Angle in Standard Position

An angle is in standard position when it is drawn on the Cartesian plane so that its vertex sits exactly at the origin, \((0,0)\), and one of its sides, called the initial side, lies along the positive x-axis. The other side of the angle, called the terminal side, is free to rotate around the origin. Where that terminal side ends up tells you everything about the angle's size, sign, and quadrant.

Placing every angle the same way, vertex at the origin and initial side on the positive x-axis, gives us a consistent starting point for measuring rotation. This is what makes it possible to compare angles, find their trig ratios, and locate coterminal angles quickly.

Three parts make up every angle in standard position:

• Vertex: always the origin, \((0,0)\).
• Initial side: the ray that stays fixed along the positive x-axis, marking a rotation of \(0^\circ\).
• Terminal side: the ray that rotates away from the initial side to form the angle \(\theta\).

The figure below shows an angle \(\theta\) of about \(120^\circ\) in standard position. The initial side runs along the positive x-axis, and the terminal side has rotated counterclockwise into the second quadrant.

The direction the terminal side rotates gives the angle its sign:

• Rotating counterclockwise from the initial side produces a positive angle.
• Rotating clockwise from the initial side produces a negative angle.

For example, \(\theta = 45^\circ\) and \(\theta = -315^\circ\) both end with the terminal side in the same place, but they were reached by rotating in opposite directions. This connects directly to the idea of coterminal angles below.

Two angles in standard position are coterminal if their terminal sides land in exactly the same spot, even though the amount of rotation is different. Because a full rotation is \(360^\circ\) (or \(2\pi\) radians), you can find a coterminal angle by adding or subtracting \(360^\circ\):

\(\theta_{coterminal} = \theta \pm 360^\circ \cdot n\), where \(n\) is any positive integer.

Example: Find a positive and a negative angle coterminal with \(\theta = 100^\circ\).

Adding \(360^\circ\): \(100^\circ + 360^\circ = 460^\circ\).
Subtracting \(360^\circ\): \(100^\circ - 360^\circ = -260^\circ\).

So \(460^\circ\) and \(-260^\circ\) are both coterminal with \(100^\circ\), since all three share the same terminal side.

Once you know where the terminal side lands, you can tell which quadrant the angle belongs to just by comparing it to the axis boundaries at \(90^\circ\), \(180^\circ\), \(270^\circ\), and \(360^\circ\).

For instance, an angle of \(200^\circ\) lies between \(180^\circ\) and \(270^\circ\), so its terminal side is in Quadrant III. This matters because the sign of each trig ratio (sine, cosine, tangent) depends entirely on the quadrant, which is exactly what the ASTC rule organizes for you.

Standard position works the same way whether the angle is measured in degrees or in radians, the vertex is still at the origin and the initial side is still along the positive x-axis. A full rotation is \(2\pi\) radians instead of \(360^\circ\), so a quadrantal boundary like \(90^\circ\) becomes \(\frac{\pi}{2}\), and \(180^\circ\) becomes \(\pi\). If a problem gives you an angle in one unit and you need the other, you can review how to convert between degrees and radians before placing the angle.

Problem: Sketch \(\theta = -150^\circ\) in standard position, name its quadrant, and give one positive coterminal angle.

Step 1: Since \(\theta\) is negative, rotate the terminal side clockwise from the positive x-axis.

Step 2: A clockwise rotation of \(150^\circ\) passes through the fourth, third quadrants and stops partway into Quadrant III, since \(-150^\circ\) corresponds to \(360^\circ - 150^\circ = 210^\circ\) measured counterclockwise.

Step 3: Because \(210^\circ\) is between \(180^\circ\) and \(270^\circ\), the terminal side lies in Quadrant III.

Step 4: A positive coterminal angle is \(-150^\circ + 360^\circ = 210^\circ\).

Once the terminal side is placed correctly, the angle's exact position often needs to be compared to the nearest x-axis, which is the idea behind the reference angle, and this quadrant and sign information is also what feeds directly into the sine and cosine relationships used in the law of cosines and the sine law when solving triangles.

• Forgetting that the initial side must be on the positive x-axis, not just any axis.
• Mixing up rotation direction: counterclockwise is positive, clockwise is negative.
• Assuming coterminal angles must be less than \(360^\circ\), when in fact you can add or subtract \(360^\circ\) as many times as needed.
• Placing a radian angle as if it were in degrees, or vice versa, without converting first.

Related lessons

• Trigonometry Help