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

120 Chapter 3 The trigonometric functions
Thecosecant,secantandcotangentratiosaredefinedasthereciprocalsofthesine,
cosine and tangent ratios.
cosec θ = 1
sinθ
secθ = 1
cos θ
cotθ = 1
tanθ
Example3.1 An angle θ issuchthatsinθ > 0 and cosθ < 0.Inwhichquadrantdoes θ lie?
Solution FromFigure3.5weseethatsinθ > 0when θ isinthefirstandsecondquadrants.Also,
cosθ < 0when θ isinthesecondandthirdquadrants.Forbothsinθ > 0andcosθ < 0
thus requires θ to be in the second quadrant. Hence sin θ > 0 and cosθ < 0 when θ is
inthe secondquadrant.
EXERCISES3.3
1 Verify usingascientificcalculator that
(a)sin30 ◦ = sin390 ◦ = sin750 ◦
(b)cos100 ◦ = cos460 ◦ = cos820 ◦
(c)tan40 ◦ = tan220 ◦ = tan400 ◦
(d)sin70 ◦ = sin(−290 ◦ ) = sin(−650 ◦ )
(e)cos200 ◦ = cos(−160 ◦ ) = cos(−520 ◦ )
(f) tan150 ◦ = tan(−30 ◦ ) = tan(−210 ◦ )
2 Verify the following usingascientificcalculator. All
angles are in radians.
(a) sin0.7 = sin(0.7 +2π) = sin(0.7 +4π)
(b) cos1.4 = cos(1.4 +8π) = cos(1.4 −6π)
(c) tan1 = tan(1 + π) = tan(1 +2π) = tan(1 +3π)
(d) sin2.3 = sin(2.3 −2π) = sin(2.3 −4π)
(e) cos2 = cos(2 −2π) = cos(2 −4π)
(f) tan4 = tan(4 − π) = tan(4 −2π) = tan(4 −3π)
3 An angle θ is suchthat cos θ > 0 andtan θ < 0.In
whichquadrant does θ lie?
4 An angle α is suchthattan α > 0 andsin α < 0.In
whichquadrant does α lie?
Solutions
3 4thquadrant 4 3rd quadrant
3.4 THESINE,COSINEANDTANGENTFUNCTIONS
The sine, cosine and tangent functions follow directly from the trigonometric ratios.
These are defined to be f (x) = sinx, f (x) = cosx and f (x) = tanx. Graphs can be
constructedfromatableofvaluesfoundusingascientificcalculator.Theyareshownin
Figures3.7 to3.9. Notethatthese functions aremany-to-one.