what angle in radians corresponds to 0.75 rotations around the unit circle?

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In the previous section, we introduced periodic functions and demonstrated how they can be used to model real life phenomena like the rotation of the London Eye. In fact, there is an intuitive connectedness between periodic functions and the rotation of a circle. In this section, we will use our intuition and formalize this connection by exploring the unit circle and its unique features that pb us into the rich world of trigonometry.

Many applications involving circles also involve a rotation of the circumvolve and so nosotros must first introduce a measure out for the rotation, or angle, between 2 rays (line segments) emanating from the eye of a circumvolve.

The measure of an angle is a measurement between 2 intersecting lines, line segments, or rays, starting at the initial side and catastrophe at the terminal side. It is a rotational measure, not a linear measure.

When measuring angles on a circle, unless otherwise directed, nosotros measure angles in standard position: starting at the positive horizontal axis with a counterclockwise rotation.

Subsection Measuring Angles in Degrees

A degree is a unit of an angle. One rotation around a circle is equal to 360 degrees.

An angle measured in degrees should always include the caste symbol \(^\circ\) or the discussion "degrees" after the number. For instance, \(90^\circ=ninety\) degrees.

Case 6

Requite the degree measure of the angle shown on the circumvolve.

Solution

The vertical and horizontal lines divide the circle into quarters. Since one total rotation is 360 degrees, each quarter rotation is \(360^\circ / iv = 90^\circ\text{.}\)

Example 7

Draw an angle of \(thirty^\circ\) on a circle.

Solution

An angle of \(30^\circ\) is \(1/three\) of \(90^\circ\) so by dividing a quarter rotation into thirds, we can draw a line showing an bending of \(30^\circ\text{.}\)

Notice that since there are 360 degrees in one rotation, an angle greater than 360 degrees would betoken more than one full rotation. Shown on a circle, the resulting direction in which this angle's terminal side points is the same every bit another for an angle betwixt 0 and 360. These angles are called coterminal.

Subsequently completing their full rotation based on the given angle, two angles are coterminal if they terminate in the same position, and then their terminal sides coincide (signal in the same direction).

Case nine

Find an bending \(\theta\) that is coterminal with \(800^\circ\) where \(0^\circ \leq \theta \lt 360^\circ \text{.}\)

Solution

Since adding or subtracting a full rotation, or 360 degrees, would result in an angle with the last side pointing in the same direction, we can observe coterminal angles past calculation or subtracting 360 degrees.

An angle of 800 degrees is coterminal with an angle of

\brainstorm{equation*} 800^\circ - 360^\circ = 440^\circ \cease{equation*}

It is also coterminal with an angle of

\brainstorm{equation*} 440^\circ - 360^\circ = 80^\circ\text{.} \end{equation*}

Finding the coterminal bending between 0 and \(360^\circ\) tin make it easier to see which direction the terminal side of an angle points in.

Example x

Find an angle \(\alpha\) that is coterminal with \(870^\circ\) where \(0^\circ \leq \blastoff \lt 360^\circ \text{.}\)

Solution

To find angles with the terminal sides pointing in the same management, we can subtract 360 degrees:

\begin{equation*} \begin{aligned} 870^\circ-360^\circ \amp= 510^\circ \\ 510^\circ-360^\circ \amp= 150^\circ \cease{aligned} \stop{equation*}

Therefore \(870\) degrees is coterminal with \(150\) degrees.

Instance 12

Depict an angle of \(-45^\circ\) on a circle and observe a positive coterminal angle \(\alpha\) where \(0^\circ \leq \alpha \lt 360^\circ \text{.}\)

Solution

Since 45 degrees is half of 90 degrees, we can start at the positive horizontal centrality and measure out clockwise one-half of a xc degree bending.

