MCR3U1
Date: __________________
Angles in Standard Position
Learning Goals:
1. Able to define what an angle in standard position is.
2. Able to determine the direction of rotation for positive and negative angles.
3. Able to express the trig. ratios in terms of x, y, and r for angles in standard position.
4. Able to determine a coterminal angle, principal angle, and related acute angle of an
angle in standard position.
-----------------------------------------------------------------------------------------------------------The measure of angle θ (theta) is the amount of rotation about
the vertex required to move the initial arm onto the terminal arm.
An angle is in standard position if it is drawn in
the x-y plane so that its vertex is at the origin and
the initial arm is on the positive x-axis.
If the initial arm is rotated counterclockwise
the angle is considered positive, whereas if
the initial arm is rotated clockwise the angle
is considered negative.
Let θ be an angle in standard position and (x, y)
be a point on its terminal arm, a distance r from
the origin O. We define the following:
CAST Rule
Angles in standard position are coterminal if their terminal arms are the same.
150o and -210o are coterminal angles
Example 1
Give the measure of a positive and negative angle that are coterminal to 60o.
Since coterminal angles have the same terminal arm, then their corresponding
trigonometric ratios are equal.
∴ cos 60o = cos (-300o) = cos 420o and so on since 60o, -300o, and 420o are coterminal
angles.
The smallest positive angle for all angles in a group of coterminal angles is called the
principal angle. For the above example, 60o is the principal angle.
The positive acute angle between the terminal arm of an angle in standard position and
the x-axis is called the related acute angle.
The principal angle is
__ .
__ is the related acute angle for 150o and -210o.
Example 2
Determine the principal angle and the related acute angle for -135o.
Example 3
(6, 8) is a point on the terminal arm of angle β (beta)
in standard position as shown.
a) Find the values of the six trigonometric ratios.
b) Find the measure of angle β to the nearest degree .
Example 4
(-7, 5) is on the terminal arm of an angle in standard position.
a) Sketch the principal angle, θ.
b) Find the values of the primary trigonometric ratios.
c) Determine the value of the related acute angle to the nearest degree.
d) What is the measure of θ to the nearest degree?