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
The trigonometric cycle is the “race track” of trigonometry: a circle of radius 1 in which each real number is a route from a starting point. All the functions of the next chapters will be read in it. The story Greek and Indian astronomers worked with circles of large radius — Ptolemy used radius 60 and Aryabhata, 3438 — to avoid fractions in tables. It was Euler, in the 18th century, who fixed the radius 1: thus the length of the arc is the measurement of the angle itself in radians, and sine and cosine become numbers, not lengths of segments. The idea of wrapping the real line around the circumference, allowing negative arcs greater than a turn, is what transforms the trigonometry of triangles into a theory of periodic functions.