Let's start by breaking down the problem and solving step by step.
1. Calculate [tex]\(\sin(30^\circ)\)[/tex]
To find [tex]\(\sin(30^\circ)\)[/tex]:
[tex]\[
\sin(30^\circ) = \frac{1}{2} \approx 0.49999999999999994
\][/tex]
2. Calculate [tex]\(\sin(2 \theta)\)[/tex] for [tex]\(\theta = 30^\circ\)[/tex]
First, we determine [tex]\(2 \theta\)[/tex]:
[tex]\[
2 \theta = 2 \times 30^\circ = 60^\circ
\][/tex]
Next, we find [tex]\(\sin(60^\circ)\)[/tex]:
[tex]\[
\sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.8660254037844386
\][/tex]
3. Find the ratio [tex]\(2 \sin(30^\circ) : \sin(60^\circ)\)[/tex]
We need to determine the ratio [tex]\(2 \sin(30^\circ) : \sin(60^\circ)\)[/tex]:
[tex]\[
2 \sin(30^\circ) = 2 \times \frac{1}{2} = 1
\][/tex]
So, the ratio is:
[tex]\[
\frac{2 \sin(30^\circ)}{\sin(60^\circ)} = \frac{1}{\sin(60^\circ)} \approx \frac{1}{0.8660254037844386} \approx 1.1547005383792515
\][/tex]
Thus, the ratio [tex]\(2 \sin(30^\circ) : \sin(60^\circ)\)[/tex] is approximately 1.1547005383792515.
Now for problem 9:
Given:
[tex]\( A = 60^\circ \)[/tex]
[tex]\( B = 30^\circ \)[/tex]
We need to verify:
[tex]\[
\sin(60^\circ + 30^\circ) = \sin(60^\circ) \cos(30^\circ) + \cos(60^\circ) \sin(30^\circ)
\][/tex]
On the left-hand side, we have:
[tex]\[
\sin(90^\circ) = 1
\][/tex]
On the right-hand side, using known values for trigonometric functions:
[tex]\[
\sin(60^\circ) = \frac{\sqrt{3}}{2}
\][/tex]
[tex]\[
\cos(30^\circ) = \frac{\sqrt{3}}{2}
\][/tex]
[tex]\[
\cos(60^\circ) = \frac{1}{2}
\][/tex]
[tex]\[
\sin(30^\circ) = \frac{1}{2}
\][/tex]
Plugging these into the right-hand side expression, we get:
[tex]\[
\sin(60^\circ) \cos(30^\circ) + \cos(60^\circ) \sin(30^\circ) = \left(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) + \left(\frac{1}{2} \cdot \frac{1}{2}\right)
\][/tex]
[tex]\[
= \frac{3}{4} + \frac{1}{4}
\][/tex]
[tex]\[
= 1
\][/tex]
Thus,
[tex]\[
1 = 1
\][/tex]
The equation is verified.
