Free Math Worksheets

- What conversion factor changes radians to degrees? - Think about what fraction of a full rotation, \(2\pi\), the given angle represents. - Recall that \(\pi\) radians equals \(180^\circ\).

1. Use the conversion formula \(\theta = x \cdot \frac{180^\circ}{\pi}\). 2. Substitute \(x = \frac{5\pi}{12}\): \(\theta = \frac{5\pi}{12} \cdot \frac{180^\circ}{\pi}\). 3. Cancel \(\pi\) and simplify: \(\theta = \frac{5 \cdot 180^\circ}{12} = 75^\circ\).

b) \(75^\circ\)

- Determine what fraction of a full rotation, \(360^\circ\), each angle represents. - How many radians are in a half rotation of \(180^\circ\)? - Simplify each fraction completely before writing the result in terms of \(\pi\).

1. Use the conversion formula \(x = \theta \cdot \frac{\pi}{180^\circ}\) for each angle. 2. For a): \(30^\circ \cdot \frac{\pi}{180^\circ} = \frac{\pi}{6}\). 3. For b): \(135^\circ \cdot \frac{\pi}{180^\circ} = \frac{3\pi}{4}\). 4. For c): \(-210^\circ \cdot \frac{\pi}{180^\circ} = -\frac{7\pi}{6}\). 5. For d): \(315^\circ \cdot \frac{\pi}{180^\circ} = \frac{7\pi}{4}\). 6. For e): \(1080^\circ \cdot \frac{\pi}{180^\circ} = 6\pi\).

a) \(\frac{\pi}{6}\) b) \(\frac{3\pi}{4}\) c) \(-\frac{7\pi}{6}\) d) \(\frac{7\pi}{4}\) e) \(6\pi\)

- Recall that \(\pi\) radians equals \(180^\circ\). - When the radian measure contains \(\pi\), it will often cancel with the \(\pi\) in the conversion factor. - When the radian measure does not contain \(\pi\), use a decimal approximation for \(\pi\). - What does a negative sign indicate about the direction of rotation?

1. Use the conversion formula \(\theta = x \cdot \frac{180^\circ}{\pi}\). 2. For a): \(\frac{2\pi}{3} \cdot \frac{180^\circ}{\pi} = 120^\circ\). 3. For b): \(\frac{7\pi}{4} \cdot \frac{180^\circ}{\pi} = 315^\circ\). 4. For c): \(-\frac{3\pi}{2} \cdot \frac{180^\circ}{\pi} = -270^\circ\). 5. For d): \(4.5\pi \cdot \frac{180^\circ}{\pi} = 810^\circ\). 6. For e): \(1 \cdot \frac{180^\circ}{\pi} \approx 57.2958^\circ\), so the angle is approximately \(57.3^\circ\).

a) \(120^\circ\) b) \(315^\circ\) c) \(-270^\circ\) d) \(810^\circ\) e) \(57.3^\circ\)

- Determine what fraction of a full rotation, \(360^\circ\) or \(2\pi\), each angle represents. - Use the fact that \(180^\circ\) equals \(\pi\) radians. - Simplify each fraction in radians completely.

1. Convert \(120^\circ\) to radians: \(120^\circ \cdot \frac{\pi}{180^\circ} = \frac{2\pi}{3}\). 2. Convert \(\frac{5\pi}{6}\) to degrees: \(\frac{5\pi}{6} \cdot \frac{180^\circ}{\pi} = 150^\circ\). 3. Convert \(225^\circ\) to radians: \(225^\circ \cdot \frac{\pi}{180^\circ} = \frac{5\pi}{4}\). 4. Convert \(\frac{5\pi}{3}\) to degrees: \(\frac{5\pi}{3} \cdot \frac{180^\circ}{\pi} = 300^\circ\).

a) \(\frac{2\pi}{3}\) b) \(150^\circ\) c) \(\frac{5\pi}{4}\) d) \(300^\circ\)

- Determine what fraction of a full rotation, \(360^\circ\), each angle represents. - Recall that \(180^\circ\) equals \(\pi\) radians. - Simplify the fraction formed by dividing the degree measure by \(180\).

