Latihan 1
1. Hitunglah dengan rumus sinus jumlah dan selisih sudut berikut.
a. \(\sin 105^o\)
b. \(\sin 75^o \cos 15^o - \cos 75^o \sin 15^o\)
2. Hitunglah dengan rumus cosinus jumlah dan selisih sudut beeikut.
a. \(\cos 195^o\)
b. \(\cos 58^o \cos 13^o -\cos 75^o \sin 15^o\)
3. Diketahui \(\sin A = \frac {3}{5}\),\(\cos B = \frac{5}{13}\) dan A dan B merupakan sudut lancip.
a. Tentukan \(\tan (A+B)\)
b. Tentukan \(\tan (A-B)\)
4. Diketahui \(\angle A\) dan \(\angle B\) adalah sudut lancip.Jika \(\cos A =\frac{4}{5}\) dan \(\cos B = \frac{24}{25}\). Tentukan :
a.\(\cos (A+B)\)
b.\(\sin (A-B)\)
5. Sederhakanlah : \(\tan (x+45^o)\).\(\tan (x-45^o)\).
Penyelesaian :
1. a. Tentukan nilai dari \( \sin 105^o \)
= \(\sin (60^o + 45^o)\)
= \( \sin 60 . cos 45 + \sin 45 \cos 60\)
= \( (\frac{1}{2} \sqrt{3})(\frac {1}{2}\sqrt{2})(\frac{1}{2})\)
= \( \frac {1}{4}\sqrt{6}+\frac{1}{4}\sqrt{2}\)
= \( \frac {1}{4}(\sqrt{6}+\sqrt{2}\)
b. \(\sin 75^o \cos 15^o - \cos75^o \sin 15^o \)
= \( \sin (45+30).\cos (45-30)- \cos (45+30)\sin (45-30)\)
= \((\sin 45 \cos 30 + \sin 30 \cos 45)(\cos 45 \cos 30- \sin 45 \sin 30)- (\cos 45 \cos 30 - \sin 30 \sin 45)(\sin 45 \cos 30- \sin 45 \sin 30)\)
= \(\frac{1}{2}\sqrt{2}. \frac {1}{2}\sqrt{3} + \frac{1}{2}. \frac{1}{2}\sqrt{2})(\frac{1}{2}\sqrt{2}.\frac {1}{2}\sqrt{3}-\frac {1}{2}\sqrt{2}.\frac {1}{2})-(\frac {1}{2}\sqrt{2}.\frac {1}{2}\sqrt{3}-\frac{1}{2}.\frac {1}{2}\sqrt{2})(\frac{1}{2}\sqrt{2}.\frac {1}{2}\sqrt{3}-\frac{1}{2}\sqrt{2}.\frac{1}{2})\)
= \(\frac{1}{4}\sqrt{6}+\frac{1}{4}\sqrt{2})(\frac{1}{4}\sqrt{6}-\frac{1}{4}\sqrt{2})-(\frac{1}{4}\sqrt{6}-\frac{1}{4}\sqrt{2})(\frac{1}{4}\sqrt{6}-\frac{1}{4}\sqrt{2})\)
= \(\frac{6}{16}-\frac{2}{16}-\frac{6}{16}-\frac{2}{16}+\frac{1}{4}\sqrt{3}\)
= \(\frac{1}{4}\sqrt{3}-\frac{4}{16}\)
2. a. \(\cos 195^o\)
= \( \cos (150 + 45)\)
= \( \cos ( \cos 150 \cos 45 - \sin 15 \sin 45\)
= \( \frac {1}{2} \sqrt{3} x \frac{1}{2} \sqrt{2}- \frac{1}{2} x \frac{1}{2}\sqrt{2}\)
