{"id":67320,"date":"2024-02-29T01:29:23","date_gmt":"2024-02-29T01:29:23","guid":{"rendered":"https:\/\/schooltutoring.com\/help\/?p=67320"},"modified":"2024-02-29T01:32:13","modified_gmt":"2024-02-29T01:32:13","slug":"what-are-radians-and-why-we-use-them","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/help\/what-are-radians-and-why-we-use-them\/","title":{"rendered":"What are Radians and Why we Use them"},"content":{"rendered":"
[vc_row][vc_column][vc_column_text]\n

Why do we use Degrees?<\/h1>\n

When we think about angles, the first thing that comes to mind is degrees. Examples are 30\u00b0, 45\u00b0, 90\u00b0, 360\u00b0, etc. However, have you ever wondered why we define angles in this way?<\/p>\n

The answer lies in the properties of 360\u00b0. This number is special as it can be divided exactly by no more than 24 numbers (ie. 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360). In consequence, if we use 360\u00b0 to represent the angle that covers the full circle, a lot of pieces of the circle will have “nice” angles.<\/p>\n

For example, half a circle is represented by 180\u00b0, a third by 120\u00b0, a quarter by 90\u00b0, a sixth by 60\u00b0 and so on. However, this choice is arbitrary. Nothing is stopping us from making the angle that covers a full circle 50 new degrees and having half being 25 new degrees. Let’s represent new degrees using \"*\"<\/p>\n

Let’s find the conversion factor to go from usual degrees \"\circ\" to new degrees \"*\" for the angle \"\theta\":<\/p>\n

  <\/span>   <\/span>\"\[\frac{\theta_{*}}{\theta_{\circ}}=\frac{50^{*}}{360^{\circ}}=\frac{5^{*}}{36^{\circ}}\]\"<\/p>\n

If we multiply this factor by 90\u00b0 we’ll see how much this angle is worth in new degrees:<\/p>\n

  <\/span>   <\/span>\"\[90^{\circ}\left( \frac{5^{*}}{36^{\circ}}\right)=12.5^{*}\]\"<\/p>\n

This value is exactly what we would expect.<\/p>\n

What are Radians and Why do we use them?<\/h1>\n

Now that we have established that we can choose how to measure our angles arbitrarily, we should be wondering if there is a superior<\/strong> way of measuring angles. The answer is, that it depends on what you are working on. Let’s see an example. Let’s say you want to find a formula to relate an angle with the arc it forms in a circle of radius r.<\/p>\n

In this graph we show the arc (in green) formed by the 45\u00b0 angle (called \"\theta_{\circ}\") in a circle of radius \"r=5\" (in blue):<\/p>\n[\/vc_column_text]

\"Example <\/div><\/div>[\/vc_column][\/vc_row][vc_row][vc_column][vc_column_text]\n

We can find a formula to calculate the arc algebraically like this:<\/p>\n

  <\/span>   <\/span>\"\begin{align*}<\/p>\n

Where \"arc\" is the arc that the angle \"\theta\" forms in a circle of radius r. This formula is extremely useful, but can this formula be simplified?<\/p>\n

Let’s see what happens if we transform the angle \"\theta\" to new units using the conversion factor \"k\" that transforms the angles from degrees to a new unknown unit. In our new units, our angle will be:<\/p>\n

  <\/span>   <\/span>\"\[\theta_{\text{new}}=k\theta_{\circ} \implies \theta_{\circ}=\frac{\theta_{\text{new}}}{k}\]\"<\/p>\n

Using this in our formula we obtain:<\/p>\n

  <\/span>   <\/span>\"\[arc=\frac{{\pi}}{180^{\circ}}\left( r\frac{\theta_{\text{new}}}{k} \right)\]\"<\/p>\n

Now we remember that we are free to choose \"k\" to be anything as we are free to choose the unit we measure our new angle with. Why not make it such that it cancels the constant in the front? Let <\/p>\n

  <\/span>   <\/span>\"\[k=\frac{\pi}{180^{\circ}}\quad \text{Then}\]\"<\/p>\n

  <\/span>   <\/span>\"\begin{align*}<\/p>\n

This formula looks cleaner! The arc length is just the radius of the circle times the angle in this new unit that transforms from degrees using the conversion factor \"k=\dfrac{\pi}{180^{\circ}}\". This means that in this new units \"\pi\" is equivalent to \"180^{\circ}\", \"2\pi\" is equivalent to \"360^{\circ}\" and \"\frac{\pi}{2}\" is equivalent to \"90^{\circ}\".<\/p>\n

