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A rectangle with dimensions 3 cm by 4 cm is inscribed in a circle. What is the circumference of the circle?
The diagonal of the rectangle is the diameter of the circle. Using the Pythagorean theorem, the diagonal \(d\) is:
d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ cm}
The radius \(r\) of the circle is half of the diameter:
r = \frac{5}{2} = 2.5 \text{ cm}
C = 2\pi r = 2\pi \times 2.5 = 5\pi \text{ cm}
Thus, the circumference of the circle is \(\boxed{5\pi}\) cm.
Compute \(\tan 45^\circ\) using the unit circle and confirm its geometric interpretation.
On the unit circle, the angle \(45^\circ\) corresponds to the point \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\). The tangent of an angle is the ratio of the y-coordinate to the x-coordinate:
\tan 45^\circ = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1