1. Use the conversion formula \(x = \theta \cdot \frac{\pi}{180^\circ}\). 2. For a): \(120^\circ \cdot \frac{\pi}{180^\circ} = \frac{2\pi}{3}\). 3. For b): \(315^\circ \cdot \frac{\pi}{180^\circ} = \frac{7\pi}{4}\). 4. For c): \(-30^\circ \cdot \frac{\pi}{180^\circ} = -\frac{\pi}{6}\). 5. For d): \(270^\circ \cdot \frac{\pi}{180^\circ} = \frac{3\pi}{2}\). 6. For e): \(72^\circ \cdot \frac{\pi}{180^\circ} = \frac{2\pi}{5}\).

a) \(\frac{2\pi}{3}\) b) \(\frac{7\pi}{4}\) c) \(-\frac{\pi}{6}\) d) \(\frac{3\pi}{2}\) e) \(\frac{2\pi}{5}\)

- First identify whether the given measure is in degrees or radians. - When a radian measure does not contain \(\pi\), use a decimal approximation for \(\pi\). - As a reasonableness check, \(1\) radian is about \(57.3^\circ\). - Keep the sign of the angle when you convert.

1. Use \(\theta = x \cdot \frac{180^\circ}{\pi}\) to convert radians to degrees and \(x = \theta \cdot \frac{\pi}{180^\circ}\) to convert degrees to radians. 2. For a): \(2.4 \cdot \frac{180^\circ}{\pi} \approx 137.51^\circ\). 3. For b): \(100^\circ \cdot \frac{\pi}{180^\circ} \approx 1.75\) radians. 4. For c): \(-\frac{3\pi}{4} \cdot \frac{180^\circ}{\pi} = -135^\circ\). 5. For d): \(6 \cdot \frac{180^\circ}{\pi} \approx 343.77^\circ\). 6. For e): \(12.5^\circ \cdot \frac{\pi}{180^\circ} \approx 0.22\) radians.

a) \(137.51^\circ\) b) \(1.75\) radians c) \(-135^\circ\) d) \(343.77^\circ\) e) \(0.22\) radians

- What are the degree and radian measures of one full rotation? - Use the relationship \(180^\circ = \pi\) radians. - Choose the conversion factor that cancels the unit you were given. - The sign of an angle does not change during conversion.

1. To convert degrees to radians, use \(x = \theta \cdot \frac{\pi}{180^\circ}\). 2. For a): \(72^\circ \cdot \frac{\pi}{180^\circ} = \frac{2\pi}{5}\). 3. For b): \(-210^\circ \cdot \frac{\pi}{180^\circ} = -\frac{7\pi}{6}\). 4. To convert radians to degrees, use \(\theta = x \cdot \frac{180^\circ}{\pi}\). 5. For c): \(\frac{4\pi}{9} \cdot \frac{180^\circ}{\pi} = 80^\circ\). 6. For d): \(-1.5\pi \cdot \frac{180^\circ}{\pi} = -270^\circ\).

a) \(\frac{2\pi}{5}\) b) \(-\frac{7\pi}{6}\) c) \(80^\circ\) d) \(-270^\circ\)

- Compare each angle with a half rotation of \(180^\circ\) or \(\pi\) radians. - Choose a conversion factor that introduces or cancels \(\pi\). - Simplify all fractions completely.

1. For a), \(150^\circ \cdot \frac{\pi}{180^\circ} = \frac{5\pi}{6}\). 2. For b), \(\frac{5\pi}{4} \cdot \frac{180^\circ}{\pi} = 225^\circ\). 3. For c), \(-40^\circ \cdot \frac{\pi}{180^\circ} = -\frac{2\pi}{9}\). 4. For d), \(-0.2\pi \cdot \frac{180^\circ}{\pi} = -36^\circ\).

a) \(\frac{5\pi}{6}\) b) \(225^\circ\) c) \(-\frac{2\pi}{9}\) d) \(-36^\circ\)

- A full rotation is \(2\pi\) radians or \(360^\circ\). - Use the fraction of a full rotation represented by each radian measure. - Multiply by \(\frac{180^\circ}{\pi}\) and round only at the end.

