(ii) Find the length of the wire from the point 'O' to the top of section 'A'.

Q: 37. (ii) Find the length of the wire from the point 'O' to the top of section 'A'.

$OA = 12\text{ m}$. Since OP = 6 m is adjacent to the 60° angle and OA is the hypotenuse, $\cos 60° = \frac{OP}{OA}$ gives $OA = \frac{6}{\cos 60°} = \frac{6}{1/2} = 12\text{ m}$.

Q: What value of cos 60° is used?

$\cos 60° = \frac{1}{2}$, so the wire = $6 \div (1/2) = 12\text{ m}$.

Some Applications of Trigonometry

Heights and distances - length of wire (60°)

Official Answer

$OA = 12\text{ m}$ . Since OP = 6 m is adjacent to the 60° angle and OA is the hypotenuse, $\cos 60° = \frac{OP}{OA}$ gives $OA = \frac{6}{\cos 60°} = \frac{6}{1/2} = 12\text{ m}$ .

12 mcos 60°wire lengthhypotenuseOA = 6/cos60°cos 60 = 1/2angle of elevation 60°section A

Marking Scheme

11 mark: correct set-up $\cos 60° = \frac{6}{OA}$ giving $OA = \frac{6}{\cos 60°}$ .
2Award full credit for the final answer 12 m; accept the working $6 \div (1/2) = 12\text{ m}$ .

Hint

The 6 m ground distance is adjacent to the 60° angle and the wire is the hypotenuse, so $\text{wire} = 6 \div \cos 60° = 6 \div (1/2)$ .

Quick Oral Answer

Using $\cos 60° = \frac{6}{OA}$ with $\cos 60° = \frac{1}{2}$ , the wire $OA = 6 \div (1/2) = 12\text{ m}$ .

Analysis & Explanation

This part repeats the cosine method at the steeper 60° angle, letting the student compare how wire length changes with the angle of elevation.

Concept

OP = 6 m is adjacent to the 60° angle, and OA is the hypotenuse, so $\cos 60° = \frac{OP}{OA}$ gives $OA = \frac{6}{1/2} = 12\text{ m}$ , a clean value since $\cos 60° = \frac{1}{2}$ .

Key points

Comparing with part (i): the higher point A (60° elevation) needs a longer wire (12 m) than the lower point B (30°, ≈6.93 m) — matching physical intuition.

Common mistakes (परीक्षा में सावधानी)

Using $\tan 60°$ to find the tower's height ( $6\sqrt{3}\text{ m}$ ) and reporting that as the wire length, or writing $\sin 60° = \frac{6}{OA}$ by mistake.

Real-world

Recognising the memorised value $\cos 60° = \frac{1}{2}$ rewards accurate recall of standard trigonometric ratios over heavy computation.

Common Mistakes

1Using $\tan 60°$ to find the height $6\sqrt{3}\text{ m}$ and reporting that as the wire length instead of the hypotenuse.
2Writing $\sin 60° = \frac{6}{OA}$ (mismatching the adjacent side) and getting $4\sqrt{3}\text{ m}$ .
3Using $\cos 60° = \frac{\sqrt{3}}{2}$ by confusing it with cos 30°, giving a wrong value.

Interesting Facts

$\cos 60° = \frac{1}{2}$ is the only standard cosine value equal to a simple fraction, which is why the wire length comes out to a neat 12 m.

The wire to the higher point A (12 m) is longer than the wire to the lower point B (≈6.93 m) — steeper elevation always means a longer guy wire from the same anchor.

$OA = 12\text{ m}$ is exactly twice the 6 m base, a direct consequence of $\cos 60° = \frac{1}{2}$ .

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Frequently Asked Questions

Why is the wire to A longer than the wire to B?

Point A is higher up the tower and seen at a steeper 60° angle, so the wire from the same anchor point O must be longer — 12 m versus about 6.93 m for B.

What value of cos 60° is used?

$\cos 60° = \frac{1}{2}$ , so the wire = $6 \div (1/2) = 12\text{ m}$ .

Which side of the triangle is the wire?

The wire OA is the hypotenuse of the right triangle, opposite the right angle at the base of the tower.