Definition of trigonometric angle, trigonometric circle

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 angle with initial ray $Oa$ and terminal ray $Ob$, denoted as $\left( Oa, Ob \right)$.
  • 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 $s\tilde{n}\left( Oa, Ob \right) = \alpha$.


  1. 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.
  2. Counterclockwise rotation is considered positive, and clockwise rotation is considered negative.
  3. 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 $-360^\circ$.
  4. 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 $\left( Oa, Ob \right) = \alpha^\circ + k \cdot 360^\circ$.
  5. 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) ${{585}^{0}}.$