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Sagot :
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.
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.
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