F is an angle in a right-angled triangle, and \( \cos F = \frac{6}{11} \) - Junior Cycle Mathematics - Question 11 - 2018

Use the Pythagorean theorem to find the opposite side:
[ \text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2 ]
[ 11^2 = \text{Opp}^2 + 6^2 ]
[ 121 = \text{Opp}^2 + 36 ]
[ \text{Opp}^2 = 121 - 36 = 85 ]
[ \text{Opp} = \sqrt{85} ]

Now that we have the length of the opposite side, we can use it to find ( \sin F ).
Using the definition:
[ \sin F = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{85}}{11} ]

Thus, the value of ( \sin F ) in surd form is:
[ \sin F = \frac{\sqrt{85}}{11} ]