= \( \frac {1}{4} \sqrt{6} - \frac{1}{4} \sqrt{2}\)
= \( \frac {1}{4} (\sqrt{6}-\sqrt{2})\)
b. \(\cos 58^o \cos 13^o -\cos 75^o \sin 15^o\)
= \( \cos (58-13)^o)\)
= \( \cos 45^o\)
= \( \frac{1}{2}\sqrt{2}\)
3. \( \sin A = \frac{3}{5}\)
sisi samping sudut = \(\sqrt{5^2-3^2} = \sqrt {25-9}= \sqrt{16} = 4\)
\( \tan A = \frac{3}{4}\)
\( \cos B = \frac{5}{13}\)
sisi samping sudut = \( \sqrt {13^2-5^2} = \sqrt{169-25}= \sqrt{144}= 12\)
\( \tan B = \frac {12}{5}\)
a. Tentukan \(\tan (A+B)\)
= \(\frac{\tan \alpha - \tan \beta}{1 - \tan\alpha. \tan \beta} \)
= \(\frac {\frac{3}{4}- \frac{12}{5}}{1 - \frac {3}{4}.\frac{12}{5}}\)
= \(\frac {(\frac{63}{20})}{(\frac{56}{20})}\)
= \(\frac {-33}{56}\)
b. Tentukan \(\tan (A-B)\)
= \(\frac{\tan \alpha - \tan \beta}{1 + \tan\alpha. \tan \beta}\)
= \(\frac {\frac{3}{4}- \frac{12}{5}}{1 + \frac {3}{4}.\frac{12}{5}}\)
= \(-\frac{33}{56}\)
4. \(\cos A = \frac{4}{5}\)
\(\cos A = \frac {3}{5}\)
\(\cos B = \frac {24}{25}\)
\(\cos B = \frac {7}{25}\)
a.\(\cos (A+B)\)
\(= \cos A.\cos B - \sin A \sin B\)
\(= \frac{96}{125} - \frac{21}{125}\)
\(= \frac{75}{125}\)
\(= \frac{3}{5}\)
b.\(\sin (A-B)\)
\(= \sin A. \cos B - \cos A \sin B\)
\(= \frac{72}{125} - \frac{28}{125}\)
\(= \frac{3}{5}\)
5. \(\tan (x+45) x \tan (x-45)\)
\(= \frac{\tan x + \tan 45}{1-\tan x \tan45} x \frac{\tan x - \tan 45}{1+\tan x \tan 45}\)
\(= \frac {\tan x + 1 }{1-\tan x X 1} X \frac{\tan x - 1}{1+ \tan x X 1}\)
\(= \frac {\tan x +1}{-(\tan x - 1} X \frac{\tan x -1}{\tan x+1}\)
\(= -1\)
Latihan 2
1. Diketahui \(\sin A = \frac{12}{13}, 0, \angle A \angle \frac{1}{2}\pi\).
a.Tentukan nilai dari \(\sin 2A\)
b.Tentukan nilai dari \(\cos 2A\)
c.Tentukan nilai dari \(\tan 2A\)
2. Tanpa Tabel logaritma dan kalkulator,Hitunglah :
a. \(2 \sin 75^o cos 15^o\)
b. \(\frac{\sin 81^o + \sin 15^o}{\sin 69^o - \sin 171^o}\)
3. Jika \(\sin A = \frac{12}{13}\) dan A terletak di kuadran II, tentukan dari :
a. \( \sin 2A\)
b. \( \cos 2A\)
4. Hitunglah :
a. \( \sin 67,5^o\)
b. \( \cos 22,5^o\)
c. \( \tan 15^o\)
