Soalan 1:
Tentukan nilai-nilai berikut tanpa menggunakan kalkulator.
(a) 2 kos 60° + tan 45°
(b) 3 kos 60° + 2 tan 45°
(c) 2 tan 45° + kos 60°
(d) 3 sin 30° – 2 kos 60°
(e) 2 sin 30° – 3 kos 60°
(f) 4 tan 45° – 2 kos 60°
(g) (2 sin 60°)(3 kos 60°) + 3 tan 30°
(h) (3 tan 45°)(4 sin 60°) – (2 kos 30°)(3 sin 30°)
(i) 4 tan 45° + (2 sin 45°)(6 kos 45°)
(j) (5 tan 60°)(2 sin 60°) – (3 sin 45°)(4 kos 45°)
Penyelesaian:
(a)
$$ \begin{aligned} & 2 \operatorname{kos} 60^{\circ}+\tan 45^{\circ} \\ & =2\left(\frac{1}{2}\right)+1 \\ & =1+1 \\ & =2 \end{aligned} $$
(b)
$$ \begin{aligned} & 3 \operatorname{kos} 60^{\circ}+2 \tan 45^{\circ} \\ & =3\left(\frac{1}{2}\right)+2(1) \\ & =\frac{3}{2}+2 \\ & =3.5 \end{aligned} $$
(c)
$$ \begin{aligned} & 2 \tan 45^{\circ}+\operatorname{kos} 60^{\circ} \\ & =2(1)+\frac{1}{2} \\ & =2+\frac{1}{2} \\ & =2.5 \end{aligned} $$
(d)
$$ \begin{aligned} & 3 \sin 30^{\circ}-2 \operatorname{kos} 60^{\circ} \\ & =3\left(\frac{1}{2}\right)-2\left(\frac{1}{2}\right) \\ & =\frac{3}{2}-1 \\ & =\frac{1}{2} \\ & =0.5 \end{aligned} $$
(e)
$$ \begin{aligned} & 2 \sin 30^{\circ}-3 \operatorname{kos} 60^{\circ} \\ & =2\left(\frac{1}{2}\right)-3\left(\frac{1}{2}\right) \\ & =1-\frac{3}{2} \\ & =-\frac{1}{2} \\ & =-0.5 \end{aligned} $$
(f)
$$ \begin{aligned} & 4 \tan 45^{\circ}-2 \operatorname{kos} 60^{\circ} \\ & =4(1)-2\left(\frac{1}{2}\right) \\ & =4-1 \\ & =3 \end{aligned} $$
(g)
$$ \begin{aligned} & \left(2 \sin 60^{\circ}\right)\left(3 \operatorname{kos} 60^{\circ}\right)+3 \tan 30^{\circ} \\ & =\left[2\left(\frac{\sqrt{3}}{2}\right)\right]\left[3\left(\frac{1}{2}\right)\right]+3\left(\frac{1}{\sqrt{3}}\right) \\ & =(\sqrt{3})\left(\frac{3}{2}\right)+\frac{3}{\sqrt{3}} \\ & =\frac{3 \sqrt{3}}{2}+\frac{3}{\sqrt{3}} \\ & =\frac{3 \sqrt{3}}{2} \times \frac{\sqrt{3}}{\sqrt{3}}+\frac{3}{\sqrt{3}} \times \frac{2}{2} \\ & =\frac{9}{2 \sqrt{3}}+\frac{6}{2 \sqrt{3}} \\ & =\frac{15}{2 \sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \\ & =\frac{15 \sqrt{3}}{2(3)} \\ & =\frac{15 \sqrt{3}}{6} \\ & =\frac{5 \sqrt{3}}{2} \end{aligned} $$
(h)
