Free math worksheets

- Think of a radian as measuring an arc by comparing it with the radius. - Compare the given arc length with the given radius. - The angle in radians has no length unit.

1. Radian measure compares an intercepted arc length with the circle's radius, so the angle is \(\frac{12}{8}=\frac{3}{2}\) radians. 2. The value \(\frac{3}{2}\) means the arc is \(1.5\) times as long as the radius.

\(\frac{3}{2}\) radians; the arc length is \(1.5\) radius-lengths.

- Compare the terminal side with the initial side on the positive x-axis. - What fraction of one full rotation is shown? - Recall the radian measure of one full rotation.

1. The terminal side points directly opposite the initial side, so the angle is one-half of a full rotation. 2. A full rotation measures \(2\pi\) radians, so one-half of a full rotation measures \(\pi\) radians.

\(\pi\) radians

- List the four axis-intersection points of the unit circle. - The x-coordinate is cosine and the y-coordinate is sine. - Apply the equation \(x^2+y^2=1\).

1. The axis angles are \(0^\circ\), \(90^\circ\), \(180^\circ\), \(270^\circ\), and \(360^\circ\). 2. The corresponding unit-circle coordinates \((\cos(\alpha), \sin(\alpha))\) are \((1, 0)\), \((0, 1)\), \((-1, 0)\), \((0, -1)\), and \((1, 0)\). 3. Thus the sine-cosine pairs are \((0, 1)\), \((1, 0)\), \((0, -1)\), \((-1, 0)\), and \((0, 1)\), respectively. 4. If both sine and cosine were \(1\), then \(\sin^2(\alpha)+\cos^2(\alpha)=1^2+1^2=2\), contradicting the unit-circle equation, which requires the sum to equal \(1\).

a) \(0^\circ, 90^\circ, 180^\circ, 270^\circ, 360^\circ\) b) \(0^\circ: (\sin(\alpha), \cos(\alpha))=(0, 1)\); \(90^\circ: (1, 0)\); \(180^\circ: (0, -1)\); \(270^\circ: (-1, 0)\); \(360^\circ: (0, 1)\) c) It is impossible because \(1^2+1^2=2 \ne 1\).

- On the unit circle, each point has the form \((\cos\theta,\sin\theta)\). - Use the exact side ratios of a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle for the first-quadrant point. - For the second-quadrant point, keep the same reference-angle magnitudes and check the signs of the coordinates.

1. Point \(P\) is at \(\frac{\pi}{3}\). Its unit-circle coordinates are \(\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)\). 2. Point \(Q\) is at \(\frac{2\pi}{3}\), in Quadrant II with reference angle \(\frac{\pi}{3}\). Its coordinates are \(\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)\). 3. On the unit circle, cosine is the x-coordinate and sine is the y-coordinate. Therefore, \(\cos\left(\frac{\pi}{3}\right)=\frac{1}{2}\), \(\sin\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{2}\), \(\cos\left(\frac{2\pi}{3}\right)=-\frac{1}{2}\), and \(\sin\left(\frac{2\pi}{3}\right)=\frac{\sqrt{3}}{2}\).

a) \(P=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)\); \(Q=\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)\) b) \(\cos\left(\frac{\pi}{3}\right)=\frac{1}{2}\), \(\sin\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{2}\); \(\cos\left(\frac{2\pi}{3}\right)=-\frac{1}{2}\), \(\sin\left(\frac{2\pi}{3}\right)=\frac{\sqrt{3}}{2}\)

- Choose a conversion factor that cancels the angle unit you start with. - For degrees to radians, compare the degree measure with a full or half rotation. - For radians to degrees, use the relationship between \(\pi\) radians and a straight angle.

1. Convert \(150^\circ\) to radians: \(150^\circ\cdot\frac{\pi}{180^\circ}=\frac{5\pi}{6}\). 2. Convert \(\frac{7\pi}{12}\) radians to degrees: \(\frac{7\pi}{12}\cdot\frac{180^\circ}{\pi}=105^\circ\).

a) \(\frac{5\pi}{6}\) b) \(105^\circ\)

- A full rotation is \(360^\circ\) or \(2\pi\) radians. - Use \(\frac{\pi}{180^\circ}\) to convert degrees to radians. - Recall exact sine values for special angles. - Check whether your calculator is in degree or radian mode.

1. For \(60^\circ\), \(x=60^\circ\cdot\frac{\pi}{180^\circ}=\frac{\pi}{3}\approx1.05\), and \(\sin(x)=\frac{\sqrt{3}}{2}\approx0.87\). 2. For \(x=\frac{3\pi}{4}\), the degree measure is \(135^\circ\), and \(\sin(x)=\frac{\sqrt{2}}{2}\approx0.71\). 3. On \(0\le x<2\pi\), \(\sin(x)=-1\) at \(x=\frac{3\pi}{2}\approx4.71\), which is \(270^\circ\). 4. For \(310^\circ\), \(x=310^\circ\cdot\frac{\pi}{180^\circ}=\frac{31\pi}{18}\approx5.41\), and \(\sin(x)\approx-0.77\).

