Soal Perkalian, Penjumlahan dan Pengurangan Sinus dan Cosinus serta Pembahasan

Hallo...! Pengunjung setia Catatan Matematika, kali ini Bang RP (Reikson Panjaitan, S.Pd) berbagi Kumpulan Soal Perkalian, Jumlah dan Selisih Sinus dan Cosinus berserta pembahasannya. Ayo... manfaatkan website Catatan Matematika ini untuk belajar matematika secara online. Untuk dapat menjawab soal-soal ini dengan baik, kalian perlu belajar materi berikut:
1. Rumus Perkalian Sinus dan Cosinus.
2. Rumus Penjumlahan/Pengurangan Sinus dan Cosinus

Tata Cara Belajar:
Cobalah mengerjakan soal-soal yang tersedia secara mandiri. Setelah itu cek jawaban kamu dengan pembahasan yang telah disediakan, dengan cara:
klik "LIHAT/TUTUP:".

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Soal No. 1

$2\sin 3a\cos a$ = ….
A. $\sin 4a+\sin 2a$
B. $\sin 4a-\sin 2a$
C. $\cos 4a+\cos 2a$
D. $\cos 4a-\cos 2a$
E. $\sin 4a+\cos 2a$

Penyelesaian: Lihat/Tutup

$2\sin 3a\cos a$
= $\sin (3a+a)+\sin (3a-a)$
= $\sin 4a+\sin 2a$
Jawaban: A

Soal No. 2

$2\sin 52,5{}^\circ \cos 7,5^\circ $ = ….
A. $\frac{1}{2}\sqrt{3}+\sqrt{2}$
B. $\frac{1}{2}(\sqrt{3}-\sqrt{2})$
C. $\frac{1}{2}(\sqrt{3}+\sqrt{2})$
D. $\frac{1}{4}(\sqrt{3}+\sqrt{2})$
E. $\frac{1}{2}\sqrt{3}-\sqrt{2}$

Penyelesaian: Lihat/Tutup

$2\sin 52,5^\circ \cos 7,5^\circ $
= $\sin (52,5^\circ +7,5^\circ )+\sin (52,5^\circ -7,5^\circ )$
= $\sin 60^\circ +\sin 45^\circ $
= $\frac{1}{2}\sqrt{3}+\frac{1}{2}\sqrt{2}$
= $\frac{1}{2}(\sqrt{3}+\sqrt{2})$
Jawaban: C

Soal No. 3

$\cot 105^\circ \tan 15^\circ $ = ….
A. $-7+4\sqrt{3}$
B. $7+4\sqrt{3}$
C. $7-4\sqrt{3}$
D. $-7-4\sqrt{3}$
E. $-7+2\sqrt{3}$

Penyelesaian: Lihat/Tutup

$\cot 105^\circ \tan 15^\circ $
= $\frac{\cos 105^\circ }{\sin 105^\circ }.\frac{\sin 15^\circ }{\cos 15^\circ }$
= $\frac{\cos 105^\circ \sin 15^\circ }{\sin 105^\circ \cos 15^\circ }$
= $\frac{\frac{1}{2}\left( \sin (105^\circ +15^\circ )-\sin (105^\circ -15^\circ ) \right)}{\frac{1}{2}\left( \sin (105^\circ +15^\circ )+\sin (105^\circ -15^\circ ) \right)}$
= $\frac{\sin 120^\circ -\sin 90^\circ }{\sin 120^\circ +\sin 90^\circ }$
= $\frac{\frac{1}{2}\sqrt{3}-1}{\frac{1}{2}\sqrt{3}+1}$
= $\frac{\sqrt{3}-2}{\sqrt{3}+2}\times \frac{\sqrt{3}-2}{\sqrt{3}-2}$
= $\frac{3-2\sqrt{3}-2\sqrt{3}+4}{3-4}$
= $\frac{7-4\sqrt{3}}{-1}$
= $-7+4\sqrt{3}$
Jawaban: A

Soal No. 4

$\cos 45^\circ \cos 15^\circ $ = ….
A. $\sqrt{3}+1$
B. $\frac{1}{2}\sqrt{3}+\frac{1}{2}$
C. $\frac{1}{4}\sqrt{3}+\frac{1}{4}$
D. $-\frac{1}{4}\sqrt{3}-\frac{1}{4}$
E. $-\frac{1}{2}\sqrt{3}-\frac{1}{2}$

Penyelesaian: Lihat/Tutup

$\cos 45^\circ \cos 15^\circ $
= $\frac{1}{2}\left( \cos (45^\circ +15^\circ )+\cos (45^\circ -15^\circ ) \right)$
= $\frac{1}{2}\left( \cos 60^\circ +\cos 30^\circ \right)$
= $\frac{1}{2}\left( \frac{1}{2}+\frac{1}{2}\sqrt{3} \right)$
= $\frac{1}{4}+\frac{1}{4}\sqrt{3}$
= $\frac{1}{4}\sqrt{3}+\frac{1}{4}$
Jawaban: C

