Answer :
Let's analyze the problem step by step to determine if Mia and Raul's statements are correct.
### Step 1: Analyzing Mia's Statement
Mia claims:
[tex]\[ \sin \theta = \frac{2 \sqrt{5}}{5} \][/tex]
### Step 2: Analyzing Raul's Statement
Raul claims:
[tex]\[ \tan \theta = 2 \][/tex]
### Step 3: Finding [tex]\(\cos \theta\)[/tex] from [tex]\(\sin \theta\)[/tex]
We know that:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Given [tex]\(\sin \theta = \frac{2 \sqrt{5}}{5}\)[/tex], we can find [tex]\(\cos \theta\)[/tex]:
[tex]\[ \cos^2 \theta = 1 - \sin^2 \theta \][/tex]
[tex]\[ \cos^2 \theta = 1 - \left( \frac{2 \sqrt{5}}{5} \right)^2 \][/tex]
[tex]\[ \cos^2 \theta = 1 - \frac{20}{25} \][/tex]
[tex]\[ \cos^2 \theta = 1 - 0.8 \][/tex]
[tex]\[ \cos^2 \theta = 0.2 \][/tex]
[tex]\[ \cos \theta = \sqrt{0.2} \][/tex]
### Step 4: Calculating [tex]\(\tan \theta\)[/tex] using [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex]
We know that:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
Given [tex]\(\sin \theta = \frac{2 \sqrt{5}}{5}\)[/tex] and [tex]\(\cos \theta = \sqrt{0.2}\)[/tex]:
[tex]\[ \tan \theta = \frac{\frac{2 \sqrt{5}}{5}}{\sqrt{0.2}} \][/tex]
We can simplify the expression:
[tex]\[ \tan \theta = 2 \][/tex]
### Conclusion
- Mia claims that [tex]\(\sin \theta = \frac{2 \sqrt{5}}{5}\)[/tex]. This value is correct.
- Raul claims that [tex]\(\tan \theta = 2\)[/tex]. Using [tex]\(\sin \theta\)[/tex] and calculating the corresponding [tex]\(\tan \theta\)[/tex], this value is also correct.
Therefore, both Mia and Raul are correct.
### Answer:
They are both correct.
### Step 1: Analyzing Mia's Statement
Mia claims:
[tex]\[ \sin \theta = \frac{2 \sqrt{5}}{5} \][/tex]
### Step 2: Analyzing Raul's Statement
Raul claims:
[tex]\[ \tan \theta = 2 \][/tex]
### Step 3: Finding [tex]\(\cos \theta\)[/tex] from [tex]\(\sin \theta\)[/tex]
We know that:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Given [tex]\(\sin \theta = \frac{2 \sqrt{5}}{5}\)[/tex], we can find [tex]\(\cos \theta\)[/tex]:
[tex]\[ \cos^2 \theta = 1 - \sin^2 \theta \][/tex]
[tex]\[ \cos^2 \theta = 1 - \left( \frac{2 \sqrt{5}}{5} \right)^2 \][/tex]
[tex]\[ \cos^2 \theta = 1 - \frac{20}{25} \][/tex]
[tex]\[ \cos^2 \theta = 1 - 0.8 \][/tex]
[tex]\[ \cos^2 \theta = 0.2 \][/tex]
[tex]\[ \cos \theta = \sqrt{0.2} \][/tex]
### Step 4: Calculating [tex]\(\tan \theta\)[/tex] using [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex]
We know that:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
Given [tex]\(\sin \theta = \frac{2 \sqrt{5}}{5}\)[/tex] and [tex]\(\cos \theta = \sqrt{0.2}\)[/tex]:
[tex]\[ \tan \theta = \frac{\frac{2 \sqrt{5}}{5}}{\sqrt{0.2}} \][/tex]
We can simplify the expression:
[tex]\[ \tan \theta = 2 \][/tex]
### Conclusion
- Mia claims that [tex]\(\sin \theta = \frac{2 \sqrt{5}}{5}\)[/tex]. This value is correct.
- Raul claims that [tex]\(\tan \theta = 2\)[/tex]. Using [tex]\(\sin \theta\)[/tex] and calculating the corresponding [tex]\(\tan \theta\)[/tex], this value is also correct.
Therefore, both Mia and Raul are correct.
### Answer:
They are both correct.
