Mathematics
Grade 11
15 min
Quadrants
Quadrants
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1
Introduction & Learning Objectives
Learning Objectives
Identify the four quadrants and their corresponding angle ranges in both degrees and radians.
Determine the specific quadrant in which the terminal side of any given angle in standard position lies.
Distinguish between positive (counter-clockwise) and negative (clockwise) angles.
Calculate the principal coterminal angle for any angle greater than 360° or less than 0°.
Classify an angle as quadrantal if its terminal side lies on an axis.
Apply the ASTC rule to determine the sign (positive or negative) of sine, cosine, and tangent functions in each quadrant.
How does a video game character know which way to turn? 🎮 Understanding quadrants is the first step in programming any kind of rotation or direction!
This tutorial moves beyond basic angles to explore...
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Key Concepts & Vocabulary
TermDefinitionExample
Standard PositionAn angle is in standard position when its vertex is at the origin (0,0) of the Cartesian plane and its initial side lies along the positive x-axis.An angle of 120° drawn with its starting point at the origin and its initial side on the positive x-axis is in standard position. Its terminal side will be in Quadrant II.
QuadrantsThe four regions into which the x-axis and y-axis divide the Cartesian plane. They are numbered with Roman numerals (I, II, III, IV) in a counter-clockwise direction, starting from the top right.Quadrant I: x > 0, y > 0. Quadrant II: x < 0, y > 0. Quadrant III: x < 0, y < 0. Quadrant IV: x > 0, y < 0.
Terminal SideThe ray of an angle in standard position that indicates the end of the rotation. The locatio...
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Core Formulas
Quadrant Angle Ranges (Degrees)
Q I: 0° < θ < 90° | Q II: 90° < θ < 180° | Q III: 180° < θ < 270° | Q IV: 270° < θ < 360°
Use these inequalities to determine the quadrant of an angle between 0° and 360°.
Quadrant Angle Ranges (Radians)
Q I: 0 < θ < π/2 | Q II: π/2 < θ < π | Q III: π < θ < 3π/2 | Q IV: 3π/2 < θ < 2π
Use these inequalities to determine the quadrant of an angle between 0 and 2π radians.
Coterminal Angle Formulas
Degrees: θ_c = θ + n(360°) | Radians: θ_c = θ + n(2π), where n is any integer (..., -2, -1, 0, 1, 2, ...)
Use these formulas to find angles that are equivalent to a given angle θ by adding or subtracting full rotations.
ASTC Rule (Trigonometric Signs)
Q I: All positive. Q II: Sine positive. Q...
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Challenging
An angle θ in standard position has its terminal side passing through the point (-5, 12). Without calculating the angle, determine the signs of sin(θ), cos(θ), and tan(θ).
A.sin(θ)>0, cos(θ)>0, tan(θ)>0
B.sin(θ)>0, cos(θ)<0, tan(θ)<0
C.sin(θ)<0, cos(θ)<0, tan(θ)>0
D.sin(θ)<0, cos(θ)>0, tan(θ)<0
Challenging
If the terminal side of an angle θ lies in Quadrant IV, which of the following expressions MUST be negative?
A.cos(θ) - sin(θ)
B.cos(θ) * tan(θ)
C.sin(θ) + tan(θ)
D.1 / cos(θ)
Challenging
An angle α is coterminal with 10π/3. An angle β is coterminal with -500°. In which quadrants do the terminal sides of α and β lie, respectively?
A.Quadrant III and Quadrant II
B.Quadrant II and Quadrant III
C.Quadrant III and Quadrant IV
D.Quadrant IV and Quadrant I
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