Soal No. 5

Nilai dari $\sin 37,5^\circ \sin 7,5^\circ $ = ….
A. $\sqrt{2}-\sqrt{3}$
B. $\frac{1}{2}\sqrt{2}-\frac{1}{2}\sqrt{3}$
C. $\frac{1}{4}\sqrt{2}-\frac{1}{4}\sqrt{3}$
D. $\frac{1}{4}\sqrt{3}-\frac{1}{4}\sqrt{2}$
E. $\frac{1}{2}\sqrt{3}-\frac{1}{2}\sqrt{2}$

Penyelesaian: Lihat/Tutup

$\sin 37,5^\circ \sin 7,5^\circ $
= $-\frac{1}{2}\left( \cos (37,5^\circ +7,5^\circ )-\cos (37,5^\circ -7,5^\circ ) \right)$
= $-\frac{1}{2}\left( \cos 45^\circ -\cos 30^\circ \right)$
= $-\frac{1}{2}\left( \frac{1}{2}\sqrt{2}-\frac{1}{2}\sqrt{3} \right)$
= $-\frac{1}{4}\sqrt{2}+\frac{1}{4}\sqrt{3}$
= $\frac{1}{4}\sqrt{3}-\frac{1}{4}\sqrt{2}$
Jawaban: D

Soal No. 6

$2\cos \left( \frac{1}{4}\pi +A \right)\cos \left( \frac{1}{4}\pi -A \right)$ = ….
A. $2\cos 2A$
B. $2\sin 2A$
C. $\cos 2A$
D. $\sin 2A$
E. $\tan 2A$

Penyelesaian: Lihat/Tutup

$2\cos \left( \frac{1}{4}\pi +A \right)\cos \left( \frac{1}{4}\pi -A \right)$
= $2.\frac{1}{2}\left( \cos \left( \frac{1}{4}\pi +A+\frac{1}{4}\pi -A \right)+\cos \left( \frac{1}{4}\pi +A-\frac{1}{4}\pi +A \right) \right)$
= $\cos \left( \frac{1}{2}\pi \right)+\cos \left( 2A \right)$
= $0+\cos 2A$
= $\cos 2A$
Jawaban: C

Soal No. 7

Nilai dari $\cos 20^\circ \cos 40^\circ \cos 80^\circ $ = ….
A. 1
B. 0,5
C. 0,25
D. 0,125
E. 0,0625

Penyelesaian: Lihat/Tutup

$\cos 20^\circ \cos 40^\circ \cos 80^\circ $
= $(\cos 80^\circ \cos 40^\circ )\cos 20^\circ $
= $\frac{1}{2}\left( \cos (80^\circ +40^\circ )+\cos (80^\circ -40^\circ ) \right)\cos 20^\circ $
= $\frac{1}{2}\left( \cos 120^\circ +\cos 40^\circ \right)\cos 20^\circ $
= $\frac{1}{2}\left( -\frac{1}{2}+\cos 40^\circ \right)\cos 20^\circ $
= $\frac{1}{2}\left( -\frac{1}{2}\cos 20^\circ +\cos 40^\circ \cos 20^\circ \right)$
= $\frac{1}{2}\left( -\frac{1}{2}\cos 20^\circ +\frac{1}{2}\left( \cos (40^\circ +20^\circ )+\cos (40^\circ -20^\circ ) \right) \right)$
= $\frac{1}{2}\left( -\frac{1}{2}\cos 20^\circ +\frac{1}{2}\left( \cos 60^\circ +\cos 20^\circ \right) \right)$
= $\frac{1}{2}\left( -\frac{1}{2}\cos 20^\circ +\frac{1}{2}\left( \frac{1}{2}+\cos 20^\circ \right) \right)$
= $\frac{1}{2}\left( -\frac{1}{2}\cos 20^\circ +\frac{1}{4}+\frac{1}{2}\cos 20^\circ \right)$
= $\frac{1}{2}.\frac{1}{4}$
= $\frac{1}{8}$
= 0,125
Jawaban: D

Soal No. 8

Nilai dari $\cos 265^\circ -\cos 95^\circ $ = ….
A. $-2$
B. $-1$
C. 0
D. $\frac{1}{2}$
E. 2

Penyelesaian: Lihat/Tutup

$\cos 265^\circ -\cos 95^\circ $
= $-2\sin \frac{1}{2}(265^\circ +95^\circ )\sin \frac{1}{2}(265^\circ -95^\circ )$
= $-2\sin 180^\circ \sin 85^\circ $
= $-2.0.\sin 85^\circ $
= 0
Jawaban: C

Soal No. 9

Nilai dari $\sin 75^\circ -\sin 165^\circ $ adalah ….
A. $\frac{1}{4}\sqrt{2}$
B. $\frac{1}{4}\sqrt{3}$
C. $\frac{1}{4}\sqrt{6}$
D. $\frac{1}{2}\sqrt{2}$
E. $\frac{1}{2}\sqrt{6}$

