Section 2.5
The Meaning of y in y = sin(x)

You now know the meaning of x in $y = \sin(x)$. It is an arc length on a circle measured in units of the circle’s radius. But what is the meaning of y? First, some background.

A chord is a line segment with endpoints on a circle. Early Greeks were interested in chords because they used inscribed polygons (each of the polygon’s sides was a chord to a circle) to approximate a circle’s circumference. They could calculate the perimeter of the polygon, thereby estimating the circumference of the circle.

According to Wikipedia, the word "sine" comes from the Latin word sinus, or curve. Sinus is "a Latin mistranslation of the Arabic jib, which [itself] is a transliteration of the Sanskrit word for half a chord, jya-ardha".

In Figure 2.5.1, the varying angle’s measure is labeled θ and the signed length of the half chord θ determines is labeled sin(θ). As θ varies, sin(θ) varies accordingly, having positive values when the half chord is above the horizontal axis and negative values when the half chord is below the horizontal axis. Thus, sin(θ), the signed length of the half chord determined by θ, is a function of θ.

Notice: if we use the old, triangle definition of sine after embedding the triangle in a circle, the sine of the central angle in Figure 2.10 is $\dfrac {\text{opposite}} {\text{hypotenuse}}$. So, $$\begin{align} \sin(\theta)&=\frac {\text{signed length of opposite side}}{\text{length of hypotenuse}}\\[1ex] &=\frac{\text{signed length of opposite side}} {1}\\[1ex] &=\text{signed length of opposite side}\end{align}$$

Our new way of thinking about sine fits the old way of thinking about it. But the new way of thinking about sine is tightly integrated with measuring angles. The old way is tightly integrated with thinking about triangles. Angle measures hardlyl played a role

Putting a triangle in a circle gave us, for free, a way to relate the sine of an angle with the measure of it. Even more comes free. The definition of cosine of an angle in a right triangle is $\cos(\theta) = \mathrm{\frac{adjacent}{hypotenuse}}$. In the unit circle, the hypotenuse is 1. So in Figure 2.5.1,

Reflection 2.5.1 Why is it important to say the numerator in definitions of sine and cosine is a signed length? Why may we not just say the numerator is a length?

Figure 2.5.2 embeds the circle into a coordinate system. Angle measures $\theta$ are still arc lengths on the circle, measured in units of the radius.

On a unit circle, meaning any circle measured in units of its radius, $\cos(\theta)$ and $\sin(\theta)$ are the horizontal and vertical coordinates of the point on the circle at which the arc terminates.

Notice that sine and cosine are each measured in units of the circle’s radius, even if the radius is measured in other units. Values of $\sin(x)$ and $\cos(x)$ are therefore each a fraction of the circle’s radius. That is, values of x, $\sin(x)$ and $\cos(x)$ are all in radians.

We now have all the pieces to understand the graphs of $y = \sin(x)$ and $y = \cos(x)$ in a rectangular coordinate system.

Examine the animation in Figure 2.5.2, focusing on how the sine of the arc length varies with the arc length. In the animation, the sine of the arc length:

• starts at 0 when the arc length is 0, increases to 1 as the arc length increases to $\frac{\pi}{2}$ (1.57) radians,
• decreases to 0 as the arc length increases to π (3.14) radians,
• decreases to -1 as the arc length increases to $\frac{3 \pi}{2}$ (4.71) radians,
• increases to 0 as the arc length increases to 2π (6.28) radians.

Figure 2.5.3 displays this covariation of arc length and sine of that arc length. Look back and forth between left and right animations to see how they show the same thing regarding the way arc length and sine of that arc length vary together.

Now examine Figure 2.5.4. It shows the covariation of arc length and cosine of that arc length. Look back and forth between left and right animations to see how they show the same thing regarding the way arc length and cosine of that arc length vary together.

Exercise Set 2.5

1. Explain how values of x and sin(x) in Figure 2.5.2 are related to values of x and sin(x) in Figure 2.5.3.
2. Explain how values of x and cos(x) in Figure 2.5.2 are related to values of x and cos(x) in Figure 2.5.4.
3. Download this radian ruler. Use it to estimate each of the following. Then compare your estimates to values from your calculator or from GC.
1. sin(2.75)
2. cos(1.8)
3. sin(4.3)
4. cos(5.9)
4. Download this radian ruler. Use it to estimate each of the following. Then compare your estimates to valuesfrom your calculator or from GC.
1. sin(-2.75)
2. cos(-1.8)
3. sin(-4.3)
4. cos(-5.9)
5. The animation below shows a ball hanging by a rubber cord from a metal plate. The transparent plane records the top of the ball at its resting position. The ball is struck hard from the top, causing it to bounce down and up repeatedly. The animation shows just an initial period of bouncing, and then repeats. Sketch a graph of the ball’s displacement from its resting position in relation to the number of seconds since it was struck. Positive displacement is up, negative displacement is down.


6. The diagram from Exercise 2.5.5 is repeated below, but with the ball’s displacement and elapsed bouncing time recorded. Explain how to think about the ball’s motion in relation to elapsed time so that  $y = -\sin(x)$ gives a conceptual model of the ball’s motion. (Hint: Imagine a timer with a second hand running counter-clockwise behind the ball.)