We can find a positive coterminal bending by adding 360 degrees.

\begin{equation*} -45^\circ+360^\circ=315^\circ \end{equation*}

Example 13

Find an bending \(\beta\) that is coterminal with \(-300^\circ\) where \(0^\circ \leq \beta \lt 360^\circ \text{.}\)

Solution

Since \(-300^\circ+360^\circ=sixty^\circ\text{,}\) \(60\) degrees is coterminal with \(-300^\circ\text{.}\)

Subsection Measuring Angles in Radians

While measuring angles in degrees may be familiar, doing so often complicates matters since the units of measure tin make it the way of calculuations. For this reason, another measure of angles is commonly used. This measure is based on the distance around the unit circle.

The Unit Circle

The unit of measurement circle is a circle of radius i, centered at the origin of the \((10,y)\) plane. When measuring an angle around the unit circumvolve, we travel in the counterclockwise direction, starting from the positive \(10\)-axis. A negative angle is measured in the opposite, or clockwise, direction. A complete trip effectually the unit of measurement circle amounts to a total of 360 degrees.

A radian is a measurement of an bending that arises from looking at angles equally a fraction of the circumference of the unit circumvolve. A complete trip around the unit circle amounts to a total of \(two\pi\) radians.

Radians are a unitless measure. Therefore, it is not necessary to write the label "radians" after a radian measure, and if you lot see an angle that is not labeled with "degrees" or the degree symbol, yous should assume that information technology is a radian measure.

Radians and degrees both measure angles. Thus, information technology is possible to convert between the two. Since one rotation around the unit circumvolve equals 360 degrees or \(2\pi\) radians, nosotros can utilize this equally a conversion cistron.

Converting Between Radians and Degrees

Since \(360 \text{ degrees} = two\pi \text{ radians}\text{,}\) nosotros tin can divide each side past 360 and conclude that

\begin{equation*} \displaystyle 1 \text{ degree} = \frac{two\pi \text{ radians}}{360} = \frac{\pi \text{ radians}}{180} \stop{equation*}

Then, to convert from degrees to radians, we can multiply past \(\displaystyle \ \frac{\pi \text{ radians}}{180^\circ}\)

Similarly, nosotros tin conclude that

\begin{equation*} \displaystyle 1 \text{ radian} = \frac{360^\circ}{2\pi} = \frac{180^\circ}{\pi} \end{equation*}

So, to convert from radians to degrees, we can multiply by \(\displaystyle \ \frac{180^\circ}{\pi \text{ radians}}\)

Instance 15

Convert \(\displaystyle \frac{\pi}{6}\) radians to degrees.

Solution

Since we are given an angle in radians and we want to convert information technology to degrees, we multiply the bending by \(180^\circ\) and then separate past \(\pi\) radians.

\begin{equation*} \frac{\pi}{6} \text{ radians} \cdot \frac{180^\circ}{\pi \text{ radians}} = 30^\circ \stop{equation*}

Example 16

Catechumen \(15^\circ\) to radians.

Solution

In this example, we beginning with an angle in degrees and want to catechumen it to radians. We multiply by \(\pi\) and divide by \(180^\circ\) so that the units of degrees abolish and we are left with the unitless measure of radians.

\begin{equation*} xv^\circ \cdot \frac{\pi}{180^\circ} = \frac{\pi}{12} \end{equation*}

Instance 17

Convert \(\displaystyle \frac{vii\pi}{10}\) radians to degrees.

Solution

Since nosotros are given an angle in radians and we desire to catechumen it to degrees, nosotros multiply by the angle \(180^\circ\) and and then separate past \(\pi\) radians.

\brainstorm{equation*} \frac{7\pi}{10} \text{ radians} \cdot \frac{180^\circ}{\pi \text{ radians}} = 126^\circ \end{equation*}

To a higher place, we explored how to discover coterminal angles for angles greater than 360 degrees and less than 0 degrees. Similarly, we can discover coterminal angles for angles greater than \(2\pi\) radians and less than 0 radians.

