Operations on Complex Numbers

Degrees, Radians, and Arc Length Angles can be measured in degree or radians We are used to degrees but trigonometry uses radians A Circle = 360° A Circle = 2π To convert degrees to radians you multiply by π/180 To convert radians to degrees you multiply by 180/π This can be done by replacing the π with 180 Arc Length: s = θr or l = θr BUT… θ must be in radians!

The Unit circle and trigonometric functions with radian measure You need to know where the sin, cos, and tan functions are positive and negative Use “All Students Take Calculus” to help you remember where the trig functions are positive. You will need to use reasoning to identify where they are negative Students Sin + All + Take Tan + Calculus Cos +

Circle definitions and equations Equation of a circle: (x – h)2 + (y – k)2 = r2 When finding h and k remember to change the sign (h, k) is the center of your circle r is your radius but remember to square root r2 to get r The radius can be found by finding the distance from the center to a point on the circle If a point is ON the circle then the x and y from that point must make the equation true

Simplification of imaginary monomial expressions The imaginary number “i” allows us to square root negative numbers You simplify the root as you would a positive number then include “i” in your answer The exponents of “i” follow a pattern: i = i i5 = i i2 = -1 i6 = -1 i3 = -i i7 = -i i4 = 1 i8 = 1 So if you remove all the multiples of 4 from the exponent you can simplify even large exponents

Arithmetic operations on complex numbers When performing operation with “i” – treat it as you would any other variable When simplifying you must not leave i2 anywhere: i2 = -1 To Add/Subtract Complex Numbers – Combine Like Pieces To Multiply Complex Numbers – FOIL and clean up To Divide Complex Numbers – Multiply both the top and bottom by the conjugate of the denominator A Conjugate of a binomial changes the middle sign Example of Conjugates: 3 + 2i and 3 – 2i