Penyelesaian: Lihat/Tutup

$\sin 75^\circ -\sin 165^\circ $
= $2\cos \frac{1}{2}(75^\circ +165^\circ )\sin \frac{1}{2}(75^\circ -165^\circ )$
= $2\cos 120^\circ \sin (-45^\circ )$
= $2.\left( -\frac{1}{2} \right).\left( -\frac{1}{2}\sqrt{2} \right)$
= $\frac{1}{2}\sqrt{2}$
Jawaban: D

Soal No. 10

Nilai dari $\cos 195^\circ +\cos 105^\circ $ adalah ….
A. $\frac{1}{2}\sqrt{6}$
B. $\frac{1}{2}\sqrt{3}$
C. $\frac{1}{2}\sqrt{2}$
D. 0
E. $-\frac{1}{2}\sqrt{6}$

Penyelesaian: Lihat/Tutup

$\cos 195^\circ +\cos 105^\circ $
= $2\cos \frac{1}{2}(195^\circ +105^\circ )\cos \frac{1}{2}(195^\circ -105^\circ )$
= $2\cos 150^\circ \cos 45^\circ $
= $2.\left( -\frac{1}{2}\sqrt{3} \right).\frac{1}{2}\sqrt{2}$
= $-\frac{1}{2}\sqrt{6}$
Jawaban: E

Soal No. 11

$\sin 4x+\sin 2x$ = ….
A. $2\sin 3x\cos x$
B. $2\cos 3x\sin x$
C. $\sin 3x\cos x$
D. $\cos 3x\sin x$
E. $\sin 6x$

Penyelesaian: Lihat/Tutup

$\sin 4x+\sin 2x$
= $2\sin \frac{1}{2}(4x+2x)\cos \frac{1}{2}(4x-2x)$
= $2\sin 3x\cos x$
Jawaban: A

Soal No. 12

$\sin (3x-30^\circ )-\sin (3x+60^\circ )$ = ….
A. $-2\cos (3x+30^\circ )$
B. $-\sqrt{2}\cos (3x+15^\circ )$
C. $-\sqrt{2}\sin (3x+15^\circ )$
D. $-\sqrt{2}\cos (3x+15^\circ )$
E. $-\sqrt{2}\sin (3x+15^\circ )$

Penyelesaian: Lihat/Tutup

$\sin (3x-30^\circ )-\sin (3x+60^\circ )$
= $2\cos \frac{1}{2}(3x-30^\circ +3x+60^\circ )\sin \frac{1}{2}(3x-30^\circ -3x-60^\circ )$
= $2\cos \frac{1}{2}(6x+30^\circ )\sin \frac{1}{2}(-90^\circ )$
= $2\cos (3x+15^\circ )\sin (-45^\circ )$
= $2.\cos (3x+15^\circ ).\left( -\frac{1}{2}\sqrt{2} \right)$
= $-\sqrt{2}\cos (3x+15^\circ )$
Jawaban: D

Soal No. 13

$\frac{\cos 3x+\cos x}{\sin 3x+\sin x}$ = ….
A. $\tan 2x$
B. $\cot 2x$
C. $\tan x$
D. $\cot x$
E. $\sec 2x$

Penyelesaian: Lihat/Tutup

$\frac{\cos 3x+\cos x}{\sin 3x+\sin x}$
= $\frac{2\cos \frac{1}{2}(3x+x)\cos \frac{1}{2}(3x-x)}{2\sin \frac{1}{2}(3x+x)\cos \frac{1}{2}(3x-x)}$
= $\frac{2\cos 2x.\cos x}{2\sin 2x.\cos x}$
= $\frac{\cos 2x}{\sin 2x}$
= $\cot 2x$
Jawaban: D

Soal No. 14

Nilai dari $\sin 145^\circ -\sin 35^\circ -\sin 45^\circ $ = ….
A. $-\frac{1}{2}\sqrt{3}$
B. $-\frac{1}{2}\sqrt{2}$
C. $\frac{1}{2}$
D. $\frac{1}{2}\sqrt{2}$
E. $\frac{1}{2}\sqrt{3}$

Penyelesaian: Lihat/Tutup

$\sin 145^\circ -\sin 35^\circ -\sin 45^\circ $
= $\sin 145^\circ -\sin 35^\circ -\frac{1}{2}\sqrt{2}$
= $2\cos \frac{1}{2}(145^\circ +35^\circ )\sin \frac{1}{2}(145^\circ -35^\circ )-\frac{1}{2}\sqrt{2}$
= $2\cos 90^\circ \sin 55^\circ -\frac{1}{2}\sqrt{2}$
= $2.0.\sin 55^\circ -\frac{1}{2}\sqrt{2}$
= $-\frac{1}{2}\sqrt{2}$
Jawaban: B

Soal No. 15

Nilai dari $\cos 25^\circ +\cos 95^\circ +\cos 145^\circ $ = ….
A. $-1$
B. $-\frac{1}{2}$
C. 0
D. $\frac{1}{2}$
E. 1

