First quadrant

Definition

The first quadrant refers to the area of the Cartesian coordinate plane where both the x-coordinates and y-coordinates are positive. In the context of the unit circle, this is where angles between 0° and 90° are located, and it plays a crucial role in defining the sine and cosine values of these angles as they relate to circular functions.

5 Must Know Facts For Your Next Test

1. In the first quadrant, both sine and cosine values are positive, which means that all trigonometric functions yield positive results.
2. The angles in the first quadrant are typically measured from the positive x-axis, moving counterclockwise.
3. Key angles in the first quadrant include 0°, 30°, 45°, 60°, and 90°, each corresponding to specific values for sine, cosine, and tangent.
4. The first quadrant is crucial for determining reference angles for any given angle, especially when considering angles in other quadrants.
5. Understanding the first quadrant is essential for mastering concepts like the Pythagorean identity, which relates sine and cosine functions.

Review Questions

• How do the sine and cosine values differ between the first quadrant and other quadrants?
  • In the first quadrant, both sine and cosine values are positive, resulting in all trigonometric functions also being positive. However, in the second quadrant, sine remains positive while cosine becomes negative. In the third quadrant, both sine and cosine values are negative, while in the fourth quadrant, cosine is positive but sine is negative. Understanding these differences is key to solving problems related to trigonometric functions across different quadrants.
• What are some key angles in the first quadrant and their corresponding sine and cosine values?
  • Key angles in the first quadrant include 0° (cos = 1, sin = 0), 30° (cos = √3/2, sin = 1/2), 45° (cos = √2/2, sin = √2/2), 60° (cos = 1/2, sin = √3/2), and 90° (cos = 0, sin = 1). These angles serve as foundational values for understanding trigonometric relationships and enable students to calculate values for angles in other quadrants using reference angles.
• Evaluate how knowledge of the first quadrant aids in solving real-world problems involving circular functions.
  • Knowledge of the first quadrant is essential for applying circular functions to real-world problems, such as those involving oscillatory motion or wave patterns. By understanding how angles correspond to positive sine and cosine values in this quadrant, one can accurately model phenomena like sound waves or harmonic motion. This foundational understanding allows for accurate predictions and calculations in various fields including physics, engineering, and even economics where cyclical trends may be analyzed.

Related terms

Unit Circle: A circle with a radius of one centered at the origin of the coordinate plane, used to define trigonometric functions based on angles.

Reference Angle: The acute angle formed by the terminal side of an angle and the x-axis, which helps determine the value of trigonometric functions in different quadrants.

Trigonometric Functions: Functions such as sine, cosine, and tangent that relate angles to side lengths in right triangles and are defined using the unit circle.