The exercises with resolution show a method; the following ask to apply it. Try each one before opening the answer. Arch measurements 1 Express $225^\circ$ in radians. See the resolutionSimple rule of three: $$\begin{array}{rcl} 180^\circ & \longleftrightarrow & \pi \text{ rad} \\ 225^\circ & \longleftrightarrow & x \end{array} \qquad\Longrightarrow\qquad x = \frac{225 \cdot \pi}{180} = \frac{5\pi}{4} \text{ rad}.$$ 2 Express in radians: a) $210^\circ$; b) $240^\circ$; c) $270^\circ$; d) $300^\circ$; e) $315^\circ$; f) $330^\circ$. Ver respostaa) $\dfrac{7\pi}{6}$ rad b) $\dfrac{4\pi}{3}$ rad c) $\dfrac{3\pi}{2}$ rad d) $\dfrac{5\pi}{3}$ rad e) $\dfrac{7\pi}{4}$ rad f) $\dfrac{11\pi}{6}$ rad
Before talking about sine or cosine, we need to know how to measure a piece of circumference. There are two rulers for this: the degree, inherited from the Babylonians, and the radian, which measures the arc with the radius itself. The story The division of the circumference into 360 parts comes from the astronomy of Mesopotamia, which counted based on 60 and followed the Sun, which covered the sky in about 360 days. In the second century BC, Hipparchus of Nicaea adopted the 360 degrees in his chord tables, and Ptolemy, in Almagest (c. 150 AD), divided each degree into 60 minutes and each minute into 60 seconds — the minutae primae and secundae parts that gave their names to the minute and the second.