https://file.aiquickdraw.com/mathbot/file/1730967690258bbdtalwg.jpg

To solve for the length \( f \) in this right-angled triangle, let's use the given information and trigonometric concepts.

### Step 1: Set up the equation using the cosine of the angle.
In a right-angled triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse. Here:
- The angle is \( 56^\circ \),
- The hypotenuse is \( 5.7 \) cm,
- The adjacent side (relative to the \( 56^\circ \) angle) is \( f \).

Using the cosine definition:
\[
\cos 56^\circ = \frac{f}{5.7}
\]

### Step 2: Solve for \( f \).
Rearrange the equation to isolate \( f \):
\[
f = 5.7 \times \cos 56^\circ
\]

### Step 3: Calculate \( \cos 56^\circ \) and find \( f \).
Using \( \cos 56^\circ \approx 0.5592 \):
\[
f = 5.7 \times 0.5592
\]
\[
f \approx 3.19 \text{ cm}
\]

### Final Answer
The length \( f \) is approximately \( 3.2 \) cm to 1 decimal place.

Would you like more details or have any questions?

Here are five related questions to consider:
1. What is the sine ratio for the same angle in this triangle?
2. How would you find the length of the other leg using trigonometry?
3. What if the angle was different? How would that change the calculation?
4. How does the accuracy of angle values affect trigonometric calculations?
5. Can you solve for the missing side using the Pythagorean theorem as a check?

**Tip:** Always check if your calculator is in the correct mode (degrees or radians) when working with trigonometric functions.

By first writing an equation involving cos 56°, work out the length f in the right-angled triangle shown.

Learn how to use trigonometry to find the length of a side in a right-angled triangle. This example involves using the cosine ratio with an angle of 56° and a hypotenuse of 5.7 cm to find the adjacent side, f.

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