1. **Stating the problem:** We need to calculate the length $f$ in a right triangle where one angle is $27^\circ$, the other non-right angle is $34^\circ$, and the base adjacent to the $27^\circ$ angle is $2.2$ cm. 2. **Formula and rules:** In a right triangle, the sum of angles is $90^\circ + 27^\circ + 34^\circ = 151^\circ$, but since it's a right triangle, the right angle is $90^\circ$, so the other two angles must sum to $90^\circ$. Given the angles $27^\circ$ and $34^\circ$, the triangle is consistent. We use the sine function to find the side opposite an angle: $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ 3. **Identify sides:** The base $2.2$ cm is adjacent to the $27^\circ$ angle, so the hypotenuse $h$ can be found using cosine: $$\cos(27^\circ) = \frac{2.2}{h} \implies h = \frac{2.2}{\cos(27^\circ)}$$ 4. **Calculate hypotenuse:** $$h = \frac{2.2}{\cos(27^\circ)}$$ 5. **Calculate $f$ (opposite side to $27^\circ$):** $$f = h \sin(27^\circ) = \frac{2.2}{\cos(27^\circ)} \sin(27^\circ)$$ 6. **Simplify expression:** $$f = 2.2 \frac{\sin(27^\circ)}{\cos(27^\circ)} = 2.2 \tan(27^\circ)$$ 7. **Calculate numerical value:** $$f = 2.2 \times \tan(27^\circ) \approx 2.2 \times 0.5095 = 1.12$$ 8. **Final answer:** $$f \approx 1.1 \text{ cm (to 2 significant figures)}$$