What’s Your Sine?

Continuing with the trigonometry theme, there are other relationships between the angles and the length of the sides of a right triangle. You may have a calculator with buttons labeled as “sin” or “sin x” or “sin θ”. There would be similar buttons using the prefix “cos” and “tan”. These are the basic trig (trigonometric, hence the abbreviation) functions. What do these functions do?

As with the Pythagorus theorem covered in my last post, there are other relationships that apply to right triangles regardless of their size. But unlike the Pythagorus theorem which relates the lengths of just the sides, the trig functions relate the side lengths with the internal angles.

But before I show these, remember that there are several ways to measure angles, degrees and radians being the most common. When using the trig functions, your calculator needs to know what measurement you are using. As we have been and will continue to use degrees, you need to make sure that in your calculator settings, you have set the degrees mode. In most calculators, this will show up as an abbreviation “deg”. Radians will display as “rad”. As a full circle angle is 360° and approximately 6.28 radians, there is quite a bit of difference between the two types of measurements. So let’s begin.

So here is your basic, everyday right triangle with the angle of interest, 𝜃, and the sides labelled with respect to that angle. The side could be adjacent to it or opposite it. The hypotenuse, as seen before, is the side opposite the right angle. For any size right triangle, the basic trig functions are defined as follows:

\[
\sin\mathit{\theta}\hspace{0.33em}{=}\hspace{0.33em}\frac{\mathrm{opp}}{\mathrm{hyp}}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\cos\mathit{\theta}\hspace{0.33em}{=}\hspace{0.33em}\frac{\mathrm{adj}}{\mathrm{hyp}}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\tan\mathit{\theta}\hspace{0.33em}{=}\hspace{0.33em}\frac{\mathrm{opp}}{\mathrm{adj}}\hspace{0.33em}{=}\hspace{0.33em}\frac{\sin\mathit{\theta}}{\cos\mathit{\theta}}
\]

So the trig functions show the ratios of the various sides. In the above formulas, the following abbreviations are used: sin = sine, cos = cosine, tan = tangent, opp = opposite, adj = adjacent, hyp = hypotenuse. If you type 45 into your calculator and hit the “sin” key, you should see a decimal number 0.7071… . This means that for a right triangle with a 45° angle, the length of the opposite side divided by the length of the hypotenuse is always 0.7071…

The power of these functions is that you only need the length of one side and an angle (other than the right angle) of a right triangle to determine the lengths of the other two sides.

Example:

What are the lengths of the other two sides of the above triangle?

\[
\begin{array}{l}
{\sin{30}\hspace{0.33em}{=}\hspace{0.33em}\frac{\mathrm{opp}}{5}\hspace{0.33em}{=}\hspace{0.33em}{0}{.}{5}}\\
{{\mathrm{opp}}\hspace{0.33em}{=}\hspace{0.33em}{0}{.}{5}\hspace{0.33em}\times\hspace{0.33em}{5}\hspace{0.33em}{=}\hspace{0.33em}{2}{.}{5}}\\
{\cos{30}\hspace{0.33em}{=}\hspace{0.33em}\frac{\mathrm{adj}}{5}\hspace{0.33em}{=}\hspace{0.33em}{0}{.}{866}}\\
{{\mathrm{adj}}\hspace{0.33em}{=}\hspace{0.33em}{0}{.}{866}\hspace{0.33em}\times\hspace{0.33em}{5}\hspace{0.33em}{=}\hspace{0.33em}{4}{.}{33}}
\end{array}
\]

So the lengths are 2.5 and 4.33. The units (centimeters, inches, etc) are whatever the units of the given hypotenuse is.

Let’s try this on a practical problem:

You are 500 meters from a tall tower. Using a sextant (a device to measure angles), you measure the angle between the ground and the line between your feet and the top of the tower to be 15°. How high is the tower?

The side adjacent to the angle is 500 meters. We need to find the side opposite (the tower). Looking at the trig functions, the tangent looks like it is the best one to use since it is the ratio of the side opposite (unknown) and the side adjacent (known). Finding the tangent of 15° on my calculator gives 0.2679. So,

\[
\begin{array}{l}
{\tan\hspace{0.33em}{15}\hspace{0.33em}{=}\hspace{0.33em}\frac{\mathrm{opp}}{\mathrm{adj}}\hspace{0.33em}{=}\hspace{0.33em}\frac{{\mathrm{tower}}\hspace{0.33em}{\mathrm{height}}}{{\mathrm{distance}}\hspace{0.33em}{\mathrm{from}}\hspace{0.33em}{\mathrm{tower}}}\hspace{0.33em}}\\
{{=}\frac{{\mathrm{tower}}\hspace{0.33em}{\mathrm{height}}}{500}\hspace{0.33em}{=}\hspace{0.33em}{0}{.}{2679}}\\
{{\mathrm{tower}}\hspace{0.33em}{\mathrm{height}}\hspace{0.33em}{=}\hspace{0.33em}{0}{.}{2679}\hspace{0.33em}\times\hspace{0.33em}{500}\hspace{0.33em}{=}\hspace{0.33em}{133}{.}{97}\hspace{0.33em}{\mathrm{meters}}}
\end{array}
\]

Now that saves a lot of effort to physically measure the height, doesn’t it?

