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To find \( \cos 66^\circ \) for the given triangle, we use the formula for cosine:

\[
\cos x = \frac{\text{length of adjacent side}}{\text{length of hypotenuse}}
\]

In this case:
- The angle \(66^\circ\) has the **adjacent side** of \(34 \, \text{cm}\).
- The **hypotenuse** is \(37 \, \text{cm}\).

Thus, we calculate \( \cos 66^\circ \) as follows:

\[
\cos 66^\circ = \frac{34}{37}
\]

So, the answer is:

\[
\cos 66^\circ = \frac{34}{37}
\]

Would you like further details or have any questions?

Here are related questions that expand this information:
1. How would you find \( \sin 66^\circ \) for the same triangle?
2. How can you express \( \tan 66^\circ \) using the triangle's sides?
3. How can the Pythagorean theorem verify the side lengths?
4. What would \( \cos 24^\circ \) be in this triangle, assuming another angle is \(24^\circ\)?
5. How do these calculations change if the triangle's sides change proportionally?

**Tip:** Remember that trigonometric ratios are consistent for a given angle, regardless of the triangle's size, as long as the triangle is similar.

Using the right-angled triangle below, write cos 66° as a fraction.

Learn how to express cos 66° as a fraction using a right-angled triangle. With the adjacent side of 34 cm and the hypotenuse of 37 cm, find the cosine ratio step-by-step. Perfect for students in Grades 8-10 covering trigonometric concepts.

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