5. Jika \(\cos 2A=\frac{8}{10}\) dan A sudut lancip. Tentukan tan A.
Pembahasan :
1. \(\sin a = \frac{12}{13}\)
samping :
\( \sqrt{13^2-12^2} \)
\(= \sqrt{168-144}\)
\(=\sqrt{25}\)
\(=5\)
a. \(\sin 2a \)
\(= 2 x \sin a \)
\(= 2 x (\frac{12}{13})\)
\(= 5\)
b. \(\cos 2a \)
\(= 2 x \cos a \)
\(= 2 x (\frac{5}{13})\)
\(=\frac{10}{13}\)
c. \(\tan 2a \)
\(= 2x \tan a\)
\(= 2 x (\frac{12}{5})\)
\(=\frac{24}{5}\)
2. a. \( 2 \sin 75 \cos 15 \)
\(= \sin (75+15)+ \sin (75-15)\)
\(= \sin 90 + \sin 60\)
\(=\frac{(2+\sqrt{3})}{2}\)
b. \(\frac{\sin 81^o + \sin 15^o}{\sin 69^o - \sin 171^o}\)
\(=\frac{2 \sin \frac{1}{2}(81 + 21) \cos \frac{1}{2}(81 - 21)}{2.\cos \frac{1}{2}(69 + 171)(\sin \frac{1}{2}(69 - 171)}\)
\(=\frac{2 \sin \frac{1}{2} (102).\cos \frac{1}{2} (60)} {2.\cos \frac{1}{2} (240).\sin \frac{1}{2} (- 102)}\)
\(=\frac {\cos \frac{1}{2} (60)} {- \cos \frac{1}{2} (240)}\)
\(=\frac{\cos 30}{-\cos120}\)
\(=\frac{\frac{1}{2}\sqrt{3}}{-(-\frac{1}{2}}\)
\(=\sqrt{3}\)
3. \(\sin A = \frac{12}{13}, \cos a = \frac {-5}{13}\)
a.\(\sin 2a = 2(\frac{12}{13})(-\frac{5}{13}=-\frac{120}{169}\)
b.\(\cos 2a = (-\frac{5}{13})^2 - (\frac{12}{13})^2 = -\frac{119}{169}\)
4.a.\( \sin 67,5 \)
\(= \sqrt{\frac{1}{2}}(1-\cos 135)\)
\(= \sqrt{\frac{1}{2}}(1-(\frac{1}{2}\sqrt{2}))\)
b.\(\cos 22,5 \)
\(= \sqrt{\frac{1}{2}}(\cos 45-1)\)
\(= \sqrt{\frac{1}{2}}(\frac{1}{2}\sqrt{2}-1)\)
\(= \sqrt{\frac{1}{4}}(\sqrt{2}-1)\)
\(= \frac{1}{2}\sqrt{(\sqrt{2}-2)}\)
c.\(\tan 15 \)
\(= \frac{1-\cos x }{\sin x}\)
\(=\frac{1-\cos30}{\sin 30}\)
\(=\frac{1-\frac{1}{2}\sqrt{3}}{\frac{1}{2}}\)
\(=2(1-\frac{1}{2}\sqrt{3})\)
\(= 2-\sqrt{3}\)
5. \(\tan A = \frac{\sqrt {1-\cos 2a}}{\sqrt{1+\cos 2a}}\)
\(\tan A = \frac{\sqrt{1-\frac{8}{10}}}{1+\frac{8}{10}}\)
\(\tan A = \frac{\sqrt{\frac{2}{10}}}{\sqrt{\frac{18}{10}}}\)
\(\tan A = \sqrt{\frac{2}{10}}x\sqrt{\frac{10}{18}}\)
\(\tan A = \frac{\sqrt{2}}{\sqrt{18}}\)
\(\tan A = \sqrt{\frac{2}{3}}\sqrt{2}\)
\(\tan A = \frac{1}{3}\)
Latihan 3
1. Tentukan nilai dari :
a. \( \cos 120^o \sin 60^o\)
b. \( \sin 75^o \cos 15^o\)
2. Tentukan Nilai dari :
a. \( 2 \sin 52 \frac{1}{2}^o \sin 7 \frac{1}{2}^o\)
b. \(2 \cos 52 \frac{1}{2}^o \cos 7 \frac{1}{2}^o\)
3. Tentukan nilai dari :
a. \( \sin \frac{5}{12}\pi \cos \frac{1}{6}\pi\)
Pembahasan :
1. a. \( \cos 120^o \sin 60^o\)
\(=\cos 120 \sin 60 \)
\(= (-\frac{1}{2})(\sqrt{\frac{3}{2}}\)
\(= \sqrt{\frac{3}{4}}\)
b. \( \sin 75^o \cos 15^o\)
\(= \sin (45+30). \cos (45-30)\)
\(= (\sin 45. \cos 30 + \sin 30. \cos 45).( \cos 45.\sin 30 + \sin 45.\cos 30)\)
