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) \(75^\circ\)

- What is the radius of a unit circle? - How is radian measure defined using arc length and radius? - Check whether the numerical value changes when the radius is \(1\).

1. Radian measure is the arc length divided by the circle's radius. 2. The radius of a unit circle is \(1\), so the angle measure is \(\frac{1.3}{1}=1.3\) radians.

\(1.3\) radians. On a unit circle, the radian measure equals the intercepted arc length.

- Start with the relationship among arc length, radius, and an angle measured in radians. - Substitute the radius of a unit circle before using the given arc length. - Explain the equality between arc length and radian measure rather than only stating the number.

1. For a central angle, arc length satisfies \(s=r\theta\), where \(\theta\) is measured in radians. 2. On a unit circle, \(r=1\), so \(s=\theta\). 3. Since \(s=\frac{5\pi}{6}\), the angle measure is \(\theta=\frac{5\pi}{6}\) radians.

\(\frac{5\pi}{6}\) radians, because \(s=r\theta\) and \(r=1\), so \(s=\theta\) on a unit circle.

- Focus on the definition of one radian rather than a degree conversion. - Ask what ratio of arc length to radius gives an angle measure of \(1\).

1. Radian measure compares intercepted arc length with the circle's radius. 2. An angle of \(1\) radian intercepts an arc whose length equals one radius. 3. On a unit circle, both lengths are \(1\) unit.

An angle of \(1\) radian intercepts an arc whose length equals the radius. On a unit circle, that arc is \(1\) unit long.

- Read the magnitude of the central angle from the diagram. - Decide what sign is used for clockwise rotation. - Combine the direction sign with the angle magnitude.

1. The marked angle has magnitude \(\frac{\pi}{3}\). 2. Clockwise rotations have negative radian measure. 3. Therefore, the signed radian measure is \(-\frac{\pi}{3}\).

\(-\frac{\pi}{3}\)

- 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) \(300^\circ\)

- Locate each measure by using familiar quarter-turn and half-turn directions on the unit circle. - Compare which terminal ray is reached later during one counterclockwise revolution. - For a decimal multiple of \(\pi\), first relate it to a familiar fraction of a full turn.

1. In a), \(120^\circ\) lies in Quadrant II, while \(\frac{5\pi}{4}\) lies in Quadrant III. During the first counterclockwise revolution, the Quadrant III ray is farther around the circle, so \(\frac{5\pi}{4}>120^\circ\). 2. In b), \(100^\circ\) lies in Quadrant II, while \(\frac{11\pi}{6}\) lies in Quadrant IV. Therefore, \(\frac{11\pi}{6}>100^\circ\). 3. In c), \(1.5\pi=\frac{3\pi}{2}\) is the downward benchmark ray. The \(280^\circ\) ray lies just past the \(270^\circ\) downward direction, so \(280^\circ>1.5\pi\).

a) \(\frac{5\pi}{4}>120^\circ\) b) \(\frac{11\pi}{6}>100^\circ\) c) \(280^\circ>1.5\pi\)

- 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) \(-36^\circ\)

- 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

- 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

- 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. - Compare your result with the benchmark \(180^\circ=\pi\) radians to check whether its size is reasonable.

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: \(5 \cdot \frac{180^\circ}{\pi} \approx 286.5^\circ\).

<table> <tr><th>Degree measure</th><th>Radian measure</th></tr> <tr><td>\(72^\circ\)</td><td>\(1.3\)</td></tr> <tr><td>\(85.9^\circ\)</td><td>\(1.5\)</td></tr> <tr><td>\(-210^\circ\)</td><td>\(-3.7\)</td></tr> <tr><td>\(135^\circ\)</td><td>\(\frac{3\pi}{4}\)</td></tr> <tr><td>\(286.5^\circ\)</td><td>\(5\)</td></tr> </table>

- Start with the circumference of a circle whose radius is \(1\). - What fraction of that circumference is intercepted? - On a unit circle, how is intercepted arc length related to radian measure?

1. The circumference of a unit circle is \(2\pi\). 2. One quarter of that circumference has length \(\frac{1}{4}\cdot 2\pi=\frac{\pi}{2}\). 3. On a unit circle, arc length equals radian measure, so the angle measures \(\frac{\pi}{2}\) radians.

\(\frac{\pi}{2}\) radians

- Use the dashed radius as the \(\frac{\pi}{2}\) benchmark in each panel. - Compare the terminal radius with that benchmark. - Decide whether the terminal radius is reached before or after the dashed radius when turning counterclockwise from the starting radius.

1. In panel \(1\), the terminal radius lies before the dashed \(\frac{\pi}{2}\) benchmark radius. Therefore, the marked central angle is less than \(\frac{\pi}{2}\). 2. In panel \(2\), the terminal radius lies beyond the dashed \(\frac{\pi}{2}\) benchmark radius. Therefore, the marked central angle is greater than \(\frac{\pi}{2}\).

a) Less than \(\frac{\pi}{2}\). The terminal radius is before the dashed \(\frac{\pi}{2}\) benchmark. b) Greater than \(\frac{\pi}{2}\). The terminal radius is beyond the dashed \(\frac{\pi}{2}\) benchmark.

