C3-Trigonometry
Prove (1 - cos 2A/sin 2A)=tan A.
=(1-(1-2sin^2 A))/2sinAcosA =2sin^2 A/2sinAcosA =sinA/cosA=tanA
Prove (1-cos^2 x)/(cos x + 1)=1-cos x.
=(1-cos x)(1+cos x)/(1+cos x) =1-cos x
Prove cosec ø/(cot ø + tan ø) = cos ø.
=(1/sin ø)/((1/(sin ø/cos ø))+(sin ø/cos ø)) =cos ø/(cos^2 ø + sin^2 ø) =cos ø
Prove (cos S + cos T)/(sin S - sin T)=cot (S-T)/2.
=(2cos ((S+T)/2)cos ((S-T)/2) / 2cos ((S+T)/2) sin ((S-T)/2) = cos ((S-T)/2) / sin ((S-T)/2)) =cot ((S-T)/2))
Prove cos A/sin A + tan A=1/(sin A cos A).
=(cos^2 A + sin^2 A)/sin A cos A =1/sin A cos A
Prove cos^4 ø - sin^4 ø +1=2 cos^2 ø.
=(cos^2 ø - sin^2 ø)(cos ^2 ø + sin^2 ø)+1 =cos^2 ø - sin^2 ø + 1 =cos^2 ø + cos^2 ø =2 cos^2 ø
Prove cos ø/(1-sin ø)+(1-sin ø)/cos ø=2 sec ø
=(cos^2+ 1 - 2 sin ø + sin^2 ø)/cos ø(1-sin ø) =2-2sin ø/cos ø(1-sin ø) =2(1- sin ø)/cos ø(1-sin ø) =2/cos ø=2 sec ø
sin 165...
=(root3 -1)/2root2
Prove sin(A+B)/sin(A-B)=(tan A + tan B)/(tan A - tan B).
=(sinAcosB + sinBcosA)/(sinAcosB - sinBcosA) =((sinAcosB/cosAcoB + sinBcosA/cosAcosB))/((sinAcosB/cosAcosB - sinBcosA/cosAcosB)) =(tan A + tan B)/(tan A - tan B)
tan 2A...
=2 tan A/1- tan^2 A (tan A + tan A/ 1- tanAtanA)
Prove (sin S - sin T)/(sin S + sin T)=cot ((S+T)/2) tan ((S-T)/2).
=2cos ((S+T)/2) sin ((S-T)/2) / 2sin ((S+T)/2) cos ((S-T)/2) =cot ((S+T)/2) tan ((S-T)/2)
sin 2A...
=2sinAcosA (sinAcosA +sinAcosA)
sin 3x...
=3 sin x -4 sin^3 x (sin (2x + x))
Express 330/(30+(3sinx + 4cosx)^2) in the form Rsin(x+a) and find the max and min values of y and the first positive x value for which this occurs.
=330/(30+25(sin(x+53))^2) max y=11, x=127 min y=6, x=37
cos 3x...
=4cos^3 x - 3 cos x (cos (2x + x))
cos (arctan 3/5)...
=5/root34
Express 56/(15-(12cosx + 5sinx)) in the form Rcos(x-a) and find the max and min values of y and the first positive x value for which this occurs.
=56/15-13cos(x-23) max y=28, x=23 min y=2, x=203
Express 3sinx + 4cosx in the form Rsin(x+a) and find the max and min values of y and the first positive x value for which this occurs.
=5sin(x+53) max=5, x=37 min=-5, x=217
Show Asinx + Bcosx = Rsin(x+a)
=R(sinxcosa + cosxsina) =Rcosasinx + Rsinacosx Let A=Rcosa B=Rsina So=Asinx + Bcosx So A^2 + B^2 = R^2(cos^2 x + sin^2 x)=R^2 So R=root(A^2 + B^2)
A cosx - B sinx...
=Rcos(x+a)
A cosx + B sin x...
=Rcos(x-a)
A sinx - B cosx...
=Rsin(x-a)
Prove cos 4ø=8cos^4ø-8cos^2ø +1.
=cos 2(2ø) =2 cos^2 2ø -1 =2(2cos^2 ø -1)^2 -1 =2(2cos^2 ø - 1)(2cos^2 ø -1) -1 =8cos^4 ø - 8cos^2 ø + 1
Prove cot(A+B)=(cotAcotB-1)/(cot A + cot B)
=cos(A+B)/sin(A+B) =(cosAcosB - sinAsinB)/(sinAcosB - sinBcosA) =((cosAcosB/sinAsinB - sinAsinB/sinAsinB))/((sinAcosB/sinAsinB - sinBcosA/sinAsinB)) =(cotAcotB-1)/(cot A + cot B)
cos(A+/-B)...