$$ \begin{aligned} & \left(3 \tan 45^{\circ}\right)\left(4 \sin 60^{\circ}\right)-\left(2 \operatorname{kos} 30^{\circ}\right)\left(3 \sin 30^{\circ}\right) \\ & =[3(1)]\left[4\left(\frac{\sqrt{3}}{2}\right)\right]-\left[2\left(\frac{\sqrt{3}}{2}\right)\right]\left[3\left(\frac{1}{2}\right)\right] \\ & =3(2 \sqrt{3})-\sqrt{3}\left(\frac{3}{2}\right) \\ & =6 \sqrt{3}-\frac{3 \sqrt{3}}{2} \\ & =\frac{12 \sqrt{3}}{2}-\frac{3 \sqrt{3}}{2} \\ & =\frac{9 \sqrt{3}}{2} \end{aligned} $$
(i)
$$ \begin{aligned} & 4 \tan 45^{\circ}+\left(2 \sin 45^{\circ}\right)\left(6 \mathrm{kos} 45^{\circ}\right) \\ & =4(1)+\left[2\left(\frac{1}{\sqrt{2}}\right)\right]\left[6\left(\frac{1}{\sqrt{2}}\right)\right] \\ & =4+\left(\frac{2}{\sqrt{2}}\right)\left(\frac{6}{\sqrt{2}}\right) \\ & =4+\frac{12}{2} \\ & =4+6 \\ & =10 \end{aligned} $$
(j)
$$ \begin{aligned} & \left(5 \tan 60^{\circ}\right)\left(2 \sin 60^{\circ}\right)-\left(3 \sin 45^{\circ}\right)\left(4 \operatorname{kos} 45^{\circ}\right) \\ & =[5(\sqrt{3})]\left[2\left(\frac{\sqrt{3}}{2}\right)\right]-\left[3\left(\frac{1}{\sqrt{2}}\right)\right]\left[4\left(\frac{1}{\sqrt{2}}\right)\right] \\ & =(5 \sqrt{3})(\sqrt{3})-\left(\frac{3}{\sqrt{2}}\right)\left(\frac{4}{\sqrt{2}}\right) \\ & =15-\frac{12}{2} \\ & =15-6 \\ & =9 \end{aligned} $$
Tentukan nilai-nilai berikut tanpa menggunakan kalkulator.
(a) 2 kos 60° + tan 45°
(b) 3 kos 60° + 2 tan 45°
(c) 2 tan 45° + kos 60°
(d) 3 sin 30° – 2 kos 60°
(e) 2 sin 30° – 3 kos 60°
(f) 4 tan 45° – 2 kos 60°
(g) (2 sin 60°)(3 kos 60°) + 3 tan 30°
(h) (3 tan 45°)(4 sin 60°) – (2 kos 30°)(3 sin 30°)
(i) 4 tan 45° + (2 sin 45°)(6 kos 45°)
(j) (5 tan 60°)(2 sin 60°) – (3 sin 45°)(4 kos 45°)
Penyelesaian:
(a)
$$ \begin{aligned} & 2 \operatorname{kos} 60^{\circ}+\tan 45^{\circ} \\ & =2\left(\frac{1}{2}\right)+1 \\ & =1+1 \\ & =2 \end{aligned} $$
(b)
$$ \begin{aligned} & 3 \operatorname{kos} 60^{\circ}+2 \tan 45^{\circ} \\ & =3\left(\frac{1}{2}\right)+2(1) \\ & =\frac{3}{2}+2 \\ & =3.5 \end{aligned} $$
(c)
$$ \begin{aligned} & 2 \tan 45^{\circ}+\operatorname{kos} 60^{\circ} \\ & =2(1)+\frac{1}{2} \\ & =2+\frac{1}{2} \\ & =2.5 \end{aligned} $$
(d)
$$ \begin{aligned} & 3 \sin 30^{\circ}-2 \operatorname{kos} 60^{\circ} \\ & =3\left(\frac{1}{2}\right)-2\left(\frac{1}{2}\right) \\ & =\frac{3}{2}-1 \\ & =\frac{1}{2} \\ & =0.5 \end{aligned} $$
(e)
