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Mia and Raul are trying to write the correct ratio to find [tex]\(\theta\)[/tex]. Mia says to use [tex]\sin \theta=\frac{2 \sqrt{5}}{5}[/tex]. Raul says to use [tex]\tan \theta=2[/tex].

Who is correct?
A. Mia is correct.
B. Raul is correct.
C. They are both incorrect.
D. They are both correct.


Sagot :

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.