1. Use the conversion formula \(\theta = x \cdot \frac{180^\circ}{\pi}\). 2. For a): \(0.5 \cdot \frac{180^\circ}{\pi} \approx 28.65^\circ\). 3. For b): \(2.4 \cdot \frac{180^\circ}{\pi} \approx 137.51^\circ\). 4. For c): \(4.8 \cdot \frac{180^\circ}{\pi} \approx 275.02^\circ\). 5. For d): \(5.9 \cdot \frac{180^\circ}{\pi} \approx 338.05^\circ\).

a) \(28.65^\circ\) b) \(137.51^\circ\) c) \(275.02^\circ\) d) \(338.05^\circ\)

- Determine what fraction of a full rotation, \(360^\circ\), each angle represents. - A full rotation equals \(2\pi\) radians. - Use the relationship \(180^\circ = \pi\) radians. - Keep full calculator precision until the final rounding step.

1. Use the conversion formula \(x = \theta \cdot \frac{\pi}{180^\circ}\). 2. For a): \(24^\circ \cdot \frac{\pi}{180^\circ} \approx 0.42\). 3. For b): \(165^\circ \cdot \frac{\pi}{180^\circ} \approx 2.88\). 4. For c): \(212.5^\circ \cdot \frac{\pi}{180^\circ} \approx 3.71\). 5. For d): \(333^\circ \cdot \frac{\pi}{180^\circ} \approx 5.81\).

a) \(0.42\) radians b) \(2.88\) radians c) \(3.71\) radians d) \(5.81\) radians

- Rewrite both angles in each pair using the same unit. - Use benchmark angles on the unit circle, such as \(45^\circ\), \(90^\circ\), and \(180^\circ\). - Recall that \(\pi\) radians equals \(180^\circ\).

1. For a), \(\frac{5\pi}{4} = 225^\circ\). Since \(240^\circ > 225^\circ\), \(240^\circ\) is greater. 2. For b), \(\frac{11\pi}{6} = 330^\circ\). Since \(330^\circ > 320^\circ\), \(\frac{11\pi}{6}\) is greater. 3. For c), \(1.5\pi = \frac{3\pi}{2} = 270^\circ\). Since \(280^\circ > 270^\circ\), \(280^\circ\) is greater.

a) \(240^\circ > \frac{5\pi}{4}\) b) \(\frac{11\pi}{6} > 320^\circ\) c) \(280^\circ > 1.5\pi\)

- A full rotation is \(360^\circ\), or \(2\pi\) radians. - Convert radians to degrees by multiplying by \(\frac{180^\circ}{\pi}\). - Add or subtract full rotations to compare terminal sides. - Reduce each angle to the interval \([0^\circ, 360^\circ)\).

1. Multiply each radian measure by \(\frac{180^\circ}{\pi}\): \(x_1=60^\circ\), \(x_2=420^\circ\), \(x_3=-300^\circ\), and \(x_4=240^\circ\). 2. Coterminal angles differ by a multiple of \(360^\circ\), or equivalently by a multiple of \(2\pi\). 3. Since \(420^\circ-360^\circ=60^\circ\) and \(-300^\circ+360^\circ=60^\circ\), the angles \(x_1\), \(x_2\), and \(x_3\) have the same terminal side. The angle \(x_4=240^\circ\) has a different terminal side.

a) \(x_1=60^\circ\), \(x_2=420^\circ\), \(x_3=-300^\circ\), \(x_4=240^\circ\) b) \(x_1\), \(x_2\), and \(x_3\) have the same terminal side because their measures differ by integer multiples of \(360^\circ\). The angle \(x_4\) does not.

- Compare each angle with a full rotation of \(360^\circ\) or \(2\pi\) radians. - Use the relationship \(180^\circ = \pi\) radians. - Check the third decimal place when rounding to the nearest hundredth. - Convert negative angles in the same way as positive angles, keeping the negative sign.

1. For \(45^\circ\): \(45^\circ \cdot \frac{\pi}{180^\circ} = \frac{\pi}{4} \approx 0.79\). 2. For \(\frac{3\pi}{5}\): \(\frac{3\pi}{5} \cdot \frac{180^\circ}{\pi} = 108^\circ\), and \(\frac{3\pi}{5} \approx 1.88\). 3. For \(240^\circ\): \(240^\circ \cdot \frac{\pi}{180^\circ} = \frac{4\pi}{3} \approx 4.19\). 4. For \(-60^\circ\): \(-60^\circ \cdot \frac{\pi}{180^\circ} = -\frac{\pi}{3} \approx -1.05\).

a) \(45^\circ=\frac{\pi}{4}\approx0.79\) radians b) \(\frac{3\pi}{5}\) radians \(=108^\circ\), and \(\frac{3\pi}{5}\approx1.88\) c) \(240^\circ=\frac{4\pi}{3}\approx4.19\) radians d) \(-60^\circ=-\frac{\pi}{3}\approx-1.05\) radians

- Convert all angle measures to the same unit before comparing them. - In this context, unitless angle measures are in radians. - Use the \(\pi\) key on your calculator before rounding.

