Step 1: Understanding the Concept:
This problem deals with basic Trigonometry, which studies the relationships between side lengths and angles of triangles.
In a right-angled triangle, trigonometric functions like sine and cosine are defined as ratios of the sides.
Sine (\(\sin\)) is the ratio of the Opposite side to the Hypotenuse.
Cosine (\(\cos\)) is the ratio of the Adjacent side to the Hypotenuse.
The identity provided in the question, \( \sin^2 \theta + \cos^2 \theta = 1 \), is known as the Pythagorean Trigonometric Identity, which is derived directly from the Pythagorean theorem.
Key Formula or Approach:
We can use the given identity to solve for the unknown value:
\[ \cos^2 \theta = 1 - \sin^2 \theta \]
Taking the square root:
\[ \cos \theta = \sqrt{1 - \sin^2 \theta} \]
Alternatively, we can use the concept of a right-angled triangle with sides \( a \), \( b \), and hypotenuse \( c \), where \( a^2 + b^2 = c^2 \).
Step 2: Detailed Explanation:
Given: \( \sin \theta = \frac{3}{5} \).
This implies that if we have a right triangle, the opposite side is 3 and the hypotenuse is 5.
Let's use the identity method first:
Substitute the value of \(\sin \theta\):
\[ \cos^2 \theta = 1 - \left(\frac{3}{5}\right)^2 \]
Squaring the fraction:
\[ \cos^2 \theta = 1 - \frac{9}{25} \]
To perform the subtraction, find a common denominator:
\[ \cos^2 \theta = \frac{25}{25} - \frac{9}{25} \]
\[ \cos^2 \theta = \frac{25 - 9}{25} \]
\[ \cos^2 \theta = \frac{16}{25} \]
Now, take the square root of both sides to find \(\cos \theta\):
\[ \cos \theta = \sqrt{\frac{16}{25}} = \frac{4}{5} \]
Another approach is to recognize the Pythagorean Triplet (3, 4, 5).
If \(\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{3}{5}\), then the Adjacent side is calculated as:
\[ \text{Adjacent} = \sqrt{\text{Hypotenuse}^2 - \text{Opposite}^2} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \]
Since \(\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\), we have \(\cos \theta = \frac{4}{5}\).
Step 3: Final Answer:
The value of \(\cos \theta\) is \(\frac{4}{5}\).
This matches Option (A).