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Sagot :
Sure, let's solve the given problem step-by-step. We are given that:
[tex]\(\tan(\theta) = \frac{4}{3}\)[/tex] and [tex]\(\sin(\theta) = -\frac{4}{5}\)[/tex].
We need to find [tex]\(\cos(\theta)\)[/tex].
### Step 1: Use the Pythagorean Identity
We know the identity for sine and cosine:
[tex]\[ \sin^2(\theta) + \cos^2(\theta) = 1 \][/tex]
Given [tex]\( \sin(\theta) = -\frac{4}{5} \)[/tex], we can square this value:
[tex]\[ \sin^2(\theta) = \left(-\frac{4}{5}\right)^2 = \frac{16}{25} \][/tex]
### Step 2: Calculate [tex]\(\cos^2(\theta)\)[/tex]
Using the Pythagorean identity:
[tex]\[ \cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \frac{16}{25} \][/tex]
We need to subtract:
[tex]\[ 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \][/tex]
So:
[tex]\[ \cos^2(\theta) = \frac{9}{25} \][/tex]
### Step 3: Determine the Sign of [tex]\(\cos(\theta)\)[/tex]
Since we have [tex]\(\cos^2(\theta) = \frac{9}{25}\)[/tex], we can find [tex]\(\cos(\theta)\)[/tex] by taking the square root:
[tex]\[ \cos(\theta) = \pm \sqrt{\frac{9}{25}} = \pm \frac{3}{5} \][/tex]
We need to decide the correct sign. We know:
[tex]\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \][/tex]
Given [tex]\(\tan(\theta) = \frac{4}{3}\)[/tex] and [tex]\(\sin(\theta) = -\frac{4}{5}\)[/tex], we set up the equality:
[tex]\[ \tan(\theta) = \frac{-\frac{4}{5}}{\cos(\theta)} \][/tex]
Since [tex]\(\tan(\theta) > 0\)[/tex] and [tex]\(\sin(\theta) < 0\)[/tex], [tex]\(\cos(\theta)\)[/tex] must also be negative (because a negative divided by a negative equals a positive).
Therefore:
[tex]\[ \cos(\theta) = -\frac{3}{5} \][/tex]
### Conclusion:
From the options given:
A. [tex]\(\frac{1}{3}\)[/tex]
B. [tex]\(-\frac{1}{3}\)[/tex]
C. [tex]\(\frac{3}{6}\)[/tex]
D. [tex]\(-\frac{3}{6}\)[/tex]
E. [tex]\(-\frac{\sqrt{3}}{5}\)[/tex]
The closest to our answer [tex]\(\cos(\theta) = -\frac{3}{5}\)[/tex] is:
[tex]\[ -\frac{3}{6} \][/tex]
which can be simplified to:
[tex]\[ -\frac{1}{2} \][/tex]
But none of the options matches our calculated value exactly except [tex]\(\frac{1}{2}\)[/tex]. Thus there might have been a typo with options expected [tex]\(-\frac{3}{5}\)[/tex].
However to match this content:
the best fit here is
Answer is not fitting exact, should have "[tex]\(\cos(\theta) = -\frac{3}{5}\)[/tex]".
Steps are crucial to understand mathematically!
[tex]\(\tan(\theta) = \frac{4}{3}\)[/tex] and [tex]\(\sin(\theta) = -\frac{4}{5}\)[/tex].
We need to find [tex]\(\cos(\theta)\)[/tex].
### Step 1: Use the Pythagorean Identity
We know the identity for sine and cosine:
[tex]\[ \sin^2(\theta) + \cos^2(\theta) = 1 \][/tex]
Given [tex]\( \sin(\theta) = -\frac{4}{5} \)[/tex], we can square this value:
[tex]\[ \sin^2(\theta) = \left(-\frac{4}{5}\right)^2 = \frac{16}{25} \][/tex]
### Step 2: Calculate [tex]\(\cos^2(\theta)\)[/tex]
Using the Pythagorean identity:
[tex]\[ \cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \frac{16}{25} \][/tex]
We need to subtract:
[tex]\[ 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \][/tex]
So:
[tex]\[ \cos^2(\theta) = \frac{9}{25} \][/tex]
### Step 3: Determine the Sign of [tex]\(\cos(\theta)\)[/tex]
Since we have [tex]\(\cos^2(\theta) = \frac{9}{25}\)[/tex], we can find [tex]\(\cos(\theta)\)[/tex] by taking the square root:
[tex]\[ \cos(\theta) = \pm \sqrt{\frac{9}{25}} = \pm \frac{3}{5} \][/tex]
We need to decide the correct sign. We know:
[tex]\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \][/tex]
Given [tex]\(\tan(\theta) = \frac{4}{3}\)[/tex] and [tex]\(\sin(\theta) = -\frac{4}{5}\)[/tex], we set up the equality:
[tex]\[ \tan(\theta) = \frac{-\frac{4}{5}}{\cos(\theta)} \][/tex]
Since [tex]\(\tan(\theta) > 0\)[/tex] and [tex]\(\sin(\theta) < 0\)[/tex], [tex]\(\cos(\theta)\)[/tex] must also be negative (because a negative divided by a negative equals a positive).
Therefore:
[tex]\[ \cos(\theta) = -\frac{3}{5} \][/tex]
### Conclusion:
From the options given:
A. [tex]\(\frac{1}{3}\)[/tex]
B. [tex]\(-\frac{1}{3}\)[/tex]
C. [tex]\(\frac{3}{6}\)[/tex]
D. [tex]\(-\frac{3}{6}\)[/tex]
E. [tex]\(-\frac{\sqrt{3}}{5}\)[/tex]
The closest to our answer [tex]\(\cos(\theta) = -\frac{3}{5}\)[/tex] is:
[tex]\[ -\frac{3}{6} \][/tex]
which can be simplified to:
[tex]\[ -\frac{1}{2} \][/tex]
But none of the options matches our calculated value exactly except [tex]\(\frac{1}{2}\)[/tex]. Thus there might have been a typo with options expected [tex]\(-\frac{3}{5}\)[/tex].
However to match this content:
the best fit here is
Answer is not fitting exact, should have "[tex]\(\cos(\theta) = -\frac{3}{5}\)[/tex]".
Steps are crucial to understand mathematically!
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