how will Fiyinfoluwa know if an angle is a cosine angle or a sine angle? give more clearer insights using quadrant determination

Answer 1

To determine whether an angle is associated with the sine or cosine function, Fiyinfoluwa should understand the relationship between angles, their corresponding sides in a right triangle, and how these relate to the four quadrants of the Cartesian coordinate system.

Quadrants and Their Properties

1. First Quadrant (0° to 90°):

   • Both sine and cosine are positive.
   • For an angle \( \theta \):
     • \( \sin \theta > 0 \)
     • \( \cos \theta > 0 \)
   • Example angle: 30°, 45°, 60°

2. Second Quadrant (90° to 180°):

   • Sine is positive, but cosine is negative.
   • For an angle \( \theta \):
     • \( \sin \theta > 0 \)
     • \( \cos \theta < 0 \)
   • Example angle: 120°, 150°

3. Third Quadrant (180° to 270°):

   • Both sine and cosine are negative.
   • For an angle \( \theta \):
     • \( \sin \theta < 0 \)
     • \( \cos \theta < 0 \)
   • Example angle: 210°, 240°

4. Fourth Quadrant (270° to 360°):

   • Cosine is positive, but sine is negative.
   • For an angle \( \theta \):
     • \( \sin \theta < 0 \)
     • \( \cos \theta > 0 \)
   • Example angle: 300°, 330°

Determining If an Angle is Associated With Sine or Cosine

1. Identify the Quadrant:

   • Use the given angle to determine which quadrant it lies in by comparing it to the standard angle measures (0°, 90°, 180°, 270°, and 360°).

2. Apply the Sign Rules:

   • Once you know the quadrant, apply the sign rules of sine and cosine for that quadrant:
     • First Quadrant: Both functions are positive.
     • Second Quadrant: Sine is positive; cosine is negative.
     • Third Quadrant: Both are negative.
     • Fourth Quadrant: Sine is negative; cosine is positive.

3. Reference Angles:

   • Remember that angles can be expressed as reference angles, which are the acute angles formed by the terminal side of the given angle and the x-axis. The sine and cosine of the reference angle can help you understand the functions' values and which is sine and cosine.

Example

• For an angle of 130°:
  1. Quadrant: 130° is in the Second Quadrant.
  2. Signs: Here, sine is positive, and cosine is negative.
  3. Conclusion: Therefore, for 130°, sine is the value you would primarily consider positive, while cosine would be negative.

By following these steps, Fiyinfoluwa can easily determine if an angle involves sine or cosine based on the quadrant in which the angle lies.