082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

Note that
tanθ = BC
AB = BC
AC × AC
AB = sinθ
cos θ
3.3 The trigonometric ratios 117
Note that when θ reduces to0 ◦ the length of the sideBC reduces tozero and so
sin0 ◦ =0, tan0 ◦ =0
Also when θ reduces to0 ◦ the lengths of AB and ACbecome equal and so
cos0 ◦ =1
Similarlywhen θ approaches90 ◦ ,thelengthsofBCandACbecomeequalandthelength
of ABshrinks tozero.Hence
sin90 ◦ = 1, cos90 ◦ = 0
Note also that the length of AB shrinks to zero as θ approaches 90 ◦ , and so tanθ approaches
infinity. We write thisas
tanθ→∞
asθ→90 ◦
The trigonometric ratios of 30 ◦ , 45 ◦ and 60 ◦ occur frequently in calculations. They can
be calculated exactly by considering the right-angled triangles shown inFigure 3.2.
sin45 ◦ = 1 √
2
, cos45 ◦ = 1 √
2
, tan45 ◦ =1
sin30 ◦ = 1 2 , cos30◦ =
sin60 ◦ =
√
3
2 , tan30◦ = 1 √
3
√
3
2 , cos60◦ = 1 2 , tan60◦ = √ 3
Mostscientificcalculatorshavepre-programmedvaluesofsinθ,cosθ andtanθ.Anglescanbemeasuredindegreesorradians.Wewilluseradiansunlessstatedotherwise.
If welet ̸ ACB = α (see Figure 3.1),then
and
sinα= AB
AC =cosθ
cosα= BC
AC =sinθ
45°
2
1
45°
1
30°
2
3
60°
1
Figure3.2
Thetrigonometric ratiosfor30 ◦ ,
45 ◦ and60 ◦ canbe foundexactly
fromthese triangles.