How to Draw an Angle in Standard Position
How does someone draw an angle in a standard position? A protractor will be needed for this.
Protractor

To draw an angle in standard position, follow these steps:
 Start by placing the baseline of a protractor on a Cartesian plane with the vertex at the origin.
 Trace the first or initial side of the angle along the horizontal xaxis to the right.
 Find the desired degree measurement on the protractor, and mark that spot.
 Draw the second side, or terminal side, of the angle extending from the vertex to the marked point.
Depending on the size of the angle, the terminal side can be in any four quadrants of the Cartesian plane. A 360 degrees rotation makes a full circle. An angle whose terminal side lies along one axis is known as a quadrantal angle. A quadrantal angle whose terminal side lies straight up along the yaxis, for example, is a 90degree angle. A quadrantal angle whose terminal side lies along the xaxis extending left of the origin is a 180degree angle.
Angles in standard position can be considered positive or negative. A positive angle in a standard position is measured in a counterclockwise direction from the initial side to the terminal side. A negative angle in standard position is measured in a clockwise direction from the initial side to the terminal side.
Positive and negative 45 degree angles

Coterminal and Reference Angles
Coterminal angles are standard positions with the same terminal side. The angles shown in the diagram, +135 and 225, are coterminal.
Coterminal angles

A reference angle is the smallest angle that can be drawn between the terminal side of an angle and the xaxis. The diagram shows a 135degree angle and its 60degree reference angle.
Reference angle

Reference angles can be used to find the trigonometric functions sine and cosine of the original angle, based on what quadrant the terminal side of the original angle lies in.
Examples of Drawing Angles in Standard Position
When drawing angles in standard position, follow the steps:
 Start by placing the baseline of a protractor on a Cartesian plane with the vertex at the origin.
 Trace the first or initial side of the angle along the horizontal xaxis to the right.
 Find the desired degree measurement on the protractor, and mark that spot.
 Draw the second side, or terminal side, of the angle extending from the vertex to the marked point.
Here are some examples of angles drawn in standard position.
Example 1
Draw a 90degree angle in a standard position.
90 degree angle in standard position

It is a positive angle, so it is measured in a counterclockwise direction from the initial side to the terminal side. The terminal side of this angle lies along the yaxis. It is a quadrantal angle.
Example 2
Draw a 210degree angle in a standard position.
210 degree angle in standard position

It is a positive angle, so it is measured in a counterclockwise direction from the initial side to the terminal side. The terminal side of this angle lies in Quadrant III.
Example 3
Draw a 45 degree angle in standard position.
45 degree angle in standard position

It is a negative angle, so it is measured in a clockwise direction from the initial side to the terminal side. The terminal side of this angle lies in Quadrant IV.
Measuring Angles in Standard Position with Degrees vs. Radians
Angles can be measured in degrees or radians. A radian is a unit of measure for angles based on the fact that the ratio of the circumference of a circle to the diameter of the circle is equal to the constant pi, or {eq}\pi {/eq}
Using Radians
An angle in standard form using radians is based on a unit circle, a circle with a center at the origin and a radius of 1. The diameter of a unit circle is twice the radius, or 2, so its circumference is {eq}2\pi {/eq}. Angle measurements in radians are based on the circumference of the unit circle, {eq}2\pi {/eq}.
 A full circle, or 360 degrees, is equal to {eq}2\pi {/eq} radians, which means
 180 degrees equals {eq}\pi {/eq} radians, and
 90 degrees equals {eq}\frac{\pi}{2} {/eq} radians.
Converting Between Radians and Degrees
A circle can be measured in degrees, 360 degrees, or radians, {eq}2\pi {/eq} radians. This equality can be used as a conversion factor to convert from degrees to radians or from radians to degrees.
Converting from degrees to radians
Convert 135 degrees to radians. Use the conversion factor 360 degrees {eq}=\ 2\pi {/eq} radians. Multiply the number of degrees times the conversion factor {eq}\frac{2\pi\ radians}{360\ degrees} {/eq}. The unit degrees will cancel, leaving the unit radians.
{eq}135\ degrees\ \times\ \frac{2\pi\ radians}{360\ degrees}\ =\ \frac{3\pi}{4}\ radians {/eq}
Converting from radians to degrees
Convert {eq}\frac{3\pi}{2} {/eq} radians to degrees. Use the conversion factor 360 degrees {eq}=\ 2\pi {/eq} radians. Multiply the number of radians times the conversion factor {eq}\frac{360\ degrees}{2\pi\ radians} {/eq}. The unit radians will cancel, leaving the unit degrees.
{eq}\frac{3\pi}{2}\ radians\ \times\ \frac{360\ degrees}{2\pi\ radians}\ =\ 270\ degrees {/eq}
Note: These examples did not use the measurement of the radius of the circle. It is irrelevant in these conversions because radians are a ratio, not a set length measurement. However, the radius can be used to find the length of an arc on the circle. The length of an arc on a circle is equal to its radius times the measurement, in radians, of the angle it creates.
Significance of Angles in Standard Position
What is significant about an angle in a standard position? In mathematics, angles in standard position are used in geometry, trigonometry, and calculus. There are also realworld uses of angles in standard positions. When measuring motion on a circular path, the calculation being performed is angular speed rather than linear speed. The angular speed of anything from a ceiling fan to an airplane propeller can be calculated using an angle measurement, {eq}\theta {/eq}, based on standard position.
Lesson Summary
Angular geometry refers to the size of an angle rotated around a circle from a starting point. That starting point is called standard position. On the Cartesian plane, an angle in standard position has its vertex at the origin, and the first side extends to the right along the xaxis. That first side, which lies between Quadrants I and IV on the Cartesian plane, is called the initial side. The other side of the angle is called the terminal side. An angle in a standard position whose terminal side lies along one of the axes is known as a quadrantal angle. Angles in standard position can be considered positive or negative. A positive angle in a standard position is measured in a counterclockwise direction from the initial side to the terminal side. A negative angle in standard position is an angle measured in a clockwise direction from the initial side to the terminal side.
Coterminal angles are angles in standard position with the same terminal side. A reference angle is the smallest angle that can be drawn between the terminal side of an angle and the xaxis. Angles can be measured in degrees or radians. A radian is a unit of measure for angles based on the ratio of the circumference to the diameter of the circle, which is equal to the constant pi, or {eq}\pi {/eq} A circle can be measured in degrees, 360 degrees, or radians, {eq}2\pi {/eq} radians. This equality can be used as a conversion factor to convert from degrees to radians or from radians to degrees.