To solve for $f$, we are working with a right-angled triangle where:

• The given angle is $34^\circ$,
• The adjacent side to the $34^\circ$ angle is $22.7 \, \text{cm}$,
• $f$ is the hypotenuse of the triangle.

We use the cosine function in trigonometry:

$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}.$

Here:

• $\theta = 34^\circ$,
• $\text{adjacent} = 22.7 \, \text{cm}$,
• $\text{hypotenuse} = f$.

Step 1: Substitute into the formula

$\cos(34^\circ) = \frac{22.7}{f}.$

Step 2: Rearrange for $f$

$f = \frac{22.7}{\cos(34^\circ)}.$

Step 3: Calculate $\cos(34^\circ)$

Using a calculator: $\cos(34^\circ) \approx 0.8290.$

Step 4: Solve for $f$

$f = \frac{22.7}{0.8290} \approx 27.4 \, \text{cm}.$

Final Answer:

The length of $f$ is 27.4 cm (to 1 decimal place).

Would you like further explanations or verification steps? Here are some follow-up questions you might consider:

1. How do you determine which trigonometric ratio to use in right-angled triangles?
2. What is the significance of the hypotenuse in a right-angled triangle?
3. Can you solve for another side of this triangle using trigonometry?
4. How does this solution change if the given angle is replaced with another?
5. What are the sine and tangent values for $34^\circ$?

Tip: Always double-check if the triangle is right-angled before applying trigonometric ratios.