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
An angle is the “opening” between two rays of the same origin. To measure it, we transport the question to an arc: the angle measures the same as the arc it determines in a circle centered at its vertex. The story The The Elements of Euclid (c. 300 BC) defines the plane angle as “the inclination, relative to each other, of two lines in a plane that meet and are not in a straight line”. The definition seems simple, but the Greeks discussed for centuries what is, after all, an angle: a quantity, which can be added and compared? a quality, like a form? or a relation between two lines? The person who tells this dispute is Proclus, in the 5th century, in his commentary on Euclid. The solution we adopted — measuring the angle by the arc it determines in a circle centered on the vertex — leaves the philosophical question aside and transforms the angle into a number.
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.