C. Rumus perkAliAn sinus dAn cosinus

Buktikan1. 2 sin A cosB = sin (A+B) + sin(A-B)2. 2 cos A sinB = sin (A+B) - sin(A-B)3. 2 cos A cosB = cos (A+B) + cos(A-B)4. -2 sinA sinB = cos (A+B) - cos(A-B)

PemBuktiAn

1. 2 sin A cos B = sin (A+B)+sin(A-B)Ruas kanan=> = sin(A+B) + sin(A-B)=sinAcosB+cosAsinB+(sinAcosB-

cosAsinB)= 2 sin A cos B (terbukti)

PemBuktiAn

2. 2 cosAsinB= sin(A+B) – sin(A-B)Ruas kanan=> = sin(A+B) – Sin(A-B) =sinAcosB+cosAsinB-

(sinAcosBcosAsinB) = 2 cos A sinB (terbukti)

PemBuktiAn

3. 2cosAcosB= cos(A+B)+cos(A-B)Ruas kanan=> =cos(A+B) + cos(A-B) =cosAcosB-sinAsinB+

(cosAcosB+sinAsinB) = 2 cos A cos B (terbukti)

PemBuktiAn

4. -2sinAsinB = cos(A+B) - cos(A-B)Ruas kanan=> =cos(A+B) – cos(A-B) =cosAcosB-sinAsinB-

(cosAcosB+sinAsinB) =-2 sin A sin B

(terbukti)

Contoh soAl

1. Carilah :a. 4Sin70 Cos30 b. 6Cos50 Sin25 c. 4Cos70 Cos15 d. -4Sin65 Sin35

JAwAB:a. 4Sin70 Cos30 = 2[sin(70+30)+sin(70-30)]

=2[sin100+sin40]

b. 6Cos50 Sin25 =3[sin(50+25) – sin(50-25)] = 3[ sin 75 – sin 25]

c. 4Cos70 Cos15 = 2[cos(70+15)+cos(70-15)]= 2[cos 85 +cos 55]

d. -4Sin65 Sin35 = 2[cos(65+35)–cos(65-35)= 2[cos 100 - cos 30]

Contoh SoAl

2. Tanpa tabel dan kalkulator hitung nilai dari :

Cos 75.Sin 15 !

JAwAB:

2. Cos75.Sin15

= ½(Sin(75+15)-Sin(75-15))

= ½(Sin 90 – Sin 60)

= ½(1- ½√3)

= ¼(2 -√3)