Question

If \(\cos A = \frac{4}{5}\), then the value of \(\tan A\) is :

• A. \(\frac{3}{5}\)
• B. \(\frac{3}{4}\)
• C. \(\frac{4}{3}\)
• D. \(\frac{5}{3}\)

Hint:

Trigonometric ratios are frequently built on the classic \((3, 4, 5)\) right-angled Pythagorean triple.
Since \(\cos A = \frac{4}{5}\), the base is 4, the hypotenuse is 5, and the missing perpendicular side must be 3.
Using \(\tan A = \frac{\text{Perpendicular}}{\text{Base}}\), you can immediately write \(\frac{3}{4}\) without drawing a triangle or writing down identities!

Answer 1

The Correct Option is B

Step 1: Use the Pythagorean identity instead of building a triangle with sides.
We know the identity that links sine and cosine of the same angle:
\[ \sin^2 A + \cos^2 A = 1 \]
This lets us find $\sin A$ directly from $\cos A$ without drawing a right triangle.

Step 2: Substitute the given value of $\cos A$.
\[ \sin^2 A = 1 - \cos^2 A = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25} \]

Step 3: Take the square root.
Since $A$ is an angle of a triangle here, we take the positive root:
\[ \sin A = \frac{3}{5} \]

Step 4: Use the quotient identity to find $\tan A$.
\[ \tan A = \frac{\sin A}{\cos A} = \frac{3/5}{4/5} \]
The two 5's in the denominators cancel when we divide the fractions:
\[ \tan A = \frac{3}{4} \]

Step 5: Final Answer.
The value of $\tan A$ is $\frac{3}{4}$, so option (B) is correct. \[ \boxed{\tan A = \frac{3}{4}} \]