CK12 - Geometry

GeneralTrendsin TrigonometricRatios

Now that you knowhow to find the trigonometricratiosas well as their inverses,it is helpfulto look at trends
in the differentvalues.Rememberthat eachratio will havea constantvaluefor a specificangle.In any right

triangle,the sine of a anglewill alwaysbe —it doesn’tmatterhow long the sidesare. You can use
that informationto find missinglengthsin triangleswhereyou knowthe angles,or to identifythe measure
of an angleif you knowtwo of the sides.

Examinethe tablebelowfor trends.It showsthe sine,cosine,and tangentvaluesfor eightdifferentangle
measures.

Sine

Cosine

Tangent

Example 7

Usingthe tableabove,whichvaluewouldyou expectto be greater:the sine of

or the cosineof

?

You can use the informationin the tableto solvethis problem.The sine of is and the sine of

is. So, the sine of will be betweenthe values and. The cosineof is

and the cosineof is So, the cosineof will be betweenthe valuesof and
Sincethe rangefor the cosineis greater, than the rangefor the sine,it can be assumedthat the cosineof

will be greaterthan the sine of

Noticethat as the anglemeasuresapproach , approaches. Similarly, as the valueof the angles

approach , the approaches. In otherwords,as the gets greater, the gets smallerfor
the anglesin this table.

The tangent,on the otherhand,increasesrapidlyfrom a smallvalueto a largevalue(infinity, in fact) as the

angleapproaches.

LessonSummary

In this lesson,we exploredhow to workwith differentradicalsboth in theoryand in practicalsituations.
Specifically, we havelearned:

• How to identifyand use the arctangentratio in a right triangle.


• How to identifyand use the arcsineratio in a right triangle.


• How to identifyand use the arccosineratio in a right triangle.