Free Math Worksheets

- 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 \(x=\frac{11\pi}{6}\)

- 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 \(1^2+1^2=2 \ne 1\).