Understanding the Value of Cos θ When Sin ?theta is -1

The cosine and sine functions are fundamental concepts in trigonometry, playing a pivotal role in various mathematical and scientific applications. These functions are bijective on specific intervals, offering unique values for any given input. The following discussion delves into the values of cos θ when sin θ equals -1, and provides a comprehensive overview of the trigonometric circle and its implications.

Trigonometric Functions and Their Domains

The cosine and sine functions are defined on the interval [0, π] with ranges of [-1, 1]. This means that for any angle θ in [0, π], the cosine function can take any value in [-1, 1], and similarly for the sine function. Conversely, any real number y in [-1, 1] corresponds to a unique angle θ in [0, π] such that cos θ y and sin θ y, respectively.

Example: Cos π -1

A specific case to consider is when sin θ -1. Given that the sine function ranges from -1 to 1, the only angle θ in the interval [0, π] for which sin θ -1 is θ π. Thus, cos π -1, and this is the unique angle in this interval that yields -1 as the cosine value.

Visualizing the Trigonometric Circle

To better understand these concepts, one can visualize the trigonometric circle, a circle of radius 1 centered at the origin (0,0) on the Cartesian plane. The circle is divided into four quadrants, each with a point that corresponds to the terminal side of an angle θ:

A (1, 0) - corresponds to 0 and 0° (East) B (0, 1) - corresponds to π/2 or 90° (North) A' (-1, 0) - corresponds to π or 180° (West) B' (0, -1) - corresponds to 3π/2 or 270° (South)

As the angle θ increases from 0 to π, the point M moves from A to B, and the abscissa (x-coordinate) of the point where OM intersects the unit circle changes from 1 to 0, and then to -1. This corresponds to the behavior of the cosine function, which decreases monotonically from 1 to 0 and then to -1 as θ progresses from 0 to π.

Scalar and Vector Projections

The cosine of an angle θ can also be interpreted as the scalar projection of a vector OM onto the unit vector along the x-axis. Specifically, cos θ is the scalar projection of the vector OM onto the unit vector i:

cos θ pri OM

This projection can be calculated as follows:

pri OM OM cos θ, where OM 1 (unit vector).

Therefore, cos θ OM cos θ pri OM, and the abscissa of the point N Proj_Ox M (the orthogonal projection of M onto the x-axis).

Periodicity of Trigonometric Functions

Both sine and cosine functions are periodic with a period of 2π. This means that for any angle θ:

cos(θ 2kπ) cos θ and sin(θ 2kπ) sin θ for all k ∈ ? (the set of all integers).

This periodicity is a key property of these functions, as illustrated in their graphs, which can be found in textbooks or manuals of calculus and trigonometry.

Conclusion

The provided information summarizes the key properties of the cosine function when sin θ equals -1. By understanding the trigonometric circle and periodicity, one can easily identify the unique angle θ for which cos θ -1. This detailed explanation illustrates the intricate relationship between trigonometry and vector algebra, highlighting the significance of these concepts in both theoretical and applied contexts.