quadrants

Examples of quadrants in the following topics:

• Special Angles

  • In quadrant II, “Smart,” only sine is positive.
  • Reference angles in quadrant I are used to identify which value any angle in quadrants II, III, or IV will take.
  • For any given angle in the first quadrant, there is an angle in the second quadrant with the same sine value.
  • For any angle in quadrants II, III, or IV, there is a reference angle in quadrant I.
  • For any angle in quadrants II, III, or IV, there is a reference angle in quadrant I.

• The Cartesian System

  • The third quadrant has both negative x and y coordinates.
  • Point $(4,0)$ is on the x-axis and not in a quadrant.  
  • Point $(0,-2)$ is on the y-axis and also not in a quadrant.
  • The four quadrants of a Cartesian coordinate system.
  • The four quadrants of a Cartesian coordinate system.

• Defining Trigonometric Functions on the Unit Circle

  • The x- and y-axes divide the coordinate plane (and the unit circle, since it is centered at the origin) into four quarters called quadrants.
  • We label these quadrants to mimic the direction a positive angle would sweep.
  • The four quadrants are labeled I, II, III, and IV.

• Sine and Cosine as Functions

  • Notice how the sine values are positive between $0$ and $\pi$, which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between $\pi$ and $2\pi$, which correspond to the values of the sine function in quadrants III and IV on the unit circle.

• Double and Half Angle Formulae

  • Recall that different signs are applied to trigonometric functions that fall in each of the four quadrants (according to the mnemonic rule "A Smart Trig Class").
  • Notice that we used only the positive root because $15^{\circ}$$$ falls in the first quadrant and $\sin(15^{\circ})$ is therefore positive.

• Pythagorean Identities

  • If we know the quadrant where the angle is, we can easily choose the correct solution.