1. Express each angle as a decimal radian measure: - \(75^\circ \cdot \frac{\pi}{180^\circ} \approx 1.31\) - \(1.2\) is already in radians. - \(\frac{2\pi}{3} \approx 2.09\) - \(210^\circ \cdot \frac{\pi}{180^\circ} \approx 3.67\) - \(4\) is already in radians. 2. Compare the values: \(1.2 < 1.31 < 2.09 < 3.67 < 4\). 3. Match the decimal values to the original angles.

\(1.2 < 75^\circ < \frac{2\pi}{3} < 210^\circ < 4\)

- A full rotation is \(360^\circ\) or \(2\pi\) radians. - Use a conversion factor that cancels the unit you were given. - Keep exact values involving \(\pi\) until you need a decimal approximation. - Check whether each calculator entry is in degrees or radians.

1. Convert \(72^\circ\) to radians: \(72^\circ \cdot \frac{\pi}{180^\circ} = \frac{2\pi}{5} \approx 1.3\). 2. Convert \(1.5\) radians to degrees: \(1.5 \cdot \frac{180^\circ}{\pi} \approx 85.9^\circ\). 3. Convert \(-210^\circ\) to radians: \(-210^\circ \cdot \frac{\pi}{180^\circ} = -\frac{7\pi}{6} \approx -3.7\). 4. Convert \(\frac{3\pi}{4}\) to degrees: \(\frac{3\pi}{4} \cdot \frac{180^\circ}{\pi} = 135^\circ\). 5. Convert \(5\) radians to degrees: \(5 \cdot \frac{180^\circ}{\pi} \approx 286.5^\circ\).

| Degree measure | Radian measure | | :--- | :--- | | \(72^\circ\) | \(1.3\) | | \(85.9^\circ\) | \(1.5\) | | \(-210^\circ\) | \(-3.7\) | | \(135^\circ\) | \(\frac{3\pi}{4}\) | | \(286.5^\circ\) | \(5\) |

- Convert every angle to the same unit before comparing. - Estimate with \(\pi \approx 3.14\) to check the relative sizes. - To convert radians to degrees, multiply by \(\frac{180^\circ}{\pi}\).

1. Convert each radian measure to degrees. 2. \(\alpha = 155^\circ\) is already in degrees. 3. \(\beta = 2.7\) radians gives \(2.7 \cdot \frac{180^\circ}{\pi} \approx 154.7^\circ\). 4. \(\gamma = \frac{5\pi}{6}\) radians gives \(\frac{5\pi}{6} \cdot \frac{180^\circ}{\pi} = 150^\circ\). 5. \(\delta = 2.8\) radians gives \(2.8 \cdot \frac{180^\circ}{\pi} \approx 160.4^\circ\). 6. Therefore, \(150^\circ < 154.7^\circ < 155^\circ < 160.4^\circ\).

\(\gamma < \beta < \alpha < \delta\)

- A point returns to the same location after a rotation of \(2\pi\). - Write a formula using integer multiples of \(2\pi\). - Test both positive and negative integer values. - Check every result against the interval endpoints.

1. All coterminal angles have the form \(\alpha=\frac{2\pi}{3}+2\pi k\), where \(k\) is an integer. 2. For \(k=-1\), \(\alpha=-\frac{4\pi}{3}\), which lies in the interval. 3. For \(k=0\), \(\alpha=\frac{2\pi}{3}\), which lies in the interval. 4. For \(k=1\), \(\alpha=\frac{8\pi}{3}\), which lies in the interval. 5. The next values, obtained with \(k=-2\) and \(k=2\), lie outside the interval.

\(\alpha \in \left\{-\frac{4\pi}{3}, \frac{2\pi}{3}, \frac{8\pi}{3}\right\}\)