Example 19

Discover an bending \(\beta\) that is coterminal with \(\displaystyle \frac{11\pi}{4}\) where \(0^\circ \leq \beta \lt 2\pi \text{.}\)

Solution

When working in degrees, we found coterminal angles by calculation or subtracting 360 degrees, a full rotation. As well, in radians, we can find coterminal angles past adding or subtracting full rotations of \(ii\pi\) radians.

An angle of \(11\pi/4\) is coterminal with an angle of

\brainstorm{equation*} \frac{11\pi}{4} - 2\pi = \frac{11\pi}{4} - \frac{8\pi}{4} = \frac{three\pi}{4} \end{equation*}

Example 20

Detect an angle \(\phi\) that is coterminal with \(\displaystyle -\frac{11\pi}{half-dozen}\) where \(0^\circ \leq \phi \lt 2\pi \text{.}\)

Solution

An angle of \(-\frac{11\pi}{6}\) is coterminal with an bending of

\begin{equation*} -\frac{11\pi}{6} + 2\pi = -\frac{11\pi}{6} + \frac{12\pi}{half dozen} = \frac{\pi}{6} \terminate{equation*}

Subsection Finding Points on the Unit Circle

While information technology is convenient to depict the location of a point on the unit of measurement circle using an angle, relating this bending to the \(x\) and \(y\) coordinates of the corresponding point is an of import application of trigonometry. To do this, we volition need to apply our cognition of triangles.

Example 21

Notice the \((x,y)\) coordinates for the point on the unit circle corresponding to an angle of 45 degrees or \(\pi/4\) radians.

Solution

Let's starting time by drawing a moving-picture show and labeling the known data.

We want to discover the \(x\) and \(y\) coordinates of the point on the unit circumvolve respective to an angle of 45 degrees or \(\pi/iv\text{.}\) To exercise this, we can draw a vertical line from the point downward to the \(x\)-axis, which forms a right triangle. The hypotenuse of this triangle is i, since it corresponds to the radius of the unit circle, and the side lengths of this triangle are equal to \(x\) and \(y\text{.}\)

Now, using the Pythagorean Theorem, we get that

\brainstorm{equation*} x^ii+y^2=1^2 \hspace{.25in} \text{ which simplifies to } \hspace{.25in} x^2+y^ii=i \terminate{equation*}

Since the triangle formed is a 45-45-90 degree triangle, side lengths \(ten\) and \(y\) must be equal. Therefore, we can substitute in \(x=y\) into the above equation.

\begin{marshal*} ten^ii+y^2 \amp = 1 \amp\amp \text{Substitute in } ten=y \\ \\ x^ii+x^ii \amp = ane \amp\amp \text{Add similar terms } \\ \\ 2x^2 \amp = 1 \amp\amp \text{Split up by two} \\ \\ x^2 \amp = \frac{1}{ii} \amp\amp \text{Have the square root} \\ \\ 10 \amp = \pm\sqrt{\vphantom{\frac{ane^2}{2}}\frac{i}{two}} \amp\amp \text{Since the } 10 \text{ value is positive, nosotros keep the positive root so} \\ \\ x \amp = \sqrt{\vphantom{\frac{1^2}{2}}\frac{ane}{2}} \amp\amp \end{align*}

Oftentimes this value is written with a rationalized denominator. Remember that to rationalize the denominator, we multiply by a term equivalent to 1 to get rid of the radical in the denominator, then

\begin{equation*} 10 = \sqrt{\vphantom{\frac{one^2}{2}}\frac{1}{2}} \, \sqrt{\vphantom{\frac{ane^ii}{2}}\frac{2}{2}} = \sqrt{\vphantom{\frac{1^2}{ii}}\frac{ii}{four}} = \frac{\sqrt{2}}{two} \end{equation*}

and since \(10\) and \(y\) are equal, \(\displaystyle y=\frac{\sqrt{two}}{two}\text{.}\)

Thus, the \((x,y)\) coordinates for the bespeak on the unit of measurement circumvolve corresponding to an angle of 45 degrees or \(\pi/4\) radians are

\begin{equation*} (x,y) = \left(\frac{\sqrt{2}}{2},\frac{\sqrt{two}}{2}\right)\text{.} \terminate{equation*}

Case 22

Find the \((x,y)\) coordinates for the point on the unit of measurement circumvolve respective to an angle of 30 degrees or \(\pi/6\) radians.

