☰ Year plan · Mathematics

Directed angles and radians

Week 11 · Directed angles, equivalent angles, and degree and radian measure · 3 lessons
1

The directed angle: initial and terminal sides, the standard position, positive and negative measure and the four quadrants

Mathematics Sec 1 Term 1, Unit 4, Lessons 1–2
GoalDraw a directed angle in standard position and say which quadrant it is in.
New words
initial side · the fixed ray where the angle starts; in standard position it is the positive x-axisterminal side · the ray where the rotation stopsstandard position · vertex at the origin and initial side along the positive x-axis
Explanation

A directed angle is made by turning a ray about its end point from the initial side to the terminal side. If the turn is anticlockwise the measure is positive; if it is clockwise the measure is negative. In standard position the vertex is at the origin O and the initial side lies on the positive x-axis, so the angle is described by its terminal side alone. The axes split the plane into four quadrants, numbered anticlockwise from the top-right one. An angle whose terminal side lies on an axis (0°, 90°, 180°, 270°) belongs to no quadrant. A full turn is 360°, so an angle can be larger than 360° or smaller than −360°.

QuadrantThe angle θ (0° to 360°)The terminal side lies
First0° < θ < 90°up and right
Second90° < θ < 180°up and left
Third180° < θ < 270°down and left
Fourth270° < θ < 360°down and right
Worked examples
xy−3−2−1123−3−2−11230135°
Where is 135°? It is a turn anticlockwise, more than 90° and less than 180°, so the terminal side is in the second quadrant.
xy−3−2−1123−3−2−11230−60°
Where is −60°? It is a clockwise turn of 60° from the positive x-axis, so the terminal side is in the fourth quadrant (the same place as 300°).
Name the quadrant of 250° and of 400°. 250° is between 180° and 270°: third quadrant. 400° = 360° + 40°, a full turn and then 40°: first quadrant.
Class activity

Human clock. Stand facing a wall with your right arm pointing along the positive x-axis. Turn your arm anticlockwise by 90°, 180° and 270°, then turn clockwise by 90° and 45°. Say each time which quadrant (or axis) the arm points to.

Practice
1
In which quadrant is 200°?
2
In which quadrant is −120°?
3
In which quadrant is 530°?
4
Which of 0°, 90°, 180°, 270° and 45° belong to no quadrant?
5
Draw the angle −90° in standard position. Where does the terminal side point?