Penyelesaian: Lihat/Tutup

$\cos 25^\circ +\cos 95^\circ +\cos 145^\circ $
= $\cos 25^\circ +2\cos \frac{1}{2}(95^\circ +145^\circ )\cos \frac{1}{2}(95^\circ -145^\circ )$
= $\cos 25^\circ +2\cos 120^\circ \cos (-25^\circ )$
= $\cos 25^\circ +2.\left( -\frac{1}{2} \right)\cos 25^\circ $
= $\cos 25^\circ -\cos 25^\circ $
= 0
Jawaban: C

Soal No. 16

Nilai dari $\cos 145^\circ +\cos 35^\circ -\cos 45^\circ $ = ….
A. $\frac{1}{2}\sqrt{3}$
B. $\frac{1}{2}\sqrt{2}$
C. $\frac{1}{2}$
D. $-\frac{1}{2}$
E. $-\frac{1}{2}\sqrt{2}$

Penyelesaian: Lihat/Tutup

$\cos 145^\circ +\cos 35^\circ -\cos 45^\circ $
= $\cos 145^\circ +\cos 35^\circ -\frac{1}{2}\sqrt{2}$
= $2\cos \frac{1}{2}(145^\circ +35^\circ )\cos \frac{1}{2}(145^\circ -35^\circ )-\frac{1}{2}\sqrt{2}$
= $2\cos 90^\circ \cos 55^\circ -\frac{1}{2}\sqrt{2}$
= $2.0.\cos 55^\circ -\frac{1}{2}\sqrt{2}$
= $-\frac{1}{2}\sqrt{2}$
Jawaban: E

Soal No. 17

Nilai dari $\sin 105^\circ -\sin 15^\circ $ sama dengan ….
A. 1
B. 0
C. $\frac{1}{4}\sqrt{2}$
D. $\frac{1}{2}\sqrt{2}$
E. $2\sqrt{6}$

Penyelesaian: Lihat/Tutup

$\sin 105^\circ -\sin 15^\circ $
= $2\cos \frac{1}{2}(105^\circ +15^\circ )\sin \frac{1}{2}(105^\circ -15^\circ )$
= $2\cos 60^\circ \sin 45^\circ $
= $2.\frac{1}{2}.\frac{1}{2}\sqrt{2}$
= $\frac{1}{2}\sqrt{2}$
Jawaban: D

Soal No. 18

Nilai dari $\frac{\sin 125^\circ +\sin 35^\circ }{\cos 125^\circ -\cos 35^\circ }$ = ….
A. $-1$
B. $-\frac{1}{2}\sqrt{2}$
C. $\frac{1}{2}\sqrt{2}$
D. 1
E. 2

Penyelesaian: Lihat/Tutup

$\frac{\sin 125^\circ +\sin 35^\circ }{\cos 125^\circ -\cos 35^\circ }$
= $\frac{2\sin \frac{1}{2}(125^\circ +35^\circ )\cos \frac{1}{2}(125^\circ -35^\circ )}{-2\sin \frac{1}{2}(125^\circ +35^\circ )\sin \frac{1}{2}(125^\circ -35^\circ )}$
= $\frac{2\sin 80^\circ \cos 45^\circ }{-2\sin 80^\circ \sin 45^\circ }$
= $-\frac{\cos 45^\circ }{\sin 45^\circ }$
= $-\frac{\frac{1}{2}\sqrt{2}}{\frac{1}{2}\sqrt{2}}$
= $-1$
Jawaban: A

Soal No. 19

Nilai dari $\tan 75^\circ -\tan 15^\circ $ adalah ….
A. 0
B. 1
C. $\sqrt{3}$
D. $2\sqrt{3}$
E. 4

Penyelesaian: Lihat/Tutup

$\tan 75^\circ -\tan 15^\circ $
= $\frac{\sin 75^\circ }{\cos 75^\circ }-\frac{\sin 15^\circ }{\cos 15^\circ }$
= $\frac{\sin 75^\circ \cos 15^\circ -\cos 75^\circ \sin 15^\circ }{\cos 75^\circ \cos 15^\circ }$
= $\frac{\sin (75^\circ -15^\circ )}{\frac{1}{2}\left( \cos (75^\circ +15^\circ )+\cos (75^\circ -15^\circ ) \right)}$
= $\frac{2\sin 60^\circ }{\cos 90^\circ +\cos 60^\circ }$
= $\frac{2.\frac{1}{2}\sqrt{3}}{0+\frac{1}{2}}$
= $2\sqrt{3}$
Jawaban: D

Soal No. 20

Nilai dari $\frac{\cos 115^\circ +\cos 5^\circ }{\sin 115^\circ +\sin 5^\circ }$ = ….
A. $-\sqrt{3}$
B. $-1$
C. $-\frac{1}{3}\sqrt{3}$
D. $\frac{1}{3}\sqrt{3}$
E. $\sqrt{3}$

