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)