Question

Calculate the following without using a calculator:

(a) \(2 \sin 15^{\circ} \cdot \cos 15^{\circ}\)

(b) \(1-2 \cos ^2 22,5^{\circ}\)

(c) \(\left(\cos 15^{\circ}+\sin 15^{\circ}\right)^2\)

Answer 1

(a)
\[
\begin{aligned}
& 2 \sin 15^{\circ} \cdot \cos 15^{\circ} \\
& =\sin \left[2\left(15^{\circ}\right)\right] \\
& =\sin 30^{\circ} \\
& =\frac{1}{2}
\end{aligned}
\]
(b)
\[
\begin{aligned}
& 1-2 \cos ^2 22,5^{\circ} \\
& =-\left(2 \cos ^2 22,5^{\circ}-1\right) \\
& =-\left[\cos \left(2\left(22,5^{\circ}\right)\right]\right. \\
& =-\cos 45^{\circ} \\
& =-\frac{\sqrt{2}}{2}
\end{aligned}
\]

(c) \(\begin{aligned} & \left(\cos 15^{\circ}+\sin 15^{\circ}\right)^2 \\ = & \cos ^2 15^{\circ}+2 \cos 15^{\circ} \cdot \sin 15^{\circ}+\sin ^2 15^{\circ} \\ = & \cos ^2 15^{\circ}+\sin ^2 15^{\circ}+2 \sin 15^{\circ} \cdot \cos 15^{\circ} \\ = & 1+\sin \left[2\left(15^{\circ}\right)\right] \\ = & 1+\sin 30^{\circ} \\ = & 1+\frac{1}{2}=\frac{3}{2}\end{aligned}\)