**1. Trigonometric Angles**

Consider two rays $Oa$ and $Ob$.

- If a ray $Om$ rotates around its origin $O$ in a fixed direction, starting from the position of ray $Oa$ and stopping at the position of ray $Ob$, we say that the ray $Om$ sweeps an
with initial ray $Oa$ and terminal ray $Ob$, denoted as $\left( Oa, Ob \right)$.*angle* - When the ray $Om$ rotates through an angle $\alpha$, we say that the measure of the angle $\left( Oa, Ob \right)$ is $\alpha$, denoted as $s\tilde{n}\left( Oa, Ob \right) = \alpha$.

**Note:**

- For two given rays $Oa$ and $Ob$, there are infinitely many angles with initial ray $Oa$ and terminal ray $Ob$. We use the notation $\left( Oa, Ob \right)$ for all of these angles.
- Counterclockwise rotation is considered positive, and clockwise rotation is considered negative.
- One full rotation in the positive direction corresponds to an angle of $360^\circ$, and one full rotation in the negative direction corresponds to an angle of $-360^\circ$.
- The measures of angles with the same initial ray $Oa$ and terminal ray $Ob$ differ by a multiple of $360^\circ$, so the general formula is $s\tilde{n}\left( Oa, Ob \right) = \alpha^\circ + k \cdot 360^\circ$ or $\left( Oa, Ob \right) = \alpha^\circ + k \cdot 360^\circ$.
- Chasles’ theorem: For any three rays $Oa$, $Ob$, and $Oc$, we have $\left( Oa, Ob \right) + \left( Ob, Oc \right) = \left( Oa, Oc \right) + k \cdot 360^\circ$.

**Example 1.**

Determine the measures of the angles $\left( Oa, Ob \right)$ in the following figures:

$\left( Oa, Ob \right) = $ | $\left( Oa, Ob \right) = $ | ||

$\left( Oa, Ob \right) = $ | $\left( Oa, Ob \right) = $ |

**2. Radian Measure**

On a circle with an arbitrary radius $R$, there is an arc at the center with a length equal to $R$, which is called an angle with a measure of 1 radian (denoted as 1 rad).

**Conversion Formulas between Degrees and Radians**

${{a}^{0}} = \dfrac{\pi a}{180}$ radians, and $\alpha$ radians $= \left( \dfrac{180\alpha}{\pi} \right)^{0}$ degrees.

**Example 2**

Convert the following angle measures from degrees to radians and from radians to degrees:

a) ${{10}^{0}}.$ b) $\dfrac{\pi }{8}.$ c) $-{{135}^{0}}.$ d) $-\dfrac{15\pi }{6}.$

**Example 3.**

Complete the conversion table for the angle measures:

Degree Measure | ${{30}^{0}}$ | ${{60}^{0}}$ | ${{180}^{0}}$ | ${{360}^{0}}$ | |||

Radian Measure | $\dfrac{\pi }{4}$ | $\dfrac{\pi }{2}$ | $\dfrac{3\pi }{2}$ |

**3. Unit Circle**

** **

In the coordinate plane $Oxy$, there is a circle centered at $O$ with a radius of 1. On this circle, choose the point $A(1, 0)$ as the origin, with counterclockwise direction as the positive direction and clockwise direction as the negative direction. The circle with center $O$ and orientation as described is called the *unit circle.*

For any angle measure $\alpha$ in radians, there exists a unique point $M$ on the unit circle such that $\left( OA, OM \right) = \alpha$. In this case, point $M$ is called the ** representation point** of the angle with a measure of $\alpha$ on the unit circle.

**Example 4.**

On the unit circle, determine point $M$ representing the following angle measures:

a) ${{0}^{0}}.$ b) $\dfrac{\pi }{2};$ c) $-\pi ;$ d) $-{{270}^{0}}.$

e) $\dfrac{2\pi }{3};$ f) $-\dfrac{11\pi }{4};$ g) $-{{150}^{0}};$ h) ${{585}^{0}}.$

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