Penyelesaian: Lihat/Tutup

$\frac{\cos 115^\circ +\cos 5^\circ }{\sin 115^\circ +\sin 5^\circ }$
=$\frac{2\cos \frac{1}{2}(115^\circ +5^\circ )\cos \frac{1}{2}(115^\circ -5^\circ )}{2\sin \frac{1}{2}(115^\circ +5^\circ )\cos \frac{1}{2}(115^\circ -5^\circ )}$
= $\frac{\cos 60^\circ \cos 55^\circ }{\sin 60^\circ \cos 55^\circ }$
= $\frac{\cos 60^\circ }{\sin 60^\circ }$
= $\frac{\frac{1}{2}}{\frac{1}{2}\sqrt{3}}$
= $\frac{1}{\sqrt{3}}$
= $\frac{1}{3}\sqrt{3}$
Jawaban: D

Soal No. 21

Nilai $\frac{\sin 81^\circ +\sin 21^\circ }{\sin 69^\circ -\sin 171^\circ }$ = ….
A. $\sqrt{3}$
B. $\frac{1}{2}\sqrt{3}$
C. $\frac{1}{3}\sqrt{3}$
D. $-\frac{1}{2}\sqrt{3}$
E. $-\sqrt{3}$

Penyelesaian: Lihat/Tutup

$\frac{\sin 81^\circ +\sin 21^\circ }{\sin 69^\circ -\sin 171^\circ }$
= $\frac{2\sin \frac{1}{2}(81^\circ +21^\circ )\cos \frac{1}{2}(81^\circ -21^\circ )}{2\cos \frac{1}{2}(69^\circ +171^\circ )\sin \frac{1}{2}(69^\circ -171^\circ )}$
= $\frac{\sin 51^\circ \cos 30^\circ }{\cos 120^\circ \sin (-51^\circ )}$
= $\frac{\sin 51^\circ \cos 30^\circ }{-\sin 51^\circ \cos 120^\circ }$
= $-\frac{\cos 30^\circ }{\cos 120^\circ }$
= $-\frac{\frac{1}{2}\sqrt{3}}{-\frac{1}{2}}$
= $\sqrt{3}$
Jawaban: A

Soal No. 22

Nilai dari $\sin 6^\circ -\sin 42^\circ -\sin 66^\circ +\sin 78^\circ $ adalah ….
A. $-1$
B. $-\frac{1}{2}$
C. 0
D. $\frac{1}{2}$
E. 1

Penyelesaian: Lihat/Tutup

$\sin 6^\circ -\sin 42^\circ -\sin 66^\circ +\sin 78^\circ $
= $\sin 6^\circ -\sin 66^\circ +\sin 78^\circ -\sin 42^\circ $
= $2\cos \frac{1}{2}(6^\circ +66^\circ )\sin \frac{1}{2}(6^\circ -66^\circ )$ + $2\cos \frac{1}{2}(78^\circ +42^\circ )\sin \frac{1}{2}(78^\circ -42^\circ )$
= $2\cos 36^\circ \sin (-30^\circ )$ + $2\cos 60^\circ \sin 18^\circ $
= $2\cos 36^\circ .\left( -\frac{1}{2} \right)$ + $2.\frac{1}{2}.\sin 18^\circ $
= $-\cos 36^\circ $ + $\sin 18^\circ $
= $-1+2{{\sin }^{2}}18^\circ +\sin 18^\circ $
= $-1+2{{\left( \frac{\sqrt{5}-1}{4} \right)}^{2}}+\frac{\sqrt{5}-1}{4}$
= $-1+2\left( \frac{6-2\sqrt{5}}{16} \right)+\frac{\sqrt{5}-1}{4}$
= $-1+2\left( \frac{3-\sqrt{5}}{8} \right)+\frac{\sqrt{5}-1}{4}$
= $-1+\frac{3-\sqrt{5}}{4}+\frac{\sqrt{5}-1}{4}$
= $-\frac{1}{2}$
(catatan: $\sin 18^\circ =\frac{\sqrt{5}-1}{4}$ pembuktiannya lihat disini).
Jawaban: B

Soal No. 23

Hasil dari $\frac{\sin 27^\circ +\sin 63^\circ }{\cos 138^\circ +\cos 102^\circ }$ = ….
A. $-\sqrt{2}$
B. $-\frac{1}{2}\sqrt{2}$
C. 1
D. $\frac{1}{2}\sqrt{2}$
E. $\sqrt{2}$

Penyelesaian: Lihat/Tutup

$\frac{\sin 27^\circ +\sin 63^\circ }{\cos 138^\circ +\cos 102^\circ }$
= $\frac{2\sin \frac{1}{2}(27^\circ +63^\circ )\cos \frac{1}{2}(27^\circ -63^\circ )}{2\cos \frac{1}{2}(138^\circ +102^\circ )\cos \frac{1}{2}(138^\circ -102^\circ )}$
= $\frac{\sin 45^\circ \cos (-18^\circ )}{\cos 120^\circ \cos 18^\circ }$
= $\frac{\sin 45^\circ \cos 18^\circ }{\cos (180^\circ -60^\circ )\cos 18^\circ }$
= $\frac{\sin 45^\circ }{-\cos 60^\circ }$
= $\frac{\frac{1}{2}\sqrt{2}}{-\frac{1}{2}}$
= $-\sqrt{2}$
Jawaban: A