\(= 2 (\frac{\sqrt{2}}{2}.\frac{\sqrt{3}}{2}+\frac{1}{2}.\frac{\sqrt{2}}{2})\)
\(= 2 (\frac{\sqrt {6}+ \sqrt{2})}{2}\)
\(= \frac{1}{2}(\sqrt{6}+\sqrt{2})\)
2. a. \( 2 \sin 52 \frac{1}{2}^o \sin 7 \frac{1}{2}^o\)
\(= -(\cos 52,5 + 7,5)-\cos(52,5-7,5)\)
\(= -(\cos 60 - \cos 45)\)
\(= -(\frac{1}{2}-\frac{1}{2}\sqrt{2})\)
\(= \frac{1}{2}(\sqrt{2}-1)\)
b. \(2 \cos 52 \frac{1}{2}^o \cos 7 \frac{1}{2}^o\)
\(= \cos (52,5 + 7,5)+\cos(52,5 + 7,5)\)
\(= \cos 60 + \cos 45\)
\(=\frac{1}{2}+\frac{1}{2}\sqrt{2}\)
\(=\frac{1}{2}(1+\sqrt{2})\)
3. a. \( \sin \frac{5}{12}\pi \cos \frac{1}{6}\pi\)
\(= (\frac{1}{2} \sin \frac{6}{12}\pi) \cos (\frac{1}{12}\pi)\)
\(=\frac{1}{2}(\sin(\frac{6}{12}\pi)+\sin (\frac{4}{12}\pi))\)
\(=\frac{1}{2}(\sin 90 + \sin 60)\)
\(=\frac{1}{2}(1+\frac{1}{2}\sqrt{3})\)
\(=\frac{1}{2}+\frac{1}{4}\sqrt{3}\)
Latihan 4
1. \( \cos 75^o - \cos 15^o = -\frac {1}{2}\sqrt{2}\)
2. \( \sin 80^o + \sin 40^o = \sqrt{3} \cos 20^o\)
3. \( \sin A + \cos A = \sqrt{2} \cos (A-45^o)\)
4. \( \tan 75^o - \tan 15^o = 2\sqrt{3}\)
5. \( \cos 10^o + \cos 110^o + \cos 130^o = 0\)
6. \( \cos 465^o + \cos 165^o + \sin 105^o + \sin 15^o = 0\)
Pembahasan :
1. \( \cos 75^o - \cos 15^o = -\frac {1}{2}\sqrt{2}\)
\(= -2 \sin 45. \sin 30\)
\(= 2(\frac{1}{2}\sqrt{2})(\frac{1}{2})\)
\(=-\frac{1}{2}\sqrt{2}\)
2. \( \sin 80^o + \sin 40^o = \sqrt{3} \cos 20^o\)
\(= 2 \cos 60 .\cos 20 = 2(\frac{1}{2}).\cos 20 = \cos 20\)
3. \(\sin A + \cos A = \sqrt{2} \cos (A-45^o\)
\(=\sqrt{2}(\sin 45 \sin A + \cos 45 \cos A)\)
\(=\sqrt{2}\cos (54-A)\)
4. \( \tan 75^o - \tan 15^o = 2\sqrt{3}\)
\(= \frac{2 \sin 60}{(\cos 90 + \cos 60)}\)
\(= \frac{2(\frac{1}{2}\sqrt{3})}{0+12}\)
\(= 2 \sqrt{3}\)
5. \( \cos 10^o + \cos 110^o + \cos 130^o = 0\)
\(=(\cos 130 + \cos 110) + \cos 10\)
\(= 2 \cos \frac{1}{2}(130+110).\cos \frac{1}{2}(130-110)+ \cos 10\)
\(= 2 (-\frac{1}{2}).\cos 10 + \cos 10\)
\(=-\cos 10+ \cos 10\)
\(=0\)
6. \( \cos 465^o + \cos 165^o + \sin 105^o + \sin 15^o = 0\)
\(= \cos 465^o + \cos 165^o) + (\sin 105^o + \sin 15^o )\)
\(= (2 \cos \frac{1}{2}(465+165)\cos \frac{1}{2}(465 - 165))+ (2 \sin \frac{1}{2}(105+15)\cos(105-15))\)
\(= (2 . \frac{1}{2}\sqrt{2}.-\frac{1}{2}\sqrt{3})(1.\frac{1}{2}\sqrt{3}.\frac{1}{2}\sqrt{2})\)
\(= (- \frac{1}{2}\sqrt{6})+(\frac{1}{2}\sqrt{6})= 0 \)
Latihan 5
1. Diketahui \(\alpha,\beta, \) dan \(\gamma\) menyatakan besar sudut- sudut dalam segitiga ABC. dengan \( \tan \alpha = -3\) dan \(\beta = 1\). Tentukan \(\tan \gamma\)