$$ \begin{aligned} & 2 \sin 30^{\circ}-3 \operatorname{kos} 60^{\circ} \\ & =2\left(\frac{1}{2}\right)-3\left(\frac{1}{2}\right) \\ & =1-\frac{3}{2} \\ & =-\frac{1}{2} \\ & =-0.5 \end{aligned} $$
(f)
$$ \begin{aligned} & 4 \tan 45^{\circ}-2 \operatorname{kos} 60^{\circ} \\ & =4(1)-2\left(\frac{1}{2}\right) \\ & =4-1 \\ & =3 \end{aligned} $$
(g)
$$ \begin{aligned} & \left(2 \sin 60^{\circ}\right)\left(3 \operatorname{kos} 60^{\circ}\right)+3 \tan 30^{\circ} \\ & =\left[2\left(\frac{\sqrt{3}}{2}\right)\right]\left[3\left(\frac{1}{2}\right)\right]+3\left(\frac{1}{\sqrt{3}}\right) \\ & =(\sqrt{3})\left(\frac{3}{2}\right)+\frac{3}{\sqrt{3}} \\ & =\frac{3 \sqrt{3}}{2}+\frac{3}{\sqrt{3}} \\ & =\frac{3 \sqrt{3}}{2} \times \frac{\sqrt{3}}{\sqrt{3}}+\frac{3}{\sqrt{3}} \times \frac{2}{2} \\ & =\frac{9}{2 \sqrt{3}}+\frac{6}{2 \sqrt{3}} \\ & =\frac{15}{2 \sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \\ & =\frac{15 \sqrt{3}}{2(3)} \\ & =\frac{15 \sqrt{3}}{6} \\ & =\frac{5 \sqrt{3}}{2} \end{aligned} $$
(h)
$$ \begin{aligned} & \left(3 \tan 45^{\circ}\right)\left(4 \sin 60^{\circ}\right)-\left(2 \operatorname{kos} 30^{\circ}\right)\left(3 \sin 30^{\circ}\right) \\ & =[3(1)]\left[4\left(\frac{\sqrt{3}}{2}\right)\right]-\left[2\left(\frac{\sqrt{3}}{2}\right)\right]\left[3\left(\frac{1}{2}\right)\right] \\ & =3(2 \sqrt{3})-\sqrt{3}\left(\frac{3}{2}\right) \\ & =6 \sqrt{3}-\frac{3 \sqrt{3}}{2} \\ & =\frac{12 \sqrt{3}}{2}-\frac{3 \sqrt{3}}{2} \\ & =\frac{9 \sqrt{3}}{2} \end{aligned} $$
(i)
$$ \begin{aligned} & 4 \tan 45^{\circ}+\left(2 \sin 45^{\circ}\right)\left(6 \mathrm{kos} 45^{\circ}\right) \\ & =4(1)+\left[2\left(\frac{1}{\sqrt{2}}\right)\right]\left[6\left(\frac{1}{\sqrt{2}}\right)\right] \\ & =4+\left(\frac{2}{\sqrt{2}}\right)\left(\frac{6}{\sqrt{2}}\right) \\ & =4+\frac{12}{2} \\ & =4+6 \\ & =10 \end{aligned} $$
(j)
$$ \begin{aligned} & \left(5 \tan 60^{\circ}\right)\left(2 \sin 60^{\circ}\right)-\left(3 \sin 45^{\circ}\right)\left(4 \operatorname{kos} 45^{\circ}\right) \\ & =[5(\sqrt{3})]\left[2\left(\frac{\sqrt{3}}{2}\right)\right]-\left[3\left(\frac{1}{\sqrt{2}}\right)\right]\left[4\left(\frac{1}{\sqrt{2}}\right)\right] \\ & =(5 \sqrt{3})(\sqrt{3})-\left(\frac{3}{\sqrt{2}}\right)\left(\frac{4}{\sqrt{2}}\right) \\ & =15-\frac{12}{2} \\ & =15-6 \\ & =9 \end{aligned} $$