InverseTrigonometricRatios
LearningObjectives
- Identifyand use the arctangentratio in a right triangle.
- Identifyand use the arcsineratio in a right triangle.
- Identifyand use the arccosineratio in a right triangle.
- Understandthe generaltrendsof trigonometricratios.
Introduction
The wordinverseis probablyfamiliarto you—oftenin mathematics,after you learnto do an operation,you
also learnhow to “undo”it. Doingthe inverseof an operationis a way to undothe originaloperation.For
example,you may rememberthat additionand subtractionare consideredinverseoperations.Multiplication
and divisionare also inverseoperations.In algebrayou usedinverseoperationsto solveequationsand in-
equalities.You may also rememberthe termadditiveinverse, or a numberthat can be addedto the original
to yield a sum of. For example, and are additiveinversesbecause.
In this lessonyou will learnto use the inverseoperationsof the trigonometricfunctionsyou havestudied
thus far. You can use inversetrigonometricfunctionsto find the measuresof angleswhenyou knowthe
lengthsof the sidesin a right triangle.
InverseTangent
Whenyou find the inverseof a trigonometricfunction,you put the wordarcin front of it. So, the inverseof
a tangentis calledthe arctangent(or arctanfor short).Thinkof the arctangentas a tool you can use like
any otherinverseoperationwhensolvinga problem.If tangenttells you the ratio of the lengthsof the sides
oppositeand adjacentto an angle,then arctantells you the measureof an anglewith a givenratio.
Suppose. The arctangentcan be usedto find the measureof on the left side of the
equation.
Wheredid that comefrom?Thereare two basicwaysto find an arctangent.Sometimesyou will be
givena tableof trigonometricvaluesand the anglesto whichthey correspond.In this scenario,find the value
that is closestto the one provided,and identifythe correspondingangle.
Another, easierway of findingthe arctangentis to use a calculator. The arctangentbuttonmay be labeled
“arctan,”“atan,”or “ .” Eitherway, selectthis button,and inputthe valuein question.In this case,
you wouldpressthe arctangentbuttonand enter (or on somecalculators,enter , then press
“arctan”).The outputwill be the valueof measure.
is about.