Question

Find the values of -

(i) 5 sin 30​^° + 3 tan 45^° (ii) $\frac{4}{5}{\tan }^2{60}^{°}+3{\sin }^2{60}^{°}$ (iii) 2sin 30^° + cos 0^° + 3sin 90°

(iv) $\frac{\tan 60}{\sin 60+\cos 60}$ (v) ${\cos }^2{45}^{°}+{\sin }^2{30}^{°}$ (vi) cos 60^°× cos 30^° + sin 60^°× sin 30^°

Answer 1

(i)
$$\begin{gathered} 5\sin 30°+3\tan 45° \\ =5\times \frac{1}{2}+3\times 1 \\ =\frac{5}{2}+3 \\ =\frac{5+6}{2} \\ =\frac{11}{2} \end{gathered}$$
(ii)
$$\begin{gathered} \frac{4}{5}{\tan }^{2}60°+3{\sin }^{2}60° \\ =\frac{4}{5}\times {\left(\sqrt{3}\right)}^2+3\times {\left(\frac{\sqrt{3}}{2}\right)}^2 \\ =\frac{4}{5}\times 3+3\times \frac{3}{4} \\ =\frac{12}{5}+\frac{9}{4} \\ =\frac{48+45}{20} \\ =\frac{93}{20} \end{gathered}$$
(iii)
$$\begin{gathered} 2\sin 30°+\cos 0°+3\sin 90° \\ =2\times \frac{1}{2}+1+3\times 1 \\ =1+1+3 \\ =5 \end{gathered}$$
(iv)
$$\begin{gathered} \frac{\tan 60°}{\sin 60°+\cos 60°} \\ =\frac{\sqrt{3}}{{\displaystyle \frac{\sqrt{3}}{2}+\frac{1}{2}}} \\ =\frac{\sqrt{3}}{{\displaystyle \frac{\sqrt{3}{\displaystyle +}{\displaystyle 1}}{2}}} \\ =\frac{2\sqrt{3}}{\sqrt{3}{\displaystyle +}{\displaystyle 1}} \end{gathered}$$
(v)
$$\begin{gathered} {\cos }^{2}45°+{\sin }^{2}30° \\ ={\left(\frac{1}{\sqrt{2}}\right)}^2+{\left(\frac{1}{2}\right)}^2 \\ =\frac{{\displaystyle 1}}{2}+\frac{1}{4} \\ =\frac{2+1}{4} \\ =\frac{3}{4} \end{gathered}$$
(vi)
$$\begin{gathered} \cos 60°\times \cos 30°+\sin 60°\times \sin 30° \\ =\frac{1}{2}\times \frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}\times \frac{1}{2} \\ =\frac{\sqrt{3}}{4}+\frac{\sqrt{3}}{4} \\ =\frac{2\sqrt{3}}{4} \\ =\frac{\sqrt{3}}{2} \end{gathered}$$