Soal No. 24

Nilai dari $\frac{\sin 75^\circ +\sin 15^\circ }{\cos 105^\circ +\cos 15^\circ }$ = ….
A. $-\sqrt{3}$
B. $-\sqrt{2}$
C. $\frac{1}{3}\sqrt{3}$
D. $\sqrt{2}$
E. $\sqrt{3}$

Penyelesaian: Lihat/Tutup

$\frac{\sin 75^\circ +\sin 15^\circ }{\cos 105^\circ +\cos 15^\circ }$
= $\frac{2\sin \frac{1}{2}(75^\circ +15^\circ )\cos \frac{1}{2}(75^\circ -15^\circ )}{2\cos \frac{1}{2}(105^\circ +15^\circ )\cos \frac{1}{2}(105^\circ -15^\circ )}$
= $\frac{\sin 45^\circ \cos 30^\circ }{\cos 60^\circ \cos 45^\circ }$
= $\frac{\frac{1}{2}\sqrt{2}.\frac{1}{2}\sqrt{3}}{\frac{1}{2}.\frac{1}{2}\sqrt{2}}$
= $\sqrt{3}$
Jawaban: E

Soal No. 25

Nilai dari $\frac{\cos 15^\circ -\cos 105^\circ }{\sin 15^\circ -\sin 75^\circ }$ = ….
A. $\sqrt{3}$
B. $\frac{1}{2}\sqrt{3}$
C. $\frac{1}{\sqrt{3}}$
D. $-\frac{1}{\sqrt{3}}$
E. $-\sqrt{3}$

Penyelesaian: Lihat/Tutup

$\frac{\cos 15^\circ -\cos 105^\circ }{\sin 15^\circ -\sin 75^\circ }$
= $\frac{-2\sin \frac{1}{2}(15^\circ +105^\circ )\sin \frac{1}{2}(15^\circ -105^\circ )}{2\cos \frac{1}{2}(15^\circ +75^\circ )\sin \frac{1}{2}(15^\circ -75^\circ )}$
= $\frac{-\sin 60^\circ \sin (-45^\circ )}{\cos 45^\circ \sin (-30^\circ )}$
= $\frac{-\sin 60^\circ .(-\sin 45^\circ )}{\cos 45^\circ .(-\sin 30^\circ )}$
= $\frac{-\frac{1}{2}\sqrt{3}.\left( -\frac{1}{2}\sqrt{2} \right)}{\frac{1}{2}\sqrt{2}.\left( -\frac{1}{2} \right)}$
= $-\sqrt{3}$
Jawaban: E

Soal No. 26

Nilai dari $\frac{\sin 130^\circ -\sin 110^\circ }{\cos 70^\circ -\cos 50^\circ }$ = ….
A. $\sqrt{3}$
B. $\frac{1}{2}\sqrt{3}$
C. $\frac{1}{3}\sqrt{3}$
D. $\frac{1}{2}\sqrt{2}$
E. $\frac{1}{3}\sqrt{2}$

Penyelesaian: Lihat/Tutup

$\frac{\sin 130^\circ -\sin 110^\circ }{\cos 70^\circ -\cos 50^\circ }$
= $\frac{2\cos \frac{1}{2}(130^\circ +110^\circ )\sin \frac{1}{2}(130^\circ -110^\circ )}{-2\sin \frac{1}{2}(70^\circ +50^\circ )\sin \frac{1}{2}(70^\circ -50^\circ )}$
= $\frac{\cos 120^\circ \sin 10^\circ }{-\sin 60^\circ \sin 10^\circ }$
= $\frac{\cos 120^\circ }{-\sin 60^\circ }$
= $\frac{-\frac{1}{2}}{-\frac{1}{2}\sqrt{3}}$
= $\frac{1}{\sqrt{3}}$
= $\frac{1}{3}\sqrt{3}$
Jawaban: C

Soal No. 27

Nilai $\frac{\cos 140^\circ -\cos 100^\circ }{\sin 140^\circ -\sin 100^\circ }$ = ….
A. $-\sqrt{3}$
B. $-\frac{1}{2}\sqrt{3}$
C. $-\frac{1}{3}\sqrt{3}$
D. $\frac{1}{3}\sqrt{3}$
E. $\sqrt{3}$

Penyelesaian: Lihat/Tutup

$\frac{\cos 140^\circ -\cos 100^\circ }{\sin 140^\circ -\sin 100^\circ }$
= $\frac{-2\sin \frac{1}{2}(140^\circ +100^\circ )\sin \frac{1}{2}(140^\circ -100^\circ )}{2\cos \frac{1}{2}(140^\circ +100^\circ )\sin \frac{1}{2}(140^\circ -100^\circ )}$
= $\frac{-\sin 120^\circ \sin 20^\circ }{\cos 120^\circ \sin 20^\circ }$
= $\frac{-\sin 120^\circ }{\cos 120^\circ }$
= $\frac{-\frac{1}{2}\sqrt{3}}{-\frac{1}{2}}$
= $\sqrt{3}$
Jawaban: E

