In this discussion, you will discuss the relationships of key points on the unit circle.
For the key points shown in the four quadrants on the unit circle, identify a significant pattern or symmetry in the angles, the coordinates, the trigonometric functions, or a relationship between them.
Post a detailed response to the discussion prompt. Then comment on at least two other posts.
To understand how you will be graded for this assignment, read the Discussion Guidelines and Rubric.
In the four quadrants of the unit circle, there are key points that have significant patterns and symmetries. Let's explore these patterns and relationships.
First, let's identify the key points in each quadrant of the unit circle. In the first quadrant (Q1), the coordinates are positive for both x and y, and the angle is between 0 and 90 degrees. The key point in Q1 is (1, 0) or (cos(0), sin(0)), which corresponds to the angle 0 degrees.
Moving to the second quadrant (Q2), the x-coordinate is negative while the y-coordinate is positive. The angle in Q2 ranges from 90 to 180 degrees. The key point in Q2 is (-1, 0) or (cos(180), sin(180)), which corresponds to the angle 180 degrees.
In the third quadrant (Q3), both the x and y-coordinates are negative, and the angle ranges from 180 to 270 degrees. The key point in Q3 is (-1, 0) or (cos(180), sin(180)), which corresponds to the angle 180 degrees.
Finally, in the fourth quadrant (Q4), the x-coordinate is positive while the y-coordinate is negative. The angle in Q4 ranges from 270 to 360 degrees. The key point in Q4 is (1, 0) or (cos(0), sin(0)), which corresponds to the angle 0 degrees.
Now, let's examine the patterns and relationships between the key points and the trigonometric functions. First, notice that the x-coordinate corresponds to the cosine of the angle, while the y-coordinate corresponds to the sine of the angle. In all quadrants, the cosine and sine values are positive except for the key points in Q2 and Q3, where the x-coordinate is negative. This means that the cosine is negative in Q2 and Q3, while the sine remains positive.
Additionally, there is a symmetry in the unit circle. If we take any angle θ in Q1, the same angle θ exists in Q3 but with opposite trigonometric signs. For example, the cosine of θ in Q1 is the same as the negative cosine of θ in Q3, and the sine of θ in Q1 is the same as the negative sine of θ in Q3. This symmetry applies to all angles.
In summary, the key points in the four quadrants of the unit circle exhibit patterns and symmetries in the coordinates and trigonometric functions. The x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine of the angle. The cosine is positive in Q1 and Q4, while it is negative in Q2 and Q3. The sine is always positive except for in Q3, where it is negative. Additionally, there is symmetry between the angles and their trigonometric values across the quadrants.