The value of cosine is zero at specific, predictable points along the unit circle, a fundamental concept for anyone studying trigonometry. This condition occurs precisely when the terminal side of an angle intersects the unit circle on the y-axis, where the x-coordinate, by definition, is zero. Understanding this scenario requires looking at both degrees and radians, as well as the periodic nature of the function.
The Core Principle: The Unit Circle
To determine when cosine equals zero, one must visualize the unit circle, a circle with a radius of one centered at the origin of a coordinate plane. The cosine of an angle corresponds to the x-coordinate of the point where the terminal ray of that angle intersects the circle. Therefore, the question "when is cosine 0" directly translates to identifying the angles where this intersection point lies exactly on the vertical axis.
Angles in Degrees
Examining the unit circle in degrees provides an intuitive entry point. The cosine value represents horizontal displacement. Starting at 0 degrees, the point is at (1, 0). As the angle increases to 90 degrees, the point moves upward along the circle to (0, 1). At this exact position, the x-coordinate is zero, making cosine of 90 degrees equal to 0. Continuing the rotation, the point reaches 270 degrees at (0, -1), where the x-coordinate is again zero. Thus, the primary angles between 0 and 360 degrees where cosine is zero are 90° and 270°.
Periodicity and the General Solution
A trigonometric function like cosine is periodic, meaning it repeats its values in a predictable cycle. The period of cosine is 360 degrees. Because of this repetition, the solutions are not limited to just 90° and 270°. The pattern continues infinitely in both directions. The general solution for angles in degrees is expressed as:
θ = 90° + n × 360°
θ = 270° + n × 360°
Here, n represents any integer (positive, negative, or zero). This formula captures every instance where the terminal side aligns with the y-axis.
Angles in Radians
Mathematically, radians are the standard unit of angular measure. One full rotation equals 2π radians. Converting the degree solutions provides the radian equivalents. 90 degrees is π/2 radians, and 270 degrees is 3π/2 radians. Following the same logic of periodicity, which is 2π radians, the general solutions in radians are:
θ = π/2 + 2nπ
θ = 3π/2 + 2nπ
These equations are essential for advanced calculus and physics applications.
Symmetry and Negative Angles
The unit circle also clarifies the behavior of cosine for negative angles, which represent clockwise rotation. The cosine function is even, meaning cos(θ) = cos(-θ). Applying this property, the angle -90° (or -π/2 radians) has the same cosine value as 90°. Therefore, -90° is also a point where cosine is zero, as the point lands at (0, 1). This symmetry confirms that the solutions occur at every odd multiple of π/2.