- Start with the circumference of the unit circle. - Take the stated fraction of that circumference. - Connect the resulting arc length to radian measure.

1. The circumference of a unit circle is \(2\pi\). 2. Three quarters of the circumference is \(\frac{3}{4}\cdot 2\pi=\frac{3\pi}{2}\). 3. On a unit circle, intercepted arc length equals radian measure, so the angle measures \(\frac{3\pi}{2}\) radians.

\(\frac{3\pi}{2}\) radians

- On a unit circle, what quantity has the same numerical value as the radian measure? - A semicircle corresponds to half of a circumference of \(2\pi\). - Compare \(2.4\) with that benchmark rather than converting units.

1. On a unit circle, arc length equals radian measure, so the intercepted arc has length \(2.4\) units. 2. A semicircle on a unit circle has arc length \(\pi\), and \(2.4<\pi\). 3. Therefore, the angle is less than a semicircle.

a) \(2.4\) units b) Less than a semicircle, because \(2.4<\pi\).

- Radian measures can be ordinary real numbers as well as multiples of \(\pi\). - Use the unit-circle definition directly. - Check what arc length would correspond to a full turn before accepting \(2\pi\).

1. A radian measure does not have to be written as a multiple of \(\pi\). 2. On a unit circle, the numerical radian measure equals the intercepted arc length. 3. Therefore, an angle of \(2\) radians intercepts an arc of length \(2\) units, not \(2\pi\) units.

The error is assuming every radian measure must contain \(\pi\). The intercepted arc length is \(2\) units.

- Convert complete rotations directly to radians using one full turn. - Think of the total swept angle, not only the final terminal position. - On a unit circle, connect total radian measure with total arc length traveled.

1. One complete rotation is \(2\pi\) radians. 2. The total angle is \(2.5\cdot 2\pi=5\pi\) radians. 3. On a unit circle, total arc length equals the total radian measure, so the traveled arc length is \(5\pi\) units.

a) \(5\pi\) radians b) \(5\pi\) units

- Use approximate sizes of the standard radian benchmarks rather than changing units. - Compare \(2.2\) first with \(\frac{\pi}{2}\) and \(\pi\). - Connect the interval between those benchmarks with a quadrant on the unit circle.

1. \(\frac{\pi}{2}\approx1.57\) and \(\pi\approx3.14\). 2. Since \(1.57<2.2<3.14\), the angle lies between \(\frac{\pi}{2}\) and \(\pi\). 3. Therefore, its terminal side is in Quadrant II.

It lies between \(\frac{\pi}{2}\) and \(\pi\), so its terminal side is in Quadrant II.

- Use the dashed half-turn radius as the \(\pi\) benchmark. - Compare the terminal radius with the downward direction corresponding to three quarters of a turn. - The angle is measured counterclockwise from the starting radius.

1. The dashed radius marks a half-turn, or \(\pi\) radians. 2. The terminal radius is past that half-turn but before the downward radius that would mark \(\frac{3\pi}{2}\). 3. Therefore, \(\pi<\theta<\frac{3\pi}{2}\).

\(\theta\in(\pi,\frac{3\pi}{2})\)

- Read the terminal radius from the diagram before considering the choices. - Identify the two standard radian benchmark directions on either side of the terminal radius. - Compare the position of each answer choice with those benchmark directions.

1. From the diagram, the terminal radius is in Quadrant III and halfway between the negative x-axis direction and the downward direction. 2. Those benchmark directions correspond to \(\pi\) and \(\frac{3\pi}{2}\). 3. The angle halfway between them is \(\frac{\pi+\frac{3\pi}{2}}{2}=\frac{5\pi}{4}\).

\(\frac{5\pi}{4}\)

- Separate the magnitude of the radian measure from its sign. - Recall which rotation direction corresponds to a negative input. - To find a positive coterminal input, change the input by one full revolution.

1. The magnitude \(\frac{5\pi}{6}\) gives the size of the turn, but the negative sign determines its direction. 2. Therefore, \(-\frac{5\pi}{6}\) means a clockwise turn, not a counterclockwise turn. 3. Add one full revolution: \(-\frac{5\pi}{6}+2\pi=\frac{7\pi}{6}\). 4. Thus the positive coterminal input is \(\frac{7\pi}{6}\), not \(\frac{5\pi}{6}\).

The error is ignoring the negative sign. The rotation is clockwise, and the positive coterminal input is \(\frac{7\pi}{6}\).

- 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.

- Convert all angle measures to the same unit before comparing them. - The values explicitly labeled in radians are already in the comparison unit. - 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.

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

- 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.

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