Question

If \(\cos A = \frac{3}{5}\), then the value of \(\tan A\) is :

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

Hint:

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

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given the trigonometric ratio \(\cos A = \frac{3}{5}\). We need to determine the value of \(\tan A\).

Step 2: Key Formula or Approach:
1. In a right-angled triangle, the basic trigonometric ratios are defined as:
\[ \cos A = \frac{\text{Base}}{\text{Hypotenuse}} \]
\[ \tan A = \frac{\text{Perpendicular}}{\text{Base}} \]
2. Alternatively, we can use the fundamental trigonometric identity:
\[ \sin^2 A + \cos^2 A = 1 \]
And the quotient relationship:
\[ \tan A = \frac{\sin A}{\cos A} \]

Step 3: Detailed Explanation:

Method 1: Right-Angled Triangle Method
1. Let \(\cos A = \frac{3}{5} = \frac{\text{Base}}{\text{Hypotenuse}}\).
Let Base = \(3k\) and Hypotenuse = \(5k\), where \(k\) is a positive constant.
2. Apply Pythagoras' theorem to find the Perpendicular:
\[ \text{Hypotenuse}^2 = \text{Base}^2 + \text{Perpendicular}^2 \]
\[ (5k)^2 = (3k)^2 + \text{Perpendicular}^2 \]
\[ 25k^2 = 9k^2 + \text{Perpendicular}^2 \]
\[ \text{Perpendicular}^2 = 25k^2 - 9k^2 = 16k^2 \]
\[ \text{Perpendicular} = 4k \]
3. Now, write the formula for \(\tan A\):
\[ \tan A = \frac{\text{Perpendicular}}{\text{Base}} = \frac{4k}{3k} = \frac{4}{3} \]


Method 2: Trigonometric Identity Method
1. Find \(\sin A\) using the identity:
\[ \sin A = \sqrt{1 - \cos^2 A} \]
\[ \sin A = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5} \]
2. Compute \(\tan A\):
\[ \tan A = \frac{\sin A}{\cos A} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \]
Both methods yield \(\tan A = \frac{4}{3}\).

Step 4: Final Answer:
The value of \(\tan A\) is \(\frac{4}{3}\), which corresponds to option (D).