Finding cot(90° - A) Using Trigonometric Identities

We are given that $\sin A = \frac{4}{5}$ and angle A is acute ($0^\circ < A < 90^\circ$). We need to find the value of $\cot(90^\circ - A)$.

Applying Complementary Angle Identity

Recall the trigonometric identity for complementary angles: $ \cot(90^\circ - A) = \tan A $ So, finding $\cot(90^\circ - A)$ is equivalent to finding $\tan A$.

Calculating cos A

We can find $\cos A$ using the fundamental Pythagorean identity $\sin^2 A + \cos^2 A = 1$. $ \cos^2 A = 1 - \sin^2 A $ Substitute the given value of $\sin A$: $ \cos^2 A = 1 - \left(\frac{4}{5}\right)^2 $ $ \cos^2 A = 1 - \frac{16}{25} $ $ \cos^2 A = \frac{25 - 16}{25} $ $ \cos^2 A = \frac{9}{25} $ Since A is an acute angle, $\cos A$ is positive. Taking the square root: $ \cos A = \sqrt{\frac{9}{25}} = \frac{3}{5} $

Calculating tan A

Now, we can calculate $\tan A$ using the definition $\tan A = \frac{\sin A}{\cos A}$. $ \tan A = \frac{4/5}{3/5} $ $ \tan A = \frac{4}{3} $

Final Result

Since $\cot(90^\circ - A) = \tan A$, the value is: $ \cot(90^\circ - A) = \frac{4}{3} $ This matches Option B.