Soal No. 28

$\frac{\sin 5x-\sin x}{\cos 4x+\cos 2x}$ = ….
A. $2\cos x$
B. $2\sin x$
C. $\cos x$
D. $\sin x$
E. $2\sec x$

Penyelesaian: Lihat/Tutup

$\frac{\sin 5x-\sin x}{\cos 4x+\cos 2x}$
= $\frac{2\cos \frac{1}{2}(5x+x)\sin \frac{1}{2}(5x-x)}{2\cos \frac{1}{2}(4x+2x)\cos \frac{1}{2}(4x-2x)}$
= $\frac{2\cos 3x\sin 2x}{2\cos 3x\cos x}$
= $\frac{\sin 2x}{\cos x}$
= $\frac{2\sin x\cos x}{\cos x}$
= $2\sin x$
Jawaban: B

Soal No. 29

Nilai dari $\frac{\cos 100^\circ +\cos 40^\circ }{\sin 130^\circ -\sin 10^\circ }$ = ….
A. $-2$
B. $-1$
C. 0,5
D. 1
E. 2

Penyelesaian: Lihat/Tutup

$\frac{\cos 100^\circ +\cos 40^\circ }{\sin 130^\circ -\sin 10^\circ }$

= $\frac{2\cos \frac{1}{2}(100^\circ +40^\circ )\cos \frac{1}{2}(100^\circ -40^\circ )}{2\cos \frac{1}{2}(130^\circ +10^\circ )\sin \frac{1}{2}(130^\circ -10^\circ )}$
= $\frac{\cos 70^\circ \cos 30^\circ }{\cos 70^\circ \sin 60^\circ }$
= $\frac{\cos 30^\circ }{\sin 60^\circ }$
= $\frac{\frac{1}{2}\sqrt{3}}{\frac{1}{2}\sqrt{3}}$
= 1
Jawaban: D

Soal No. 30

Nilai dari $\frac{\sin 135^\circ -\sin 15^\circ }{\cos 135^\circ +\cos 15^\circ }$ = ….
A. $\sqrt{3}$
B. $\frac{1}{2}\sqrt{2}$
C. $\frac{1}{2}$
D. $-\frac{1}{2}$
E. $-\frac{1}{2}\sqrt{3}$

Penyelesaian: Lihat/Tutup

$\frac{\sin 135^\circ -\sin 15^\circ }{\cos 135^\circ +\cos 15^\circ }$
= $\frac{2\cos \frac{1}{2}(135^\circ +15^\circ )\sin \frac{1}{2}(135^\circ -15^\circ )}{2\cos \frac{1}{2}(135^\circ +15^\circ )\cos \frac{1}{2}(135^\circ -15^\circ )}$
= $\frac{\cos 75^\circ \sin 60^\circ }{\cos 75^\circ \cos 60^\circ }$
= $\frac{\sin 60^\circ }{\cos 60^\circ }$
= $\frac{\frac{1}{2}\sqrt{3}}{\frac{1}{2}}$
= $\sqrt{3}$
Jawaban: A

Soal No. 31

Nilai $\frac{\sin 75^\circ +\sin 15^\circ }{\cos 105^\circ -\cos 15^\circ }$ = ….
A. $-\frac{1}{3}\sqrt{3}$
B. $-\frac{1}{2}\sqrt{2}$
C. $-1$
D. $\frac{1}{2}$
E. 1

Penyelesaian: Lihat/Tutup

$\frac{\sin 75^\circ +\sin 15^\circ }{\cos 105^\circ -\cos 15^\circ }$
= $\frac{2\sin \frac{1}{2}(75^\circ +15^\circ )\cos \frac{1}{2}(75^\circ -15^\circ )}{-2\sin \frac{1}{2}(105^\circ +15^\circ )\sin \frac{1}{2}(105^\circ -15^\circ )}$
= $\frac{\sin 45^\circ \cos 30^\circ }{-\sin 60^\circ \sin 45^\circ }$
= $-\frac{\cos 30^\circ }{\sin 60^\circ }$
= $-\frac{\frac{1}{2}\sqrt{3}}{\frac{1}{2}\sqrt{3}}$
= $-1$
Jawaban: C

Soal No. 32

Bentuk $\frac{\sin 3A-\sin A}{\cos 3A-\cos A}$ ekuivalen dengan ….
A. $\tan 2A$
B. $-\tan 2A$
C. $-\cot 2A$
D. $\cot 2A$
E. $\sec 2A$

Penyelesaian: Lihat/Tutup

$\frac{\sin 3A-\sin A}{\cos 3A-\cos A}$
= $\frac{2\cos \frac{1}{2}(3A+A)\sin \frac{1}{2}(3A-A)}{-2\sin \frac{1}{2}(3A+A)\sin \frac{1}{2}(3A-A)}$
= $\frac{\cos 2A\sin A}{-\sin 2A\sin A}$
= $-\frac{\cos 2A}{\sin 2A}$
= $-\cot 2A$
Jawaban: C

