Question

If \(\tan \theta = \frac{12}{5}\) then the value of \(\sin \theta\) is

• A. \(\frac{5}{12}\)
• B. \(\frac{12}{13}\)
• C. \(\frac{5}{13}\)
• D. \(\frac{12}{5}\)

Hint:

Recognize common Pythagorean triples like (3, 4, 5), (5, 12, 13), and (8, 15, 17). If you see two of the numbers, you can instantly know the third without calculation. Here, we have 5 and 12, so the hypotenuse is 13.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Given one trigonometric ratio, we can find any other trigonometric ratio by constructing a right-angled triangle or by using trigonometric identities.

Step 2: Key Formula or Approach:
We use the definitions of trigonometric ratios in a right-angled triangle:
\[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}, \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \] We can find the hypotenuse using the Pythagorean theorem: \(h^2 = p^2 + b^2\).

Step 3: Detailed Explanation:
We are given \(\tan \theta = \frac{12}{5}\).
From this, we can consider a right-angled triangle where:
Opposite side (\(p\)) = 12
Adjacent side (\(b\)) = 5
Now, we find the hypotenuse (\(h\)) using the Pythagorean theorem:
\[ h = \sqrt{p^2 + b^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \] Now we can find \(\sin \theta\):
\[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{12}{13} \]

Step 4: Final Answer:
The value of \(\sin \theta\) is \(\frac{12}{13}\).