Question

Given \(\tan A = \frac{4}{3}\), then \(\cos A\) is :

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

Hint:

Remember the standard Pythagorean triplet \((3, 4, 5)\).
Since \(\tan A = \frac{4}{3} = \frac{\text{Opposite}}{\text{Adjacent}}\), the hypotenuse must be \(5\).
Therefore, \(\cos A\), which is \(\frac{\text{Adjacent}}{\text{Hypotenuse}}\), is immediately \(\frac{3}{5}\).

Answer 1

The Correct Option is B

Step 1: Reach for an identity instead of a triangle.
Rather than drawing a right triangle, we can go straight from $\tan A$ to $\cos A$ using the identity $\sec^2 A = 1 + \tan^2 A$.
Step 2: Plug in the given value.
With $\tan A = \frac{4}{3}$, \[ \sec^2 A = 1 + \left(\frac{4}{3}\right)^2 = 1 + \frac{16}{9} = \frac{25}{9} \]
Step 3: Take the square root and invert.
\[ \sec A = \frac{5}{3} \quad \Rightarrow \quad \cos A = \frac{1}{\sec A} = \frac{3}{5} \]
Step 4: State the answer.
Since A is an acute angle here, we take the positive root, giving the same value we would get from the 3-4-5 right triangle.
\[ \boxed{\cos A = \dfrac{3}{5}} \]