For many, Trigonometry is a bad school memory. It’s basic enough most students encounter it in a math class somewhere along the way. Often to their dismay. A big issue is that, as happens with science, teachers may be outside their comfort zone, even in unfamiliar waters. The result is awkward for everyone and may account for trigonometry’s bad reputation.
But trigonometry (“trig”), is simple. It’s all about a triangle.
The diagram above is the playing field. It’s a two-dimensional (Euclidean) plane with X and Y axes. Figure 1 shows the area close to the origin, the (0, 0) point. The diagram shows all the important parts that go into trig. The main feature is the dark gray triangle. The upside-down light grey triangle reflects the dark one. Together they form a rectangle with one corner always in the center and the other corner always touching the surrounding circle.
The other two corners of the rectangle, labeled “x” and “y” in Figure 1, are the cornerstones of trig. Most often, we know one and want to find the value of the other.
In trig, the triangle is always a right-triangle. This means one of its three angles is 90°. In Figure 1, it’s the corner angle above the red “x”. The other two angles sum to 90° because the angles in triangles on flat planes always sum to 180°. With right-triangles, because we know the other two angles sum to 90°, knowing one gives us the other one.
In a sense, this is what trig is about. Because the parts of a triangle relate to each other, given one piece of information, we can get another. More precisely, given two pieces of information, we can determine a third of interest. For example, if we know how far we are from a tower, a distance easily paced off, and if we can take a sighting on the angle to the tower’s peak, also fairly easy to do, trig tells us how tall the tower is.
Figure 1 shows the trig triangle in the upper-right quadrant of the circle. We’ll come back and explore this in more detail, but first we need to appreciate that the triangle can appear in any quadrant, depending on angle a:
Everything in the upper-right quadrant is reflected in the other three quadrants, as if the X and Y axes were mirrors. Figure 2 also shows how we measure angle a counterclockwise from the “3 o’clock” position — 0°. That means straight up (“12 o’clock”) is 90°, the far left of the circle (“9 o’clock”) is 180°, and straight down (“6 o’clock”) is 270°.
The four angles illustrated by the dark gray triangle are, from left to right, 45°, 135°, 225°, and 315°. At the same time, the inner angle of the gray triangle is 45° in all four cases (because all four are mirror versions of each other). We’ll come back to why both angles are correct. (The short answer is that while angle a determines the quadrant, trig only cares about the 90° span within the quadrant.)
Speaking of angles, there are four special “degenerate” cases (in math, “degenerate” usually means some value became meaningless — for example, at the North and South poles, longitude becomes degenerate):
In the four cases shown in Figure 3, the triangle collapses to a line. One of its dimensions, either its height or width, has degenerated to zero. As it turns out, these give us four easy-to-remember cases that can nudge our memory about sines and cosines.
We’ll come back to this, but let’s look more closely at the triangle:
Here are the parts:
Firstly, the inside angle (lower left, with the blue arc), labeled a (for angle). We always measure it counterclockwise from the “3 o’clock” position. Often in trig, this angle is one of the two pieces of information we know. It’s instrumental in Figure 2 and Figure 3. As mentioned above, the outside angle (upper right) is 90° minus angle a. So, if angle a is 30°, then the other angle (call it b) is 90° - 30° = 60°.
Secondly, the hypotenuse, the thick blue line from the circle’s center to its rim, which it touches at point p. This is always the longest side of the triangle because of Pythagoras’s famous formula:
Where r is the length of the hypotenuse, and x and y are the lengths of the two sides (red and green lines). We label the hypotenuse r because it’s also the radius of the circle. To simplify our math, we set the radius of the circle — and therefore the hypotenuse — to 1. And because 1²=1, the formula becomes:
Which is one definition of a circle. All x-y pairs with squares summing to 1 comprise the points of a circle with a radius of one.
