{"id":1791,"date":"2025-07-28T19:39:41","date_gmt":"2025-07-28T19:39:41","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1791"},"modified":"2025-12-03T00:05:44","modified_gmt":"2025-12-03T00:05:44","slug":"angles-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/angles-learn-it-2\/","title":{"raw":"Angles: Learn It 2","rendered":"Angles: Learn It 2"},"content":{"raw":"

Radians<\/h2>\r\nDividing a circle into 360 parts is an arbitrary choice, although it creates the familiar degree measurement. We may choose other ways to divide a circle.\r\n\r\n
An arc<\/strong>\u00a0may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire circle is called the\u00a0circumference<\/strong> of that circle.<\/section>The circumference of a circle is [latex]C=2\\pi r[\/latex]. If we divide both sides of this equation by [latex]r[\/latex], we create the ratio of the circumference to the radius, which is always [latex]2\\pi[\/latex] regardless of the length of the radius.\r\n

This brings us to our new angle measure. One radian<\/span> is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals [latex]2\\pi [\/latex] times the radius, a full circular rotation is [latex]2\\pi [\/latex] radians. So<\/p>\r\n\r\n

[latex]\\begin{gathered} 2\\pi \\text{ radians}={360}^{\\circ } \\\\ \\pi \\text{ radians}=\\frac{{360}^{\\circ }}{2}={180}^{\\circ } \\\\ 1\\text{ radian}=\\frac{{180}^{\\circ }}{\\pi }\\approx {57.3}^{\\circ } \\end{gathered}[\/latex]<\/div>\r\n
\r\n

radian<\/h3>\r\nOne radian<\/strong> is the measure of the central angle of a circle such that the length of the arc between the initial side and the terminal side is equal to the radius of the circle. A full revolution (360\u00b0) equals [latex]2\\pi [\/latex] radians. A half revolution (180\u00b0) is equivalent to [latex]\\pi [\/latex] radians.\r\n\r\n \r\n\r\nThe radian measure<\/strong> of an angle is the ratio of the length of the arc subtended by the angle to the radius of the circle. In other words, if [latex]s[\/latex] is the length of an arc of a circle, and [latex]r[\/latex] is the radius of the circle, then the central angle containing that arc measures [latex]\\frac{s}{r}[\/latex] radians. In a circle of radius 1, the radian measure corresponds to the length of the arc.\r\n\r\n<\/section>
When an angle is described without a specific unit, it refers to radian measure. For example, an angle measure of 3 indicates 3 radians. In fact, radian measure is dimensionless, since it is the quotient of a length (circumference) divided by a length (radius) and the length units cancel out.<\/section>
\r\n\r\n[caption id=\"attachment_4985\" align=\"aligncenter\" width=\"391\"]\"Illustration The angle t<\/i> sweeps out a measure of one radian. Note that the length of the intercepted arc is the same as the length of the radius of the circle.[\/caption]\r\n\r\n<\/section><\/div>\r\n

Using Radians<\/h2>\r\nConsidering the most basic case, the unit circle<\/strong> (a circle with radius 1), we know that 1 rotation equals 360 degrees, 360\u00b0. We can also track one rotation around a circle by finding the circumference, [latex]C=2\\pi r[\/latex], and for the unit circle [latex]C=2\\pi [\/latex]. These two different ways to rotate around a circle give us a way to convert from degrees to radians.\r\n
[latex]\\begin{gathered}1\\text{ rotation }=360^\\circ =2\\pi \\text{radians} \\\\ \\frac{1}{2}\\text{ rotation}=180^\\circ =\\pi \\text{radians} \\\\ \\frac{1}{4}\\text{ rotation}=90^\\circ =\\frac{\\pi }{2} \\text{radians} \\end{gathered}[\/latex]<\/div>\r\n
Find the radian measure of one-third of a full rotation.[reveal-answer q=\"791790\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"791790\"]For any circle, the arc length along such a rotation would be one-third of the circumference. We know that\r\n