Solution

Let'due south start by drawing a picture and labeling the known information.

We want to find the \(x\) and \(y\) coordinates of the signal on the unit circumvolve corresponding to an bending of xxx degrees or \(\pi/6\text{.}\) To practise this, we tin can describe a triangle inside the unit circumvolve with one side at an bending of thirty degrees and some other at an bending of \(-30\) degrees.

Notice that if we combine the resulting two right triangles to course one large triangle, then all three angles of the larger triangle are equal to 60 degrees.

Since all of the angles in this triangle are equal, the sides will all exist equal likewise. Ii of the side lengths of this triangle are equal to ane because they correspond to the radius of the unit of measurement circumvolve. Thus, the other side length must also be equal to 1. This side also has a length of \(2y\text{.}\) Therefore, we can conclude that \(2y = i\) so

\begin{equation*} y=\frac{1}{2} \terminate{equation*}

At present, we can employ the Pythagorean Theorem to one of the right triangles to find the \(x\) value. Nosotros get that

\brainstorm{align*} x^ii+y^2 \amp = ane^ii \\ \\ ten^ii+\left(\frac{one}{two}\right)^2 \amp = one \\ \\ x^ii+\frac{1}{4} \amp = 1 \\ \\ x^2 \amp = \frac{three}{4} \\ \\ x \amp = \pm\sqrt{\vphantom{\frac{3^2}{four}}\frac{3}{four}} \\ \\ x \amp = \sqrt{\vphantom{\frac{3^two}{four}}\frac{3}{4}} = \frac{\sqrt{3}}{two} \end{align*}

Thus, the \((x,y)\) coordinates for the signal on the unit of measurement circle corresponding to an angle of 30 degrees or \(\pi/6\) are

\begin{equation*} (ten,y) = \left(\frac{\sqrt{iii}}{2},\frac{one}{2}\correct)\text{.} \stop{equation*}

In the previous case, we applied our knowledge of triangles to discover the \(x\) and \(y\) coordinates of the point on the unit circle corresponding to 30 degrees. We can now apply symmetry to find the \(x\) and \(y\) coordinates corresponding to an angle of 60 degrees or \(\pi/3\text{.}\) Starting time, we draw a film of the triangle respective to this point on the unit of measurement circle.

Notice that the triangle shown above is like to the one formed by the 30 caste angle since the hypotenuse is the aforementioned length and both are 30-60-90 degree triangles. Therefore, the \((10,y)\) coordinates for lx degrees are the aforementioned as the \((x,y)\) coordinates for thirty degrees, only switched.

Thus, the \((10,y)\) coordinates for the point on the unit circle corresponding to an angle of 60 degrees or \(\pi/3\) are

\begin{equation*} (x,y) = \left(\frac{1}{2},\frac{\sqrt{3}}{2}\correct)\text{.} \cease{equation*}

We take now found the coordinates of the points corresponding to all the commonly encountered angles in the kickoff quadrant of the unit circle. Using symmetry, we tin can find the rest of the coordinates corresponding to common angles on the unit circle. A labeled picture of the unit of measurement circle is shown beneath.

Subsection Supplemental Videos

• Measuring Angles
• Coordinates on the Unit Circle
• Examples

Subsection Exercises

i Converting Degrees to Radians

two Converting Radians to Degrees

3 Converting Between Degrees and Radians

4 Practical Converting Between Degrees and Radians

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Source: https://mathbooks.unl.edu/PreCalculus/unit-circle.html

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