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Sagot :
Sure! Let's break it down step by step.
### Part (a)
Let's begin by analyzing the given information and solving for the required quantities.
#### (i) Finding [tex]$\cos A$[/tex] given that [tex]$\sin A = \frac{3}{5}$[/tex]
Given:
[tex]\[ \sin A = \frac{3}{5} \][/tex]
We know from trigonometric identities that:
[tex]\[ \sin^2 A + \cos^2 A = 1 \][/tex]
Plugging in the given value of [tex]$\sin A$[/tex]:
[tex]\[ \left(\frac{3}{5}\right)^2 + \cos^2 A = 1 \][/tex]
[tex]\[ \frac{9}{25} + \cos^2 A = 1 \][/tex]
[tex]\[ \cos^2 A = 1 - \frac{9}{25} \][/tex]
[tex]\[ \cos^2 A = \frac{25}{25} - \frac{9}{25} \][/tex]
[tex]\[ \cos^2 A = \frac{16}{25} \][/tex]
[tex]\[ \cos A = \sqrt{\frac{16}{25}} \][/tex]
[tex]\[ \cos A = \frac{4}{5} \][/tex]
Therefore,
[tex]\[ \cos A = 0.8 \][/tex]
#### (ii) Calculating [tex]\(\frac{\tan A - \sin A}{1 + 2 \cos A}\)[/tex]
We also need to find [tex]$\tan A$[/tex] to evaluate the given expression. We know:
[tex]\[ \tan A = \frac{\sin A}{\cos A} \][/tex]
Given:
[tex]\[ \sin A = \frac{3}{5} \][/tex]
From (i):
[tex]\[ \cos A = 0.8 \][/tex]
So:
[tex]\[ \tan A = \frac{\frac{3}{5}}{\frac{4}{5}} \][/tex]
[tex]\[ \tan A = \frac{3}{4} \][/tex]
Now, we plug [tex]$\tan A$[/tex], [tex]$\sin A$[/tex], and [tex]$\cos A$[/tex] into the given expression:
[tex]\[ \frac{\tan A - \sin A}{1 + 2 \cos A} \][/tex]
Which translates to:
[tex]\[ \frac{\frac{3}{4} - \frac{3}{5}}{1 + 2 \times 0.8} \][/tex]
[tex]\[ = \frac{\frac{15 - 12}{20}}{1 + 1.6} \][/tex]
[tex]\[ = \frac{\frac{3}{20}}{2.6} \][/tex]
[tex]\[ = \frac{3}{20 \times 2.6} \][/tex]
[tex]\[ = \frac{3}{52} \][/tex]
[tex]\[ = 0.057692307692307654 \][/tex]
### Part (b)
We now look at the problem involving the height of the building.
We have a man who is [tex]$3 \,m$[/tex] tall, standing [tex]$20 \,m$[/tex] away from the wall of the building, with an angle of elevation to the top of the building of [tex]$27^\circ$[/tex].
Given:
- Height of the man = 3 m
- Distance to the wall = 20 m
- Angle of elevation = [tex]$27^\circ$[/tex]
We can use trigonometric ratios to determine the total height of the building. Using the tangent function:
[tex]\[ \tan (\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Let [tex]\( h_b \)[/tex] be the height of the building. Therefore:
[tex]\[ \tan 27^\circ = \frac{h_b - 3}{20} \][/tex]
First, we convert the angle to radians (which is how angles are used in most trigonometric calculations):
[tex]\[ \text{angle in radians} = 27^\circ \approx 0.471239 \text{ radians} \][/tex]
Using the tangent function:
[tex]\[ \tan(27^\circ) = 0.5095254494944298 \][/tex]
Therefore:
[tex]\[ 0.5095254494944298 = \frac{h_b - 3}{20} \][/tex]
[tex]\[ h_b - 3 = 20 \times 0.5095254494944298 \][/tex]
[tex]\[ h_b - 3 = 10.190508989888596 \][/tex]
[tex]\[ h_b = 10.190508989888596 + 3 \][/tex]
[tex]\[ h_b = 13.190508989888576 \][/tex]
Thus, the height of the building is approximately:
[tex]\[ \text{Height of the building} \approx 13.19 \, m \][/tex]
### Part (a)
Let's begin by analyzing the given information and solving for the required quantities.
