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

118 Chapter 3 The trigonometric functions
But
α+θ= π 2
Hence,
( ) π
sinθ = cos
2 − θ
( ) π
cos θ = sin
2 − θ
Since θ is an angle in a right-angled triangle it cannot exceed π/2. In order to define
the sine, cosine and tangent ratios for angles larger than π/2 we introduce an extended
definition which isapplicable toangles of any size.
Consider an arm,OP, fixed atO, which can rotate (see Figure 3.3). The angle, θ, in
radians,betweenthearmandthepositivexaxisismeasuredanticlockwise.Thearmcan
beprojectedontoboththexandyaxes.TheseprojectionsareOAandOB,respectively.
Whetherthearmprojectsontothepositiveornegativexandyaxesdependsuponwhich
quadrant the armissituatedin. The length of the armOPisalways positive. Then,
sinθ= projectionofOPontoyaxis
OP
cosθ=
tanθ=
projection ofOPontoxaxis
OP
= OB
OP
= OA
OP
projection ofOP ontoyaxis
projection ofOPontoxaxis = OB
OA
In the first quadrant, that is 0 θ < π/2, both thexandyprojections are positive,
so sinθ, cos θ and tanθ are positive. In the second quadrant, that is π/2 θ < π, the
x projection,OA, is negative and theyprojection,OB, positive. Hence sinθ is positive,
and cosθ and tanθ are negative. Both thexandyprojections are negative for the third
quadrantandsosinθ andcosθ arenegativewhiletanθ ispositive.Finally,inthefourth
quadrant, thexprojection is positive and theyprojection is negative. Hence, sinθ and
tan θ are negative, and cosθ is positive (see Figure 3.4). The sign of the trigonometric
ratios can be summarized by Figure 3.5.
For angles greater than 2π, the armOP simply rotates more than one revolution before
coming to rest. Each complete revolution bringsOP back to its original position.
Second quadrant
P
y
B
u
First quadrant
A
O
x
Third quadrant
Fourth quadrant
Figure3.3
An arm,OP,fixed atO, which can rotate.