L3 Trigonometric Equations - Part 1
Completion requirements
Unit 4A
Trigonometry Part 1
Lesson 3: Trigonometric Equations
To solve an equation algebraically, inverse operations are often used.
Notice for each of the first three operations listed in the table, the inverse operation corresponds to a single value.
For example, to solve «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«mo»=«/mo»«mn»7«/mn»«/mrow»«/mstyle»«/math», subtract «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math» from both sides of the equation to get «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math».
Similarly, to solve «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»=«/mo»«mn»16«/mn»«/mrow»«/mstyle»«/math», use logarithms, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mn»16«/mn»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math», to show «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math».
The last operation and its inverse are a little different.
To solve something «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»=«/mo»«mn»9«/mn»«/mrow»«/mstyle»«/math», attention must be given to the number of possible solutions. Both «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math» satisfy this equation.
The focus of this Lesson is on solving trigonometric equations. Like square roots, the inverse trigonometric operations will often lead to more than one solution. In fact, many trigonometric equations have an infinite number of solutions!
| Operation | Inverse Operation
|
| Addition | Subtraction |
| Multiplication | Division |
| Exponent | Logarithm |
| Squaring | Square Root
|
Notice for each of the first three operations listed in the table, the inverse operation corresponds to a single value.
For example, to solve «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«mo»=«/mo»«mn»7«/mn»«/mrow»«/mstyle»«/math», subtract «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math» from both sides of the equation to get «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math».
Similarly, to solve «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»=«/mo»«mn»16«/mn»«/mrow»«/mstyle»«/math», use logarithms, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mn»16«/mn»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math», to show «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math».
The last operation and its inverse are a little different.
To solve something «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»=«/mo»«mn»9«/mn»«/mrow»«/mstyle»«/math», attention must be given to the number of possible solutions. Both «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math» satisfy this equation.
The focus of this Lesson is on solving trigonometric equations. Like square roots, the inverse trigonometric operations will often lead to more than one solution. In fact, many trigonometric equations have an infinite number of solutions!