Therefore, the trig triangle is locked inside this unit circle. Angle a rotates counterclockwise around the circle’s center with point p always on the circle (seen in Figure 2 and Figure 3). Most importantly, point p projects onto the X and Y axes at 90° angles (thus guaranteeing the triangle is always a right triangle).
Trigonometry is largely about these two projections. The projection of p down (or up) onto the X axis gives us an x value. And the projection of p left (or right) onto the Y axis gives us a y value. Note that point p depends on angle a, so angle a determines the values of x and y.
That’s a big chunk of trig right there. Angle a determines the values of x and y.
Let’s return to the height of that tower. We start some measured distance from the tower — that distance is our x value. From that position we sight on the top of the tower using an instrument that tells us what angle (from the horizon) the tower’s top is — that’s angle a. Using these two pieces of information — the value of a and x — we can determine the value of y, the tower’s height.
In this example, we measure our straight-line distance from the tower as 1851 feet. So, x=1851, and this will determine our scale factor. Using a protractor and sighting through a drinking straw, we determine that tower’s peak is 28° above the horizon. So, a=28°.
To determine the tower’s height, we use this formula:
Where y will be the tower’s height. We need to multiply the sin
value (the Y axis projection) by a scaling factor to get a height in feet. That factor is the measured x value divided by the cos
value (the X axis projection). This gives us a ratio for expanding the triangle to fit the measured distance.1
We know x and a, so we plug them in:
I’ll explain the sin(28)
and cos(28)
parts below. What’s important for now is that they have the respective values here of 0.4694716… and 0.8829476… (the decimal digits continue). Trig assumes the hypotenuse is 1, but here it’s the (unknown!) distance from our position 1851 feet away to the tip of the tower.
Plugging in the sin
and cos
values, we have:
Which might seem a strange height — why 984? Turns out it’s exactly 300 meters.
Let’s dig a little deeper into what’s going on.
Functions
Mathematical functions take one or more numbers as input and return an output number. A simple example is an add function taking two input numbers and outputting their sum.
Trig has two central functions, sine (sin
, “sin”) and cosine (cos
, “kōs”). A third function, tangent (tan
, “tan”) provides the final leg to the trig stool. The names in parentheses are the names used in equations (as seen above) along with typical pronunciations in quotes.2
All three take one input, the angle a, and output a number. The sin
function returns the projection of point p onto the Y axis. The cos
function returns its projection onto the X axis. Because of this, we generally associate cosine with the X axis and sine with the Y axis.
This gives us two important equalities:
Using parentheses after a function name to enclose the inputs is a standard notation but be aware that sometimes the parentheses are left off. This is especially common with authors who use Greek letters theta (θ) or phi (φ) to represent the angle:
This is a bit cleaner but only works when input terms are simple. More complex terms require parentheses anyway, so I recommend always using them to avoid confusion.
Given this way of identifying x and y, we can define point p based on angle a:
We define all the points of the trig triangle (and its reflection) with angle a. The inner corner is always (0,0), while x, y, and p (the outer corner), are defined by the cosine and sine of angle a.
The Tangent Function
The third function, tan
, outputs the slope of the triangle’s hypotenuse. The X (red) and Y (green) sides of the triangle are always horizontal and vertical respectively, but the slope of the hypotenuse (blue) depends on angle a.3
Slope is also called run over rise referring to its definition as horizontal distance divided by vertical distance. Mathematically, x ÷ y or x / y. That’s exactly what the tan
function gives us, x over y. So rather than a distance, like the sin
or cos
functions, tan
returns the ratio of the x and y sides of the triangle.
In fact, sin
and cos
, also return ratios (which are also distances along the respective axes). The sine function returns the ratio of the y side of the triangle to the hypotenuse. The latter is always one, so the ratio is y/1, or just y. The cosine returns the ratio of the x side to the hypotenuse, the ratio x/1 or just x. Both ratios vary between -1 and +1. When graphed they look like this:
The repeating wave traced out by the plot is called a sine wave. The name “cosine” is because the x side is “co” (next to) angle a. The sine value is always the triangle side opposite the angle.
