Question

In \(\triangle ABC, \angle B = 90^\circ\). If \( \frac{AB}{AC} = \frac{1}{2} \), then cos C is equal to

• A. \( \frac{3}{2} \)
• B. \( \frac{1}{2} \)
• C. \( \frac{\sqrt{3}}{2} \)
• D. \( \frac{1}{\sqrt{3}} \)

Answer 1

The Correct Option is C

Given:
- In \(\triangle ABC\), \(\angle B = 90^\circ\) (right-angled at \(B\)).
- Ratio of sides: \(\frac{AB}{AC} = \frac{1}{2}\).
- Find \(\cos C\).

Step 1: Identify sides
- Since \(\angle B = 90^\circ\), \(AC\) is the hypotenuse.
- \(AB\) and \(BC\) are the legs.

Step 2: Assign variables
Let:
\[AB = x\] \[AC = 2x \quad (\text{from } \frac{AB}{AC} = \frac{1}{2})\]

Step 3: Use Pythagoras to find \(BC\)
\[AC^2 = AB^2 + BC^2\] Substitute:
\[(2x)^2 = x^2 + BC^2\] \[4x^2 = x^2 + BC^2\] \[BC^2 = 3x^2\] \[BC = \sqrt{3}x\]

Step 4: Find \(\cos C\)
- \(\cos C = \frac{\text{adjacent side to } C}{\text{hypotenuse}} = \frac{BC}{AC}\)
\[\cos C = \frac{\sqrt{3}x}{2x} = \frac{\sqrt{3}}{2}\]

Final Answer:
\[\boxed{\frac{\sqrt{3}}{2}}\]