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To solve for \( f \) in this right-angled triangle, we can use trigonometry, specifically the sine function, since we are given an angle and the side opposite to it (7 cm) and are asked to find the hypotenuse \( f \).

The sine of an angle in a right triangle is given by:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

In this case:

\[
\sin(40^\circ) = \frac{7}{f}
\]

Rearranging to solve for \( f \):

\[
f = \frac{7}{\sin(40^\circ)}
\]

I'll calculate that now.The value of \( f \), correct to two decimal places, is \( 10.89 \) cm.

Would you like more details or have any other questions? 

Here are 5 related questions for further exploration:
1. How would you calculate the length of the adjacent side in this triangle?
2. What trigonometric ratio would you use if you wanted to find the adjacent side instead of the hypotenuse?
3. How would the solution change if the angle was \(50^\circ\) instead of \(40^\circ\)?
4. Can you use the cosine function to find the hypotenuse in this scenario?
5. What is the relationship between the sine and cosine of complementary angles in a right triangle?

**Tip**: In any right triangle, the sum of the angles always adds up to \(180^\circ\), and one of them is always \(90^\circ\).

Find the value of f, correct to two decimal places, in a right triangle where the opposite side to a 40° angle is 7 cm, and the hypotenuse is f.

Learn how to find the hypotenuse in a right-angled triangle using the sine function. Given a 40° angle and a 7 cm opposite side, solve for the hypotenuse to two decimal places. Perfect for Grades 9-11 students studying trigonometry.

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