2. Diketahui \(\tan x = \frac{4}{3}, \pi \angle x \angle \frac{3}{2}\pi\).Tentukan \(\cos 3x - \cos x\).
3. Jika \( \sin x = \alpha, \frac{\pi}{2} \angle x \angle \pi\) Tentukan \(\cos x - \tan x\)
4. jika \( 0 \angle A \angle \pi \)dan \(0\angle B \angle \pi\) memenuhi \( A + B = \frac {2}{3}\pi\) dan \(sin A = 2 \sin B\), Tentukan (A-B).
5. Diketahui \( \cos (A-B)=\frac{1}{2}\sqrt 3\) dan \(\cos A \cos B = \frac {1}{2}\) dengan \(A, B \) sudut lancip. Tentukan nilai \(\frac {\cos (A-B)}{\cos (A+B)}\)
Pembahasan :
1. \(\alpha +\beta +\gamma= 180\)
\(\gamma = 180 - (\alpha + \beta)\)
\(\gamma = - tan (\alpha + \beta)\)
\(\gamma = -\frac{[-3+1]}{1-(-3)(1)}\)
\(\gamma = -frac{-2}{4}\)
\(\gamma =\frac{1}{2}\)
2. \( y = \cos 3x + \cos x \)
\( y = \cos 2x \cos x \)
\( y = 2(\cos^2x - \sin^2 x)\cos x \)
\( y = -\frac{42}{125}\)
3. \(\cos x - \tan x = - \sqrt{(1-a^2)}+\frac{a}{\sqrt{(1-a^2)}}\)
\(\cos x - \tan x = \frac{[-(1-a^2)+a]}{\sqrt{(1-a^2)}}\)
\(\cos x - \tan x = \frac{(a^2+a-1)}{\sqrt(1-a^2)}\)
\(\cos x - \tan x = \frac{(a^2+a-1)(\sqrt{(1-a^2)})}{(1-a^2)}\)
4. \(\sin (120-b)= 2 \sin b \)
\(\sin 120. \cos b - \cos 120.\sin b = 2 \sin b\)
\(\sin 120. \cot b - \cos 120 \)
\(=\frac{1}{2}\sqrt{3}.\cot b -(-\sqrt{1}{2})=2\)
\(\frac{1}{2}\sqrt{3}. \cot b+\frac{1}{2}=2\)
\(\frac{1}{2}\sqrt{3}. \cot b=2-\frac{1}{2}\)
\(\frac{1}{2}\sqrt{3} \cot b =\frac{3}{2}\)
\(\cot b = \frac{3}{2}.\frac{2}{\sqrt{3}}=\frac{3}{\sqrt{3}}=\sqrt{3}\)
\(\tan b = \frac{1}{\cot b}=\frac{1}{\sqrt{3}}=\frac{1}{3}=\sqrt{3}\)
\( b = 30\)
\( a = 120-30=90\)
\(\tan (a-b) = \tan (90-30)= \tan 60 = \sqrt{3}\)
5. \(\cos (a-b)=\frac{1}{2}\)
\(\cos a \cos b + \sin a \sin b =\frac{1}{2}\sqrt{3}\)
\(\frac{1}{2}+ \sin a \sin b =\frac{1}{2}\sqrt{3}\)
\(\sin a \sin b =\frac{1}{2}(\sqrt{3}-1)\)
\(\frac{\cos (a-b)}{\cos (a+b)}\)
\(= \frac{\frac{1}{2}\sqrt{3}}{\frac{1}{2}-\frac{1}{2}(\sqrt{3}-1)}\)
\(=\frac{\frac{1}{2}\sqrt{3}}{(1-\frac{1}{2}\sqrt{3})}\)
\(=\frac{1}{2}\sqrt{3}x\frac{2}{(2-\sqrt{3})}\)
\(=\frac{\sqrt{3}}{(2-\sqrt{3})}\)
\(= 3 + 2 \sqrt{3}\)
