L2 Trigonometric Identities - Part 1
Completion requirements
Unit 4A
Trigonometry Part 1
Lesson 2: Trigonometric Identities
One way of organizing equations is to split them into three categories: conditional equations, identities, and inconsistent equations. Most equations encountered in math classes are conditional equations. Conditional equations are ones that are true for
some values of the variables, but not all.
Here are some examples of conditional equations.
Identities are equations that are true for all permissible values of the variables.
Here are some examples of identities.
Inconsistent equations are never true, regardless of the value of the variable.
Here are some examples of inconsistent equations.
In this Lesson, the focus will be on trigonometric identities, equations involving trigonometric ratios that are true for all permissible values of the variables. Most of the fundamental trigonometric identities have been introduced in previous courses. A formula sheet is provided for use throughout this course that includes all of the identities discussed in this Lesson.
Here are some examples of conditional equations.
- «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»5«/mn»«mo»+«/mo»«mi»x«/mi»«mo»=«/mo»«mn»12«/mn»«/mrow»«/mstyle»«/math» is true when «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»7«/mn»«/mrow»«/mstyle»«/math»
- «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»=«/mo»«mn»16«/mn»«/mrow»«/mstyle»«/math» is true when «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mo»§#177;«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math»
Identities are equations that are true for all permissible values of the variables.
Here are some examples of identities.
- «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mi»x«/mi»«mo»=«/mo»«mn»0«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»25«/mn»«mi»x«/mi»«/mrow»«/mstyle»«/math»
is true for all values of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»
- «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math»
is true for all values of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»
- «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mfenced»«msqrt»«mi»x«/mi»«/msqrt»«/mfenced»«mn»2«/mn»«/msup»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math» is true for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8805;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math»
Inconsistent equations are never true, regardless of the value of the variable.
Here are some examples of inconsistent equations.
- «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» is not true for any value of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math»
- «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mo»=«/mo»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math» is not true for any value of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»
In this Lesson, the focus will be on trigonometric identities, equations involving trigonometric ratios that are true for all permissible values of the variables. Most of the fundamental trigonometric identities have been introduced in previous courses. A formula sheet is provided for use throughout this course that includes all of the identities discussed in this Lesson.