Soal-Soal Matematika/Trigonometri

Trigonometri

1. $sinA=\frac{\text{sisi depan}}{\text{sisi miring}}=\frac{y}{r}$
2. $cosA=\frac{\text{sisi samping}}{\text{sisi miring}}=\frac{x}{r}$
3. $tanA=\frac{\text{sisi depan}}{\text{sisi samping}}=\frac{y}{x}$
4. $cscA=\frac{\text{sisi miring}}{\text{sisi depan}}=\frac{r}{y}=\frac{1}{sinA}$
5. $secA=\frac{\text{sisi miring}}{\text{sisi samping}}=\frac{r}{x}=\frac{1}{cosA}$
6. $cotA=\frac{\text{sisi samping}}{\text{sisi depan}}=\frac{x}{y}=\frac{1}{tanA}$

Sudut Istimewa

Trigonometri

Nama sudut	0°	30°	37°	45°	53°	60°	90°	Hasil interval
Sin A	0	$\frac{1}{2}$	$\frac{3}{5}$	$\frac{\sqrt{2}}{2}$	$\frac{4}{5}$	$\frac{\sqrt{3}}{2}$	1	$-1\le y\le 1$
Cos A	1	$\frac{\sqrt{3}}{2}$	$\frac{4}{5}$	$\frac{\sqrt{2}}{2}$	$\frac{3}{5}$	$\frac{1}{2}$	0	$-1\le y\le 1$
Tan A	0	$\frac{\sqrt{3}}{3}$	$\frac{3}{4}$	$1$	$\frac{4}{3}$	$\sqrt{3}$	$\mathrm{\infty }$	$-\mathrm{\infty }\le y\le \mathrm{\infty }$
Cot A	$\mathrm{\infty }$	$\sqrt{3}$	$\frac{4}{3}$	$1$	$\frac{3}{4}$	$\frac{\sqrt{3}}{3}$	0	$-\mathrm{\infty }\le y\le \mathrm{\infty }$
Sec A	1	$\frac{2\sqrt{3}}{3}$	$\frac{5}{4}$	$\sqrt{2}$	$\frac{5}{3}$	$2$	$\mathrm{\infty }$	$y\le -1\text{ atau }y\ge 1$
Csc A	$\mathrm{\infty }$	$2$	$\frac{5}{3}$	$\sqrt{2}$	$\frac{5}{4}$	$\frac{2\sqrt{3}}{3}$	1	$y\le -1\text{ atau }y\ge 1$

Rumus lainnya

pythagoras trigonometri

• sin²A + cos²A = 1
• tan²A + 1 = sec²A
• 1 + cot²A = csc²A

jumlah dan selisih sudut

• sin (A+B) = sin A cos B + cos A sin B
• sin (A-B) = sin A cos B - cos A sin B
• cos (A+B) = cos A cos B - sin A sin B
• cos (A-B) = cos A cos B + sin A sin B
• tan (A+B) = $\frac{tanA+tanB}{1-tanA\cdot tanB}$
• tan (A-B) = $\frac{tanA-tanB}{1+tanA\cdot tanB}$

perkalian trigonometri

• 2 sin A cos B = sin (A+B) + sin (A-B)
• 2 cos A sin B = sin (A+B) - sin (A-B)
• 2 cos A cos B = cos (A+B) + cos (A-B)
• -2 sin A sin B = cos (A+B) - cos (A-B)

jumlah dan selisih trigonometri

• sin A + sin B = $2sin(\frac{A+B}{2})cos(\frac{A-B}{2})$
• sin A - sin B = $2cos(\frac{A+B}{2})sin(\frac{A-B}{2})$
• cos A + cos B = $2cos(\frac{A+B}{2})cos(\frac{A-B}{2})$
• cos A - cos B = $-2sin(\frac{A+B}{2})sin(\frac{A-B}{2})$

rangkap dua sudut

• sin 2A = 2 sin A cos A
• cos 2A = cos²A-sin²A = 2 cos²A-1 = 1-2 sin²A
• tan 2A = $\frac{2tanA}{1-ta{n}^{2}A}$

rangkap tiga sudut

• sin 3A = 3 sin A - 4 sin³A
• cos 3A = 4 cos³A - 3 cos A
• tan 3A = $\frac{3tanA-ta{n}^{3}A}{1-3ta{n}^{2}A}$

setengah sudut

• sin 1/2A = $\sqrt{\frac{1-cosA}{2}}$
• cos 1/2A = $\sqrt{\frac{1+cosA}{2}}$
• tan 1/2A = $\sqrt{\frac{1-cosA}{1+cosA}}$ = $\frac{sinA}{1+cosA}$ = $\frac{1-cosA}{sinA}$

aturan sinus

$\frac{A}{sinA}=\frac{B}{sinB}=\frac{C}{sinC}=2R$

$L=\frac{a\cdot b}{2}sinC=\frac{a\cdot c}{2}sinB=\frac{b\cdot c}{2}sinA$

aturan kosinus

$a^2=b^2+c^2-2\cdot b\cdot c\cdot cosA$

$b^2=a^2+c^2-2\cdot a\cdot c\cdot cosB$

$c^2=a^2+b^2-2\cdot a\cdot b\cdot cosC$