[See this section of the Sine and cosine Wiki page for a great animation.]
The Reverse Functions
The three functions each have an inverse function. They’re called arcsine, arccosine, and arctangent. As inverses, their inputs are the numeric values output by the sine, cosine, and tangent functions. All three inverse functions output the angle a.
The inverse functions are part of a more advanced look at trig, so I’ll say no more about them here. A basic understanding can rest on just the sin
and cos
functions.
Putting It All Together
We have everything we need now to understand the playing field.
Figure 6 shows the triangle with three different angles (all in the first quadrant — anything we do there translates to the other three. Recall also the two examples on the left in Figure 3. The table below starts with the left-most example in Figure 3. Next, the three examples in Figure 6 (from left to right). Lastly, the second example in Figure 3. Combined they represent sweeping angle a from zero to ninety degrees in 22.5-degree steps.
Notice how the y value, sin(a)
, starts at 0 and ends at +1. Meanwhile, the x value, cos(a)
, starts at +1 and ends at 0. Because x² + y² = 1 is always true, if y gets bigger x must get smaller in balance. The two mirror each other’s values exactly.
Note how the values for 0° and 90° (both represented in Figure 3) are easy to remember, since they’re either 0 or 1. The other two examples in Figure 3 show the values for 180° (x=-1, y=0) and 270° (x=0, y=-1) are also easy, especially when you keep the triangle-circle combo in mind.
The tan(a)
values increase from 0° as the angle approaches 90° (where it becomes vertical, and slope is undefined). As the angle gets very close to 90°, the values tend towards infinity.
As the angle increases past 90° (Figure 7), the sine value decreases back to 0 while the cosine value increases towards negative 1. When the angle reaches 180°, we have the third example in Figure 3.
As the angle increases past 180° (Figure 8), we enter the lower half of the circle, and now the sine value grows towards negative 1, which it reaches at 270°. Meanwhile, the cosine decreases from -1 to 0.
Lastly, the angle increases past 270° to complete the final quadrant. The sine value decreases back to 0 while the cosine increases to 1.
In all cases, Figure 6 through Figure 9, show the same three triangles in the four quadrants. The sin
and cos
values are the same (though they reverse every other quadrant), but they have different signs.
Note the similarity of the first example in Figure 6 and Figure 8 (a short wide triangle) with the last example in Figure 7 and Figure 9. Likewise note the similarity between the last example in Figure 6 and Figure 8 (a tall thin triangle) with the first example in Figure 7 and Figure 9.4 The middle triangles are similar in all four.
This is the axes mirroring I mentioned. The sine, cosine, and tangent values are the same for all these triangles, except for the order and sign. The same three numbers in Table 1 (0.382, 0.707, and 0.923) are repeated over and over here because angle a repeats the same three angles (22.5°, 45°, and 67.5°) within each quadrant.
This was a lot, but I wanted to see if I could fit an overview of trigonometry into one post. You’ll have to be the judge of how well I succeeded. I’d be especially curious about the reactions of those who aren’t familiar with the topic (but have some interest in it). Does it at least make more sense now?
The ratio is the actual measured x value over the trig x value, cos(a)
, that assumes the hypotenuse is 1. This ratio is the same as for the y value (which we don’t know) over the trig y value, sin(a)
, so using it as a scale factor gives us the actual y value in feet. In this case cos(28)
≈ 0.88 and x=1851, so the scaling factor is 1851 ÷ 0.88 ≈ 2096.
The proper names, “sine”, “cosine”, and “tangent” are pronounced as spelled.
Be aware that, because vertical slope is undefined (like division by zero), the tangent function is undefined for the angles 90° and 270° (the second and fourth cases in Figure 3).
The tall thin triangle is in fact the same as the short wide one but with a different orientation. That’s why the values in Table 1 just swap values for a=22.5° and a=67.5°.