The Correct Angle is the Right One

In my last post, you saw that a 90° angle is called a right angle. This is the angle made by the two lines at the corner of a square. Now a triangle is a shape that has three angles inside. A basic property of any triangle is that all the internal angles add up to 180°:

But this post is about triangles where one of its angles is 90°, that is a right angle. Such triangles are called right triangles.

Below is a right triangle where I have labelled the sides as a, b, and c. Side c is the side opposite the right angle. This side is called the hypotenuse. The hypotenuse is always the longest side of any right triangle.

Right triangles have another famous property that relates the lengths of the three sides. This property is called the Pythagorus Theorem. This is named after the Greek mathematician Pythagorus who lived 570 to 495 BCE. This theorem was used before his time but he is credited with providing the first proof. Given the sides as labelled above, the following is true for any size right triangle:

c² = a² + b²

This means that if you know any two sides of a right triangle, you can calculate the third side using this equation. Let’s do an example:

So we now know that c² = 4² + 3² = 16 + 9 = 25. You can now find c by taking the square root of both side of the equation. Square roots have been covered in previous posts:

\[{c}^{2}\hspace{0.33em}{=}\hspace{0.33em}{25}\hspace{0.33em}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\hspace{0.33em}\sqrt{{c}^{2}}\hspace{0.33em}{=}\hspace{0.33em}\sqrt{25}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\hspace{0.33em}{c}\hspace{0.33em}{=}\hspace{0.33em}{5}\]

In my posts on square roots, I did say that taking a square root results in two solutions, one positive and one negative. But since we are solving for a physical length, we can ignore the negative solution. So the hypotenuse for this triangle is 5.

Not all right triangle problems work out so well. Most square roots are decimal numbers and you have to either round the number or leave the answer as a square root.

The Pythagorus Theorem can also be used to find one of the non-hypotenuse sides as well:

You can rearrange the theorem’s equation to solve for the unknown side:

\[\begin{array}{l}{{c}^{2}\hspace{0.33em}{=}\hspace{0.33em}{a}^{2}\hspace{0.33em}{+}\hspace{0.33em}{b}^{2}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\hspace{0.33em}{b}^{2}\hspace{0.33em}{=}\hspace{0.33em}{c}^{2}\hspace{0.33em}{-}\hspace{0.33em}{a}^{2}}\\{{b}^{2}\hspace{0.33em}{=}\hspace{0.33em}{9}^{2}\hspace{0.33em}{-}\hspace{0.33em}{5}^{2}\hspace{0.33em}{=}\hspace{0.33em}{81}\hspace{0.33em}{-}\hspace{0.33em}{25}\hspace{0.33em}{=}\hspace{0.33em}{56}}\\{\sqrt{{b}^{2}}\hspace{0.33em}{=}\hspace{0.33em}\sqrt{56}\hspace{0.33em}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\hspace{0.33em}{b}\hspace{0.33em}\approx\hspace{0.33em}{7}{.}{48}}\end{array}\]

So b approximately equals 7.48. That is what the symbol “≈” means. The exact answer cannot be written as a decimal number as the decimal part goes on forever.

What’s my Angle?

This is the first post in the trigonometry category. Trigonometry is the study of the properties of triangles: the lengths of their sides and the relationships with their angles. So the first topic in trigonometry is to define what an angle is.

An angle is a measure of rotation of one line from another where these lines are connected at one end. Like x is used to commonly refer to an unknown or general number in algebra, 𝜃, which is the greek letter “theta”, is commonly used to represent a generic angle. I presume this is so because the ancient Greeks did a lot of work in the fields of trigonometry and geometry. Below shows a generic angle between two lines:

Now there are several units used to measure angles. The one most known to most non-maths people is degrees. There are other measures: radians which is used frequently in maths, and gradians which is sometimes used in engineering and surveying. Here, we will use degrees.

The point where the lines join is called the vertex. You can imagine it as a flexible joint where the top line pivots around the horizontal line from being right on top of the horizontal line to swinging all the way around so that it again is superimposed on the horizontal line. When the lines are first superimposed, this is a 0 degree angle. The notation for degrees is a small circle at the upper right of the number: 0°. When the line does a full rotation so that it is again superimposed, that angle is 360°.

Now you can continue rotating the line and think of angles greater than 360°, but this post will limit itself to angles between 0° and 360°. I am specifically interested in the angle when the rotating line is pointing straight up, that is becomes perfectly vertical. Well this is a quarter of the way around to 360° so a quarter of 360 is 90. The angle is now:

Note that it is common to add lines in this angle to form a small square when the angle is 90°. This angle holds special interest in maths and is given a name: right angle. The term “right” comes from latin meaning “upright”.

This angle will be used in my next post on triangles.