Understand coterminal and standard position angles. Math 20-1 Alberta Grade 11.
Lesson 4.2 of Trigonometry in Math 20-1 — Alberta curriculum lessons.
Definition — Standard Position. An angle is in standard position when three conditions are met:
1. The vertex is at the origin ( 0, 0 ).
2. The initial arm lies along the positive x-axis.
3. The terminal arm is rotated to its final position.
The angle θ is the rotation measured from the initial arm to the terminal arm.
Positive vs. Negative Angles. Positive angle — rotated counter-clockwise from the initial arm.
Negative angle — rotated clockwise from the initial arm.
Memory Tip. Counter-clockwise = positive. Think of a clock running backwards — that is the "positive" direction in mathematics.
Quadrants and Sign Patterns. The x-axis and y-axis divide the coordinate plane into four quadrants. Each quadrant has a specific range of angles and a specific sign pattern for coordinates.
Definition — Principal Angle. The principal angle θ is the angle between the initial arm and the terminal arm, measured from 0° to 360°.
If you are given an angle that is negative or larger than 360°, find the equivalent principal angle by adding or subtracting multiples of 360° until the result is between 0° and 360°.
Definition — Reference Angle. A reference angle (written _R) is the smallest angle between the terminal arm and the closest x-axis.
Three important properties:
1. A reference angle is always positive.
2. A reference angle is always between 0° and 90° (an acute angle).
3. The reference angle uses the closest x-axis — never the y-axis.
Why Are Reference Angles Useful?. Every angle in standard position has a reference angle that is acute (between 0° and 90°). Acute angles are exactly the angles we already know how to handle from right-triangle trigonometry. Reference angles connect angles of any size back to familiar territory.
Example 1: Label the Quadrants and Axes. The four quadrants are numbered counter-clockwise starting from the top-right.
Axis Angles. The axis lines correspond to specific angles:
0° (or 360°): positive x-axis
90°: positive y-axis
180°: negative x-axis
270°: negative y-axis
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