#### (i) Finding [tex]$\cos A$[/tex] given that [tex]$\sin A = \frac{3}{5}$[/tex]
Given:
[tex]\[ \sin A = \frac{3}{5} \][/tex]
We know from trigonometric identities that:
[tex]\[ \sin^2 A + \cos^2 A = 1 \][/tex]
Plugging in the given value of [tex]$\sin A$[/tex]:
[tex]\[ \left(\frac{3}{5}\right)^2 + \cos^2 A = 1 \][/tex]
[tex]\[ \frac{9}{25} + \cos^2 A = 1 \][/tex]
[tex]\[ \cos^2 A = 1 - \frac{9}{25} \][/tex]
[tex]\[ \cos^2 A = \frac{25}{25} - \frac{9}{25} \][/tex]
[tex]\[ \cos^2 A = \frac{16}{25} \][/tex]
[tex]\[ \cos A = \sqrt{\frac{16}{25}} \][/tex]
[tex]\[ \cos A = \frac{4}{5} \][/tex]
Therefore,
[tex]\[ \cos A = 0.8 \][/tex]
#### (ii) Calculating [tex]\(\frac{\tan A - \sin A}{1 + 2 \cos A}\)[/tex]
We also need to find [tex]$\tan A$[/tex] to evaluate the given expression. We know:
[tex]\[ \tan A = \frac{\sin A}{\cos A} \][/tex]
Given:
[tex]\[ \sin A = \frac{3}{5} \][/tex]
From (i):
[tex]\[ \cos A = 0.8 \][/tex]
So:
[tex]\[ \tan A = \frac{\frac{3}{5}}{\frac{4}{5}} \][/tex]
[tex]\[ \tan A = \frac{3}{4} \][/tex]
Now, we plug [tex]$\tan A$[/tex], [tex]$\sin A$[/tex], and [tex]$\cos A$[/tex] into the given expression:
[tex]\[ \frac{\tan A - \sin A}{1 + 2 \cos A} \][/tex]
Which translates to:
[tex]\[ \frac{\frac{3}{4} - \frac{3}{5}}{1 + 2 \times 0.8} \][/tex]
[tex]\[ = \frac{\frac{15 - 12}{20}}{1 + 1.6} \][/tex]
[tex]\[ = \frac{\frac{3}{20}}{2.6} \][/tex]
[tex]\[ = \frac{3}{20 \times 2.6} \][/tex]
[tex]\[ = \frac{3}{52} \][/tex]
[tex]\[ = 0.057692307692307654 \][/tex]
### Part (b)
We now look at the problem involving the height of the building.
We have a man who is [tex]$3 \,m$[/tex] tall, standing [tex]$20 \,m$[/tex] away from the wall of the building, with an angle of elevation to the top of the building of [tex]$27^\circ$[/tex].
Given:
- Height of the man = 3 m
- Distance to the wall = 20 m
- Angle of elevation = [tex]$27^\circ$[/tex]
We can use trigonometric ratios to determine the total height of the building. Using the tangent function:
[tex]\[ \tan (\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Let [tex]\( h_b \)[/tex] be the height of the building. Therefore:
[tex]\[ \tan 27^\circ = \frac{h_b - 3}{20} \][/tex]
First, we convert the angle to radians (which is how angles are used in most trigonometric calculations):
[tex]\[ \text{angle in radians} = 27^\circ \approx 0.471239 \text{ radians} \][/tex]
Using the tangent function:
[tex]\[ \tan(27^\circ) = 0.5095254494944298 \][/tex]
Therefore:
[tex]\[ 0.5095254494944298 = \frac{h_b - 3}{20} \][/tex]
[tex]\[ h_b - 3 = 20 \times 0.5095254494944298 \][/tex]
[tex]\[ h_b - 3 = 10.190508989888596 \][/tex]
[tex]\[ h_b = 10.190508989888596 + 3 \][/tex]
[tex]\[ h_b = 13.190508989888576 \][/tex]
Thus, the height of the building is approximately:
[tex]\[ \text{Height of the building} \approx 13.19 \, m \][/tex]
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