Question:
If \(\cos A = \frac{4}{5}\), then the value of \(\tan A\) is :
If \(\cos A = \frac{4}{5}\), then the value of \(\tan A\) is :
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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!
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!
Updated On: Jul 7, 2026
- \(\frac{3}{5}\)
- \(\frac{3}{4}\)
- \(\frac{4}{3}\)
- \(\frac{5}{3}\)
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Question:
We are given the trigonometric ratio \(\cos A = \frac{4}{5}\). We need to determine the value of the trigonometric ratio \(\tan A\) for the same angle \(A\).
Step 2: Key Formula or Approach:
1. In a right-angled triangle, the basic trigonometric ratios are defined as:
\[ \cos A = \frac{\text{Base (B)}}{\text{Hypotenuse (H)}} \]
\[ \tan A = \frac{\text{Perpendicular (P)}}{\text{Base (B)}} \]
2. The Pythagorean identity connects the three sides of a right triangle:
\[ H^2 = P^2 + B^2 \]
Step 3: Detailed Explanation:
1. Let \(\cos A = \frac{4}{5} = \frac{\text{Base}}{\text{Hypotenuse}}\).
Let Base = \(4k\) and Hypotenuse = \(5k\), where \(k\) is a positive constant.
2. Apply Pythagoras' theorem to find the Perpendicular side of the right-angled triangle:
\[ \text{Hypotenuse}^2 = \text{Base}^2 + \text{Perpendicular}^2 \]
\[ (5k)^2 = (4k)^2 + \text{Perpendicular}^2 \]
\[ 25k^2 = 16k^2 + \text{Perpendicular}^2 \]
Subtract \(16k^2\) from both sides:
\[ \text{Perpendicular}^2 = 25k^2 - 16k^2 = 9k^2 \]
Taking the square root on both sides:
\[ \text{Perpendicular} = 3k \]
3. Now, write the formula for \(\tan A\):
\[ \tan A = \frac{\text{Perpendicular}}{\text{Base}} \]
\[ \tan A = \frac{3k}{4k} = \frac{3}{4} \]
This gives the value of \(\tan A\) as \(\frac{3}{4}\).
Step 4: Final Answer:
The value of \(\tan A\) is \(\frac{3}{4}\), which corresponds to option (B).
We are given the trigonometric ratio \(\cos A = \frac{4}{5}\). We need to determine the value of the trigonometric ratio \(\tan A\) for the same angle \(A\).
Step 2: Key Formula or Approach:
1. In a right-angled triangle, the basic trigonometric ratios are defined as:
\[ \cos A = \frac{\text{Base (B)}}{\text{Hypotenuse (H)}} \]
\[ \tan A = \frac{\text{Perpendicular (P)}}{\text{Base (B)}} \]
2. The Pythagorean identity connects the three sides of a right triangle:
\[ H^2 = P^2 + B^2 \]
Step 3: Detailed Explanation:
1. Let \(\cos A = \frac{4}{5} = \frac{\text{Base}}{\text{Hypotenuse}}\).
Let Base = \(4k\) and Hypotenuse = \(5k\), where \(k\) is a positive constant.
2. Apply Pythagoras' theorem to find the Perpendicular side of the right-angled triangle:
\[ \text{Hypotenuse}^2 = \text{Base}^2 + \text{Perpendicular}^2 \]
\[ (5k)^2 = (4k)^2 + \text{Perpendicular}^2 \]
\[ 25k^2 = 16k^2 + \text{Perpendicular}^2 \]
Subtract \(16k^2\) from both sides:
\[ \text{Perpendicular}^2 = 25k^2 - 16k^2 = 9k^2 \]
Taking the square root on both sides:
\[ \text{Perpendicular} = 3k \]
3. Now, write the formula for \(\tan A\):
\[ \tan A = \frac{\text{Perpendicular}}{\text{Base}} \]
\[ \tan A = \frac{3k}{4k} = \frac{3}{4} \]
This gives the value of \(\tan A\) as \(\frac{3}{4}\).
Step 4: Final Answer:
The value of \(\tan A\) is \(\frac{3}{4}\), which corresponds to option (B).
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