This new unit is so important that it received its own name the radian<\/strong>. Using our new improved formula we can see that for 1 radian:<\/p>\n

  <\/span>   <\/span>\"\[arc=r\]\"<\/p>\n

So 1 radian is the angle that forms an arc equivalent to the radius.[\/vc_column_text]

\"The <\/div><\/div>[vc_column_text]Radians are really important in Calculus and Physics so it is imperative that students learn to do everything they can do with usual degrees with radians and to naturally move from one to the other. Let’s see some examples of how to transform some angles.<\/p>\n

Transforming from Degrees to Radians:<\/h2>\n

To transform from degrees to radians, we simply need to multiply the angle in degrees times the transformation factor \"\dfrac{\pi}{180^{\degree}}\"<\/p>\n

Transform \"120^{\circ}\" to radians: <\/strong><\/p>\n

  <\/span>   <\/span>\"\[\theta_{rad}= 120^{\circ}\left(\dfrac{\pi}{180^{\degree}}\right)=\frac{2\pi}{3}\]\"<\/p>\n

Transform \"60^{\circ}\" to radians: <\/strong><\/p>\n

  <\/span>   <\/span>\"\[\theta_{rad}= 60^{\circ}\left(\dfrac{\pi}{180^{\degree}}\right)=\frac{\pi}{3}\]\"<\/p>\n

Transform \"45^{\circ}\" to radians: <\/strong><\/p>\n

  <\/span>   <\/span>\"\[\theta_{rad}= 45^{\circ}\left(\dfrac{\pi}{180^{\degree}}\right)=\frac{\pi}{4}\]\"<\/p>\n

Transforming from Radians to Degrees:<\/h2>\n

To transform from degrees to radians, we simply need to multiply the angle in radians times the inverse of the previous transformation factor \"\dfrac{180^{\degree}}{\pi}\"<\/p>\n

Transform \"\dfrac{\pi}{6}\" radians to degrees: <\/strong><\/p>\n

  <\/span>   <\/span>\"\[\theta_{\circ}= \frac{\pi}{6}\left(\dfrac{180^{\degree}}{\pi}\right)=30^{\circ}\]\"<\/p>\n

Transform \"\dfrac{3\pi}{2}\" radians to degrees: <\/strong><\/p>\n

  <\/span>   <\/span>\"\[\theta_{\circ}= \frac{3\pi}{2}\left(\dfrac{180^{\degree}}{\pi}\right)=270^{\circ}\]\"<\/p>\n

The study of radians is extensive and we could talk about them for a long time like learning how to use trigonometric rations like \"\sin\" and \"\cos\" with them, but for now, knowing where they come from and how to transform from degrees to radians is enough to get started.<\/p>\n

This has been a brief introduction to the concepts of quadratic equations. For more information on this topic as well as assistance with homework and test preparation, feel free to reach out to an Academic Director toll-free at 1 (877) 545-7737<\/strong> or via our Contact Us<\/strong><\/a> page.<\/p>\n

Powered by WordPress<\/a> with QuickLaTeX<\/a><\/p>\n[\/vc_column_text][\/vc_column][\/vc_row]\n<\/div>","protected":false},"excerpt":{"rendered":"

A radian is the natural unit of angles, but why do we use them instead of degrees. In this post we discuss the relationship between angles, radians and degrees.<\/p>\n","protected":false},"author":35,"featured_media":67322,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[2,11,14,18],"tags":[75,3027,467,1086,2722,1456,1457,3823,2660],"class_list":["post-67320","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra","category-math-fundamentals","category-pre-calculus","category-trigonometry","tag-algebra-2","tag-algebraic-equations","tag-degree-of-a-polynomial","tag-math","tag-quadratic-equation","tag-quadratic-equations","tag-quadratic-formula","tag-trigonometry","tag-using-the-quadratic-formula"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/67320","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/users\/35"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/comments?post=67320"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/67320\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media\/67322"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=67320"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=67320"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=67320"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}