Question

If \(\cos A = \frac{4}{5}\), then find the values of \(\cot A\) and \(\csc A\).

Hint:

Recognizing Pythagorean triplets like (3, 4, 5) can save you time. When you see two of the numbers, you can immediately identify the third without calculation.

Answer 1

Step 1: Understanding the Concept:
Given one trigonometric ratio, we can find the others by constructing a right-angled triangle and using the Pythagorean theorem, or by using trigonometric identities. The triangle method is often more intuitive.

Step 2: Key Formula or Approach:
We know that \(\cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}}\).
From this, we can find the Opposite side using Pythagoras theorem: \((\text{Opposite})^2 + (\text{Adjacent})^2 = (\text{Hypotenuse})^2\).
Then we use the definitions: \(\cot A = \frac{\text{Adjacent}}{\text{Opposite}}\) and \(\csc A = \frac{\text{Hypotenuse}}{\text{Opposite}}\).

Step 3: Detailed Explanation:
We are given \(\cos A = \frac{4}{5}\).
Let's consider a right-angled triangle where for angle A:
Adjacent side = 4
Hypotenuse = 5
Let the opposite side be 'p'. By the Pythagorean theorem: \[ p^2 + 4^2 = 5^2 \] \[ p^2 + 16 = 25 \] \[ p^2 = 25 - 16 = 9 \] \[ p = \sqrt{9} = 3 \] So, the Opposite side = 3.
Now we can find the required trigonometric ratios:
1. To find \(\cot A\): \[ \cot A = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{4}{3} \] 2. To find \(\csc A\): \[ \csc A = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{5}{3} \]

Step 4: Final Answer:
The values are \(\cot A = \frac{4}{3}\) and \(\csc A = \frac{5}{3}\).