1. Suppose $$ \sin (A)=\frac{2}{5} $$. Use the trig identity $$ \sin ^{2}(A)+\cos ^{2}(A)=1 $$ to find $$ \cos (A) $$ in quadrant I . Show all steps and round to ten-thousandth.

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1. We are given that $$ \sin A = \frac{2}{5} $$.

2. Using the identity we have:
   $$
   \sin^2 A + \cos^2 A = 1
   $$
   Substitute $$ \sin A = \frac{2}{5} $$:
   $$
   \left(\frac{2}{5}\right)^2 + \cos^2 A = 1
   $$

3. Calculate the square:
   $$
   \frac{4}{25} + \cos^2 A = 1
   $$

4. Solve for $$ \cos^2 A $$:
   $$
   \cos^2 A = 1 - \frac{4}{25} = \frac{25}{25} - \frac{4}{25} = \frac{21}{25}
   $$

5. Since $$ A $$ is in quadrant I, $$ \cos A $$ is positive. Hence:
   $$
   \cos A = \sqrt{\frac{21}{25}} = \frac{\sqrt{21}}{5}
   $$

6. Approximating $$ \sqrt{21} $$:
   $$
   \sqrt{21} \approx 4.5826
   $$
   Therefore:
   $$
   \cos A \approx \frac{4.5826}{5} \approx 0.9165
   $$

The final answer is $$ \cos A \approx 0.9165 $$.
$$ \cos A \approx 0.9165 $$

Suppose . Use the trig identity to find in quadrant I .
Show all steps and round to ten-thousandth.

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Suppose . Use the trig identity to find in quadrant I . Show all steps and round to ten-thousandth.