[latex]1\\text{ rotation}=2\\pi r[\/latex]<\/p>\r\nSo,\r\n

[latex]\\begin{align} s&=\\frac{1}{3}\\left(2\\pi r\\right) \\\\ &=\\frac{2\\pi r}{3} \\end{align}[\/latex]<\/p>\r\nThe radian measure would be the arc length divided by the radius.\r\n

[latex]\\begin{align}\\text{radian measure}&=\\frac{\\frac{2\\pi r}{3}}{r} \\\\ &=\\frac{2\\pi r}{3r} \\\\ &=\\frac{2\\pi }{3} \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

Find the radian measure of three-fourths of a full rotation.[reveal-answer q=\"558820\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"558820\"][latex]\\frac{3\\pi }{2}[\/latex][\/hidden-answer]<\/section>","rendered":"

Radians<\/h2>\n

Dividing a circle into 360 parts is an arbitrary choice, although it creates the familiar degree measurement. We may choose other ways to divide a circle.<\/p>\n

An arc<\/strong>\u00a0may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire circle is called the\u00a0circumference<\/strong> of that circle.<\/section>\n

The circumference of a circle is [latex]C=2\\pi r[\/latex]. If we divide both sides of this equation by [latex]r[\/latex], we create the ratio of the circumference to the radius, which is always [latex]2\\pi[\/latex] regardless of the length of the radius.<\/p>\n

This brings us to our new angle measure. One radian<\/span> is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals [latex]2\\pi[\/latex] times the radius, a full circular rotation is [latex]2\\pi[\/latex] radians. So<\/p>\n

[latex]\\begin{gathered} 2\\pi \\text{ radians}={360}^{\\circ } \\\\ \\pi \\text{ radians}=\\frac{{360}^{\\circ }}{2}={180}^{\\circ } \\\\ 1\\text{ radian}=\\frac{{180}^{\\circ }}{\\pi }\\approx {57.3}^{\\circ } \\end{gathered}[\/latex]<\/div>\n
\n
\n

radian<\/h3>\n

One radian<\/strong> is the measure of the central angle of a circle such that the length of the arc between the initial side and the terminal side is equal to the radius of the circle. A full revolution (360\u00b0) equals [latex]2\\pi[\/latex] radians. A half revolution (180\u00b0) is equivalent to [latex]\\pi[\/latex] radians.<\/p>\n

 <\/p>\n

The radian measure<\/strong> of an angle is the ratio of the length of the arc subtended by the angle to the radius of the circle. In other words, if [latex]s[\/latex] is the length of an arc of a circle, and [latex]r[\/latex] is the radius of the circle, then the central angle containing that arc measures [latex]\\frac{s}{r}[\/latex] radians. In a circle of radius 1, the radian measure corresponds to the length of the arc.<\/p>\n<\/section>\n

When an angle is described without a specific unit, it refers to radian measure. For example, an angle measure of 3 indicates 3 radians. In fact, radian measure is dimensionless, since it is the quotient of a length (circumference) divided by a length (radius) and the length units cancel out.<\/section>\n
\n
\"Illustration
The angle t<\/i> sweeps out a measure of one radian. Note that the length of the intercepted arc is the same as the length of the radius of the circle.<\/figcaption><\/figure>\n<\/section>\n<\/div>\n

Using Radians<\/h2>\n

Considering the most basic case, the unit circle<\/strong> (a circle with radius 1), we know that 1 rotation equals 360 degrees, 360\u00b0. We can also track one rotation around a circle by finding the circumference, [latex]C=2\\pi r[\/latex], and for the unit circle [latex]C=2\\pi[\/latex]. These two different ways to rotate around a circle give us a way to convert from degrees to radians.<\/p>\n

[latex]\\begin{gathered}1\\text{ rotation }=360^\\circ =2\\pi \\text{radians} \\\\ \\frac{1}{2}\\text{ rotation}=180^\\circ =\\pi \\text{radians} \\\\ \\frac{1}{4}\\text{ rotation}=90^\\circ =\\frac{\\pi }{2} \\text{radians} \\end{gathered}[\/latex]<\/div>\n
Find the radian measure of one-third of a full rotation.<\/p>\n