Soal No. 33

$\sin 40^\circ +\cos 50^\circ +\sin 80^\circ -\cos 190^\circ $ = ….
A. $\sqrt{3}\cos 20^\circ $
B. $2\sqrt{3}\cos 20^\circ $
C. $4\sqrt{3}\cos 20^\circ $
D. $\sqrt{3}\sin 20^\circ $
E. $2\sqrt{3}\sin 20^\circ $

Penyelesaian: Lihat/Tutup

$\sin 40^\circ +\cos 50^\circ +\sin 80^\circ -\cos 190^\circ $
= $\sin 40^\circ +\sin 80^\circ $ + $\cos 50^\circ -\cos 190^\circ $
= $2\sin \frac{1}{2}(40^\circ +80^\circ )\cos \frac{1}{2}(40^\circ -80^\circ )$ + $-2\sin \frac{1}{2}(50^\circ +190^\circ )\sin \frac{1}{2}(50^\circ -190^\circ )$
= $2\sin 60^\circ \cos (-20^\circ )-2\sin 120^\circ \sin (-70^\circ )$
= $2.\frac{1}{2}\sqrt{3}\cos 20^\circ -2.\frac{1}{2}\sqrt{3}.(-\sin 70^\circ )$
= $\sqrt{3}\cos 20^\circ +\sqrt{3}\sin 70^\circ $
= $\sqrt{3}(\cos 20^\circ +\sin 70^\circ )$
= $\sqrt{3}(\cos 20^\circ +\sin (90^\circ -20^\circ ))$
= $\sqrt{3}(\cos 20^\circ +\cos 20^\circ )$
= $\sqrt{3}(2\cos 20^\circ )$
= $2\sqrt{3}\cos 20^\circ $
Jawaban: B

Soal No. 34

Himpunan penyelesaian dari persamaan $\cos (x+210^\circ )+\cos (x-210^\circ )=\frac{1}{2}\sqrt{3}$ untuk $0^\circ \le x\le 360^\circ $ adalah ….
A. $\{150^\circ ,210^\circ \}$
B. $\{210^\circ ,300^\circ \}$
C. $\{210^\circ ,330^\circ \}$
D. $\{300^\circ ,330^\circ \}$
E. $\{120^\circ ,240^\circ \}$

Penyelesaian: Lihat/Tutup

Ingat: Rumus trigonometri jumlah/selisih sinus dan cosinus.
$\cos \alpha +\cos \beta =2\cos \frac{1}{2}(\alpha +\beta )\cos \frac{1}{2}(\alpha -\beta )$ maka:
$\begin{align}\cos (x+210^\circ )+\cos (x-210^\circ ) &= \frac{1}{2}\sqrt{3} \\ 2\cos x\cos 210^\circ &= \frac{1}{2}\sqrt{3} \\ 2\cos x.\left( -\frac{1}{2}\sqrt{3} \right) &= \frac{1}{2}\sqrt{3} \\ 2\cos x &= -1 \\ \cos x &= -\frac{1}{2} \end{align}$
maka $x=120^\circ $ atau $x=240^\circ $
HP = $\{120^\circ ,240^\circ \}$
Jawaban: E

Soal No. 35

Himpunan penyelesaian dari persamaan $\sin (x+210^\circ )+\sin (x-210^\circ )=\frac{1}{2}\sqrt{3}$ untuk $0^\circ \le x\le 360^\circ $ adalah ….
A. $\{120^\circ ,240^\circ \}$
B. $\{210^\circ ,300^\circ \}$
C. $\{210^\circ ,330^\circ \}$
D. $\{300^\circ ,330^\circ \}$
E. $\{120^\circ ,240^\circ \}$

Penyelesaian: Lihat/Tutup

Ingat:
Rumus Trigonometri Jumlah/Selisih Sinus dan Cosinus.
$\sin \alpha +\sin \beta =2\sin \frac{1}{2}(\alpha +\beta )\cos \frac{1}{2}(\alpha -\beta )$ maka:
$\begin{align}\sin (x+210^\circ )+\sin (x-210^\circ ) &= \frac{1}{2}\sqrt{3} \\ 2\sin x\cos 210^\circ &= \frac{1}{2}\sqrt{3} \\ 2\sin x.\left( -\frac{1}{2}\sqrt{3} \right) &= \frac{1}{2}\sqrt{3} \\ 2\sin x &= -1 \\ \sin x &= -\frac{1}{2} \end{align}$
diperoleh $x=210^\circ $ atau $x=330^\circ $
HP = $\{210^\circ ,330^\circ \}$
Jawaban: C

Semoga postingan: Soal Perkalian, Penjumlahan dan Pengurangan Sinus dan Cosinus serta Pembahasan ini bisa bermanfaat. Mohon keikhlasan hatinya, membagikan postingan ini di media sosial bapak/ibu guru dan adik-adik sekalian. Terima kasih.

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