Trigonometry Through Wayfinding and Navigation Across the Pacific

Let $t$ be a real number. Recall from Definition 1.2.19 that the radian measure of a central angle, $t\text{,}$ is defined as the ratio of the arc length $s$ to the radius $r\text{.}$ In other words, $t=\frac{s}{r}\text{.}$ In the unit circle, the radius is $r=1\text{,}$ and the angle in radians is equal to the arc length, $t=s\text{.}$ We will let $t$ be in radians. The circumference of the unit circle is $2\pi r=2\pi \cdot 1=2\pi \text{.}$

If $t\ge 0\text{,}$ we can imagine wrapping a line segment around the unit circle, marking off a distance of $t$ in the counterclockwise direction, and labeling that endpoint $P(x,y)\text{,}$ which becomes the terminal point. If $t<0\text{,}$ we would wrap in the clockwise direction.

If $t>2\pi$ or $t<-2\pi \text{,}$ then the length is longer than the circumference of the unit circle and we will need to travel around the unit circle more than once before arriving at the point $P(x,y)\text{.}$ Therefore, we can conclude that regardless of the value of $t\text{,}$ we have a unique point $P(x,y)$ that lies on the unit circle. We call $P(x,y)$ the point on the unit circle that corresponds to $t$.

The $x$- and $y$-coordinates for $P(x,y)$ can then be used to define the six trigonometric functions of a real number $t\text{:}$

which are abbreviated as sin, cos, tan, csc, sec, and cot, respectively.

We now summarize what we know about the six trigonometric functions for special angles. Note the trigonometric functions for $\theta =\frac{\pi }{2}$ and $\theta =\frac{\pi }{3}$ are left as exercises.

This symmetry within the unit circle resembles the pattern observed in the Star Compass. When a star emerges in the eastern sky, it will eventually descend and set in the corresponding house of the western sky. For instance, if a star rises above the horizon in the Nālani house of the Ko‘olau quadrant (northeast), it will journey across the sky and set in the equivalent house within the Ho‘olua quadrant (northwest). This similarity aligns with the symmetry between points $P(x,y)$ and $Q(-x,y)\text{.}$ Additionally, if an ocean swell or wind originates from the Nālani house in the Malanai quadrant (southeast), it will pass the wa‘a and exit in the opposite direction toward the Ho‘olua quadrant (northwest), still within the Nālani house. This mirrors the symmetry between points $S(x,-y)$ and $Q(-x,y)\text{.}$

A fourth form of symmetry involves reflecting points across the diagonal line $y=x\text{,}$ where the $x$- and $y$-values are equal.

$T(y,x)\text{:}$ Symmetry about the line $y=x\text{.}$ This is accomplished by interchanging the $x$- and $y$-values.

Notice on the Unit Circle that the radius extending from the center at an angle of ${30}^{\circ }$ to the point $T(x,y)=(\frac{\sqrt{3}}{2},\frac{1}{2})$ is symmetric about the line $y=x\text{,}$ in relation to the radius extending from the center at an angle of ${60}^{\circ }$ to the point $P(x,y)=(\frac{1}{2},\frac{\sqrt{3}}{2})\text{.}$

Using symmetry about the $x$-axis, symmetry about the $y$-axis, and symmetry about the origin, we can complete the unit circle, as long as we remember that the $x$-values in Quadrants II and III are negative while the $y$-values in Quadrants III and IV are negative.

Finally, we tie everything together and look at the entire Unit Circle. At first glance, it may seem intimidating; however, similar to the Star Compass, there is a lot of symmetry ($x$-axis, $y$-axis, origin, and the line $y=x$). A helpful approach is to focus on one quadrant and use symmetry to complete the rest of the circle.

Until now, computing the exact values of trigonometric functions of an angle $\theta$ required us to locate the corresponding point $P(x,y)$ on the unit circle. However, we can use any circle with center at the origin, that is, any circle of the form $x^2+y^2=r^2\text{,}$ where $r>0$ is the radius. Note that if $r=1\text{,}$ then the cirlce is the unit circle.