<table> <tr> <td>Degree measure</td> <td>Radian measure \(x\)</td> <td>\(\sin(x)\)</td> </tr> <tr> <td>\(60^\circ\)</td> <td>\(\frac{\pi}{3}\approx1.05\)</td> <td>\(\frac{\sqrt{3}}{2}\approx0.87\)</td> </tr> <tr> <td>\(135^\circ\)</td> <td>\(\frac{3\pi}{4}\approx2.36\)</td> <td>\(\frac{\sqrt{2}}{2}\approx0.71\)</td> </tr> <tr> <td>\(270^\circ\)</td> <td>\(\frac{3\pi}{2}\approx4.71\)</td> <td>\(-1\)</td> </tr> <tr> <td>\(310^\circ\)</td> <td>\(\frac{31\pi}{18}\approx5.41\)</td> <td>\(-0.77\)</td> </tr> </table>

- Approximately how many degrees are in \(1\) radian? - Use the behavior of cosine in Quadrant I. - For part c), identify the quadrants in which sine is negative. - Use a reference angle and unit-circle symmetry.

1. Convert \(225^\circ\): \(225^\circ\cdot\frac{\pi}{180^\circ}=\frac{5\pi}{4}\). 2. One radian is approximately \(57.3^\circ\). Cosine decreases on \([0^\circ, 90^\circ]\), and \(1^\circ<57.3^\circ\), so \(\cos(1^\circ)>\cos(1)\). 3. The reference angle for \(\sin(x)=-\frac{1}{2}\) is \(\frac{\pi}{6}\). On \([\pi, 2\pi]\), the solutions are \(\pi+\frac{\pi}{6}=\frac{7\pi}{6}\) and \(2\pi-\frac{\pi}{6}=\frac{11\pi}{6}\).

a) \(\frac{5\pi}{4}\) b) \(\cos(1^\circ)>\cos(1)\) c) \(x=\frac{7\pi}{6}\) or \(x=\frac{11\pi}{6}\)

- Choose one sign convention for clockwise and counterclockwise rotation and use it consistently. - Combine the two rotations before finding a coterminal angle. - One full rotation changes the angle by \(2\pi\) without changing the terminal position. - Relate signed angle measure to signed distance traveled along the rim.

1. Treat counterclockwise rotation as positive and clockwise rotation as negative. The net angle is \(-\frac{11\pi}{6}+\frac{\pi}{3}=-\frac{11\pi}{6}+\frac{2\pi}{6}=-\frac{3\pi}{2}\). 2. Add one full rotation: \(-\frac{3\pi}{2}+2\pi=\frac{\pi}{2}\). Thus, \(\frac{\pi}{2}\) is the coterminal angle in \([0,2\pi)\). 3. Signed arc displacement equals radius times signed angle, so \(8\left(-\frac{3\pi}{2}\right)=-12\pi\,\text{cm}\).

a) \(-\frac{3\pi}{2}\) b) \(\frac{\pi}{2}\) c) \(-12\pi\,\text{cm}\)

- In radian measure, arc length depends on both the radius and the central angle. - For part a), decide which quantity in the arc-length relationship is unknown. - Use the radius you find in part a) for the new central angle in part b). - Check that each requested length has length units.

1. For an angle measured in radians, arc length satisfies \(s=r\theta\). Thus, \(15\pi=r\left(\frac{5\pi}{6}\right)\), so \(r=18\,\text{cm}\). 2. For the second angle, \(s=18\left(\frac{7\pi}{12}\right)=\frac{21\pi}{2}\,\text{cm}\).

a) \(18\,\text{cm}\) b) \(\frac{21\pi}{2}\,\text{cm}\)

- Compare arc length with radius to recover the radian measure in Circle A. - The same central angle has the same radian measure in both circles. - For part c), write the arc-length relationship once for each circle and compare the two expressions. - Focus on which factor is common when the central angles are equal.

1. In Circle A, \(\theta=\frac{s}{r}=\frac{4\pi}{6}=\frac{2\pi}{3}\). 2. In Circle B, \(s=r\theta=15\left(\frac{2\pi}{3}\right)=10\pi\,\text{cm}\). 3. For a fixed radian angle \(\theta\), each arc length has the form \(s=r\theta\). Therefore, for radii \(r_1\) and \(r_2\), \(\frac{s_2}{s_1}=\frac{r_2\theta}{r_1\theta}=\frac{r_2}{r_1}\). The common angle cancels, so arc length scales directly with radius.

a) \(\theta=\frac{2\pi}{3}\) b) \(10\pi\,\text{cm}\) c) For the same radian angle, \(s=r\theta\), so the common factor \(\theta\) cancels in the ratio and \(\frac{s_2}{s_1}=\frac{r_2}{r_1}\).