=cosAcosB -/+ sinAsinB
cos 2A...
=cos^2 A - sin^2 A (cosAcosA - sinAsinA) =2cos^2 A - 1 =1 - 2sin^2 A
cos (arcsin x)...
=root(1-x^2)
If A is an obtuse angle and B a reflex angle and sin A=3/5 and cos B=5/13, find sin (A-B) and tan (A+B).
sin (A-B)=-33/65 tan (A+B)=63/16
Show that 1 + tan^2 ø = sec^2 ø.
sin^2 ø/cos^2 ø + cos^2 ø/cos^2 ø=1/cos^2 ø So tan^2 ø + 1=sec^2 ø
Show that 1 + cot^2 ø=cosec^2 ø.
sin^2 ø/sin^2 ø + cos^2 ø/sin^2 ø=1/sin^2 ø So 1+ cot^2 ø=cosec^2 ø
Solve sin 2x = sin x for 0<=x<=360.
x=0, 60, 180, 300, 360
Solve 3 tan x= tan 2x for 0<=x<=2TT.
x=0, TT/6, 5TT/6, TT, 7TT/6, 11TT/6, 2TT
Solve 3 sin x/2 + cos x = 2 for 0<=x<=360.
x=60, 180, 300
Solve 2 sec^2 ø + tan ø = 3 for 0<=ø<=2TT.
ø=0.5, 3TT/4, 3.6, 7TT/4
Solve 3 cos ø - 4 sin ø=2 for 0<=ø<=360.
ø=13.3, 240.4
Solve 2 sin 2ø + 8 cos^2 2ø=5 for 0<=ø<=360.
ø=24.3, 65.7, 204.3, 245.7, 105, 165, 285, 345
Solve 2 sin ø - 3 cos ø=1 for 0<=ø<=360.
ø=72.4, 220.2
Solve 2/(tan^2 ø) + 8=7 cosec ø for 0<=ø<=TT.
ø=TT/6, 0.7, 2.4, 5TT/6
Solve 2 sin(ø+60)=cos(ø- 45).
ø=tan^-1 (1-root6)/(root2-1)
Prove cosec ø + cot ø = cot ø/2.
(1+cosø)/sinø=(2cos^2 ø/2) / (2 sin ø/2 cos ø/2) =(cos ø/2) / (sin ø/2)= cot ø/2
tan(A+/-B)...
=(tan A +/- tan B)/(1 -/+ tanAtanB)
tan (arcsin -5/6)...
=-5/root11
Prove (1- cos ø)/sin ø = 1/(cosec ø + cot ø).
=1/((1/sin ø)+ (cos ø/sin ø)) =sin ø/(1+ cos ø) =sin ø(1-cos ø)/(1-cos^2 ø) =sin ø(1-cos ø)/sin^2 ø =(1-cos ø)/sin ø
sec 60...
=1/cos 60=1/0.5=2
sec x...
=1/cos x
cosec x...
=1/sin x
cot x...
=1/tan x
cot ø...
=1/tan ø=cos ø/sin ø
Express 15-(12cosx + 5sinx) in the form Rcos(x-a) and find the max and min values of y and the first positive x value for which this occurs.
=15-13cos(x-23) max y=28, x=203 min y=2, x=23
Prove cos 3x + cos 5x + cos 7x = cos 5x (2cos 2x +1)
=2 cos ((7x + 3x)/2) cos ((7x - 3x)/2) + cos 5x = 2 cos 5x cos 2x + cos 5x =cos 5x (2 cos 2x + 1)
Prove sin ø/(1+cos ø)=tan ø/2.
=2 sin ø/2 cos ø/2 / (1 + 2cos^2 ø/2 -1) =tan ø/2
tan (arccos x)...
=root(1-x^2)/x
cos 75...
=root2(root3-1)/4
sin (arccos 3/4)...
=root7/4
sin(A+/-B)...
=sinAcosB +/- sinBcosA
Prove (sin A/1 + cos A) + (1 +cos A/sin A)=2/sin A.
=sin^2 A + (1+cos A)^2/sin A(1 + cos A) =sin^2 A + cos^2 A + 1 + 2cos A/sin A(1 + cos A) =2+2cos A/sin A(1 + cos A) =2(1+cos A)/sin A(1 + cos A) =2/sin A
Prove tan ø + cot ø = 2cosec 2ø
=sinø/cosø + cosø/sinø =(sin^2 ø + cos^2 ø)/sinøcosø =2/2sinøcosø =2/sin 2ø =2cosec 2ø
Prove sec^2 ø+ cot^2 ø=tan^2 ø + cosec^2 ø
=tan^2 ø + 1 + cosec^2 ø -1 =tan^2 ø + cosec^2 ø
sin (arctan x)...
=x/root(1+x^2)