Angles where Sine is Less than Cosine

In this article, we will explore the specific angles where the sine function is less than the cosine function. This involves an in-depth analysis of the trigonometric functions sine and cosine, and understanding the behavior of these functions over different intervals.

Basic Understanding

The sine function is less than the cosine function when the following inequality holds true:

(sin x

This can be rewritten as:

(tan x

by dividing both sides by cosine, assuming that the cosine function is not zero. This is a crucial step in simplifying the inequality and understanding its intervals.

Finding the Angles

The tangent function is less than 1 in certain intervals where the sine function falls below the cosine function. Specifically, these intervals are:

From (0) to (frac{π}{4}) From (frac{5π}{4}) to (frac{3π}{2})

To generalize these results, the general solutions for the angles (x) where (sin x (x in left[2kπ, 2kπ frac{π}{4}right)) (x in left(2kπ frac{5π}{4}, 2kπ frac{3π}{2}right])

Here, (k) is any integer.

General Solution

Combining these intervals, we can summarize that sine is less than cosine in the following intervals:

From (0) to (45°) From (225°) to (270°)

This pattern repeats every (2π) radians (or 360°), meaning the same behavior occurs in each full rotation of the unit circle.

Additional Insights

Another way to understand these angles is by visualizing the unit circle. The sine and cosine functions represent the (y) and (x) coordinates of a point on the unit circle, respectively. When the sine value is less than the cosine value, the point lies on the lower half of the circle or in the fourth quadrant.

The line (y x) is a diagonal line that bisects the first and third quadrants. On the unit circle, the intersection points of this line with the circle ((45°) and (225°) or (5pi/4)) are where sine equals cosine. On the lower half of the circle, sine is less than cosine, as the y-coordinate is less than the x-coordinate.

This alternative approach further reinforces the intervals where (sin x

Conclusion

The sine function is less than the cosine function in the intervals specified above. By understanding these intervals, we can accurately determine the angles in which this relationship holds. The general patterns and periodicity of these functions make it easy to calculate sine values less than cosine values for any given angle.

Keywords: sine, cosine, trigonometric functions