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	$-1\leq y\leq 1$
Cos A	1	${\frac {\sqrt {3}}{2}}$	${\frac {4}{5}}$	${\frac {\sqrt {2}}{2}}$	${\frac {3}{5}}$	${\frac {1}{2}}$	0	$-1\leq y\leq 1$
Tan A	0	${\frac {\sqrt {3}}{3}}$	${\frac {3}{4}}$	$1$	${\frac {4}{3}}$	${\sqrt {3}}$	$\infty$	$-\infty \leq y\leq \infty$
Cot A	$\infty$	${\sqrt {3}}$	${\frac {4}{3}}$	$1$	${\frac {3}{4}}$	${\frac {\sqrt {3}}{3}}$	0	$-\infty \leq y\leq \infty$
Sec A	1	${\frac {2{\sqrt {3}}}{3}}$	${\frac {5}{4}}$	${\sqrt {2}}$	${\frac {5}{3}}$	$2$	$\infty$	$y\leq -1{\text{ atau }}y\geq 1$
Csc A	$\infty$	$2$	${\frac {5}{3}}$	${\sqrt {2}}$	${\frac {5}{4}}$	${\frac {2{\sqrt {3}}}{3}}$	1	$y\leq -1{\text{ atau }}y\geq 1$

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 = ${\frac {2tanA}{1-tan^{2}A}}$

rangkap tiga sudut

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

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

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

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

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