Sine-Cosine equivalence

The sine-cosine equivalence highlights a fundamental relationship between the sine and cosine trigonometric functions. This relationship underscores the phase shift property, where the cosine function of any angle (\( \htmlId{tooltip-angle}{\theta} \)) is equivalent to the sine function of that angle shifted by \(\frac{\htmlId{tooltip-pi}{\pi}}{2}\htmlId{tooltip-u-radians}{rad}\) (or \(90 \htmlId{tooltip-u-degrees}{\degree}\)). You can see this in the red arrow in the diagram below.

Essentially, this means that the cosine graph is the same as the sine graph, but it is shifted to the left by \(\frac{\htmlId{tooltip-pi}{\pi}}{2}\) radians on the unit circle, indicating a quarter of a circle ahead in terms of angle. This equivalence is pivotal in trigonometry, facilitating transformations and simplifications of trigonometric expressions and equations. Understanding this relationship enhances the comprehension of wave functions in physics and engineering, where sine and cosine functions model periodic phenomena, showing that these functions are essentially the same shape but differ in phase.

1. In the unit circle(see figure below), each point on the circle's circumference can be described by coordinates (\( \htmlId{tooltip-cosine}{\cos} \)(\( \htmlId{tooltip-angle}{\theta} \)),\( \htmlId{tooltip-sine}{\sin} \)(\( \htmlId{tooltip-angle}{\theta} \))), where \( \htmlId{tooltip-angle}{\theta} \) is the angle from the positive x-axis to the line connecting the origin to the point. The x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine of the angle.
2. When you add \(\frac{\htmlId{tooltip-pi}{\pi}}{2}\)​ (or 90 degrees) to an angle \( \htmlId{tooltip-angle}{\theta} \), you effectively rotate the point on the unit circle counterclockwise by a quarter turn. This rotation leads to a new position on the circle.
3. Before the addition of \(\frac{\htmlId{tooltip-pi}{\pi}}{2}\), the point (\( \htmlId{tooltip-cosine}{\cos} \)(\( \htmlId{tooltip-angle}{\theta} \)),\( \htmlId{tooltip-sine}{\sin} \)(\( \htmlId{tooltip-angle}{\theta} \))) represents the cosine and sine of \( \htmlId{tooltip-angle}{\theta} \), respectively. After adding \(\frac{\htmlId{tooltip-pi}{\pi}}{2}\) to \( \htmlId{tooltip-angle}{\theta} \), the original x-coordinate (cosine) of the point becomes the new y-coordinate, and the original y-coordinate (sine) becomes the negative of the new x-coordinate due to the 90-degree rotation. However, since we're interested in the positive shift, we focus on how the x-coordinate \( \htmlId{tooltip-cosine}{\cos} \)(\( \htmlId{tooltip-angle}{\theta} \))) aligns with what becomes the sine function value after the rotation.
4. After rotating the angle \( \htmlId{tooltip-angle}{\theta} \) by ​\(\frac{\htmlId{tooltip-pi}{\pi}}{2}\) radians, the point's new coordinates reflect the values of sine and cosine with the roles reversed. The new y-coordinate (which was the x-coordinate, or \( \htmlId{tooltip-cosine}{\cos} \)(\( \htmlId{tooltip-angle}{\theta} \)), before the rotation) now represents \( \htmlId{tooltip-sine}{\sin} \)(\( \htmlId{tooltip-angle}{\theta} \) + \(\frac{\htmlId{tooltip-pi}{\pi}}{2}\))
5. This geometric perspective shows that after rotating by \(\frac{\htmlId{tooltip-pi}{\pi}}{2}\)​, the cosine of the original angle (\( \htmlId{tooltip-cosine}{\cos} \)(\( \htmlId{tooltip-angle}{\theta} \))) directly corresponds to the sine of the angle plus \(\frac{\htmlId{tooltip-pi}{\pi}}{2}\)\( \htmlId{tooltip-sine}{\sin} \)(\( \htmlId{tooltip-angle}{\theta} \) + \(\frac{\htmlId{tooltip-pi}{\pi}}{2}\)). Therefore, we derive the equation \( \htmlId{tooltip-cosine}{\cos} \)(\( \htmlId{tooltip-angle}{\theta} \)) = \( \htmlId{tooltip-sine}{\sin} \)(\( \htmlId{tooltip-angle}{\theta} \) + \(\frac{\htmlId{tooltip-pi}{\pi}}{2}\)) as a natural consequence of the rotational symmetry in the unit circle.

Question:

Given the function \(f(x) = \htmlId{tooltip-cosine}{\cos}(x)\), use the sine-cosine equivalence to express \(f(x)\) as a sine function. Then, find the value of \(f(x)\) when \(x = \frac{\htmlId{tooltip-pi}{\pi}}{6}\)

Solution:

Using the sine-cosine equivalence \( \htmlId{tooltip-cosine}{\cos} \)(\( \htmlId{tooltip-angle}{\theta} \)) = \( \htmlId{tooltip-sine}{\sin} \)(\( \htmlId{tooltip-angle}{\theta} \) + \( \htmlId{tooltip-pi}{\pi} \)/2), we can express the given function \(f(x) = \htmlId{tooltip-cosine}{\cos}(x)\) in terms of sine as:

\(f(x) = \htmlId{tooltip-sine}{\sin}(x + \frac{\htmlId{tooltip-pi}{\pi}}{2}\))

To find the value of \(f(x)\) when \(x = \frac{\htmlId{tooltip-pi}{\pi}}{6}\)​, we substitute \(x = \frac{\htmlId{tooltip-pi}{\pi}}{6}\)​ into the sine expression:

\(f(\frac{\htmlId{tooltip-pi}{\pi}}{6}) = \htmlId{tooltip-sine}{\sin}(\frac{\htmlId{tooltip-pi}{\pi}}{6} + \frac{\htmlId{tooltip-pi}{\pi}}{2}) = \htmlId{tooltip-sine}{\sin}(\frac{\htmlId{tooltip-pi}{\pi}}{6} + \frac{3\htmlId{tooltip-pi}{\pi}}{6}) = \htmlId{tooltip-sine}{\sin}(\htmlId{tooltip-sine}{\sin}(\frac{4\htmlId{tooltip-pi}{\pi}}{6}) = \htmlId{tooltip-sine}{\sin}(\frac{2\htmlId{tooltip-pi}{\pi}}{3}) \)

The value of \(\htmlId{tooltip-sine}{\sin}(\frac{\htmlId{tooltip-pi}{\pi}}{2})\) can be found using the unit circle or trigonometric tables:

\(\htmlId{tooltip-sine}{\sin}(\frac{2\htmlId{tooltip-pi}{\pi}}{3}) = \frac{\sqrt(3)}{2})\)

Therefore, \(f(\frac{\htmlId{tooltip-pi}{\pi}}{6}) = \frac{\sqrt(3)}{2}\) .