Question

If 15 cot \(\theta\) = 8, then the value of sin \(\theta\) will be

• A. \( \frac{17}{15} \)
• B. \( \frac{15}{8} \)
• C. \( \frac{8}{17} \)
• D. \( \frac{15}{17} \)

Hint:

Memorizing common Pythagorean triplets can save a lot of time. (8, 15, 17) is a common triplet. When you see \( \cot \theta = 8/15 \), you can immediately identify the sides as 8 and 15, and the hypotenuse as 17, making the calculation for \( \sin \theta \) (Opposite/Hypotenuse = 15/17) very fast.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem involves finding the value of one trigonometric ratio (\(\sin \theta\)) given the value of another (\(\cot \theta\)). We can solve this by constructing a right-angled triangle or by using trigonometric identities.
Step 2: Key Formula or Approach:
From the given equation, we find \( \cot \theta \). We know that \( \cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} \). We can then use the Pythagorean theorem (\( \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 \)) to find the hypotenuse and subsequently \( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \).
Step 3: Detailed Explanation:
Given the equation:
\[ 15 \cot \theta = 8 \] \[ \cot \theta = \frac{8}{15} \] In a right-angled triangle, \( \cot \theta \) is the ratio of the adjacent side to the opposite side.
Let the adjacent side = 8k and the opposite side = 15k, where k is a positive constant.
Using the Pythagorean theorem to find the hypotenuse (H):
\[ H^2 = (\text{Opposite})^2 + (\text{Adjacent})^2 \] \[ H^2 = (15k)^2 + (8k)^2 \] \[ H^2 = 225k^2 + 64k^2 \] \[ H^2 = 289k^2 \] \[ H = \sqrt{289k^2} = 17k \] Now, we can find \( \sin \theta \), which is the ratio of the opposite side to the hypotenuse.
\[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{15k}{17k} \] \[ \sin \theta = \frac{15}{17} \] Step 4: Final Answer:
The value of \( \sin \theta \) is \( \frac{15}{17} \).