Trigonometric functions

Trigonometric functions were used to represent the ratio of sides of right triangles, but now they are used not only for geometric problems. The main trigonometric functions are sine, cosine, tangent, and cotangent. These functions are useful not only in mathematics but also in many other areas like physics, engineering, navigation, and even video game development! For example, in physics trigonometric functions are used to describe oscillations in systems like spring-mass systems, alternative currents, and even light propagation.

Right triangles and trigonometric functions

A right triangle is a triangle that has one right angle ($90\degree$). The hypotenuse is the side opposite the right angle, and the legs are two other sides of this triangle.

Let the angle $\angle A$ be $\alpha$. Then the side $b$ between this angle and the right angle will be the adjacent leg and the side a will be the opposite leg.

Now, we can define trigonometric functions — sine ($\sin$), cosine ($\cos$), tangent ($\tan$) and cotangent ($\cot$) using this right triangle.

$\sin(\alpha) = \frac{\text{opposite leg}}{\text{hypotenuse}} = \frac{a}{c}$

$\cos(\alpha) = \frac{\text{adjacent leg}}{\text{hypotenuse}} = \frac{b}{c}$

$\tan(\alpha) = \frac{\text{opposite leg}}{\text{adjacent leg}} = \frac{a}{b}$

$\cot(\alpha) = \frac{\text{adjacent leg}}{\text{opposite leg}} = \frac{b}{a}$

As you can see, cotangent is reciprocal to tangent because $\cot(\alpha) = \frac{1}{\tan(\alpha)}$. Let's take a look at the right triangle with one leg of $3$, another leg of $4$ and the hypotenuse of $5$. This triangle is also known as the Egyptian triangle because ancient Egyptians used it to construct a right angle.

Let's denote the angle $\angle E$ as $\alpha$ and compute values of its trigonometric functions:

$\sin(\alpha) = \frac{\text{FD}}{\text{ED}} = \frac{4}{5}$

$\cos(\alpha) = \frac{\text{EF}}{\text{ED}} = \frac{3}{5}$

$\tan(\alpha) = \frac{\text{FD}}{\text{EF}} = \frac{4}{3}$

$\cot(\alpha) = \frac{\text{EF}}{\text{FD}} = \frac{3}{4}$

Algebraic values of trigonometric functions

There are some angles that are essential for trigonometry: $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi.$ It's better to remember the algebraic values of their trigonometric functions.
Here is a trigonometric table of these angles, which you should remember to make the application of trigonometric functions easier. You can find values of trigonometric functions of all other angles using either calculators or special tables.

Basic trigonometric identities

Here are two simple trigonometric identities:

$$\sin(\alpha+2\pi) = \sin(\alpha)$$

$$\cos(\alpha+2\pi) = \cos(\alpha)$$

This means that values of sine and cosine repeat with the period of $2\pi$ and so they are periodical functions. As you can see from the trigonometric table in the previous paragraph, angles of $0$ and $2\pi$ are the simplest example of these identities.

And here are some more complex trigonometric identities:

$$\sin(\frac{\pi}{2} \! - \! \alpha) = \cos(\alpha)$$

$$\cos(\frac{\pi}{2} \!- \!\alpha) = \sin(\alpha)$$

$$\tan(\frac{\pi}{2} \! - \!\alpha) = \cot(\alpha)$$

$$\cot(\frac{\pi}{2} \!- \!\alpha) = \tan(\alpha)$$

Let's derive now our first Pythagorean identity:

$\sin^2(\alpha) + \cos^2(\alpha) = 1$

In order to do it, we'll have to come back to the right triangle.

Let's once more denote the angle $\angle E$ as $\alpha$ and write down definitions of its sine and cosine:

$\sin(\alpha) = \frac{\text{FD}}{\text{ED}}$

$\cos(\alpha) = \frac{\text{EF}}{\text{ED}}$

According to the Pythagorean theorem:

$\sin^2(\alpha) + \cos^2(\alpha)=(\frac{\text{FD}}{\text{ED}})^2 + (\frac{\text{EF}}{\text{ED}})^2=\frac{\text{FD}^2 + \text{EF}^2}{\text{ED}^2} = \frac {\text{ED}^2}{\text{ED}^2}=1$

There's also a Pythagorean identity for both tangent and cotangent. Let's derive them.

$$1 + \tan^2(x) = 1 + \frac{\sin^2(x)}{\cos^2(x)} = \frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)} = \frac{1}{\cos^2(x)}$$

$$1 + \cot^2(x) = 1 + \frac{\cos^2(x)}{\sin^2(x)} = \frac{\sin^2(x) + \cos^2(x)}{\sin^2(x)} = \frac{1}{\sin^2(x)}$$

You can use Pythagorean identities for tangent and cotangent together with Pythagorean identities for sine and cosine to find values of all trigonometric functions of an angle. For that, you must know one of them and the location of this angle at the unit circle.

For example, you know that $\tan(x) = 2$ and that $0\leq x \leq\frac{\pi}{2}$. In this case, the cosine and sine of the angle are non-negative.

$\cos(x) = \sqrt{\frac{1}{1+\tan^2(x)}} = \sqrt{\frac{1}{1+2^2}} =\frac{1}{\sqrt{5} }$

$\sin(x) = \sqrt{1-\cos^2(x)} = \sqrt{1 - \frac{1}{5}} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}}$

A real-life application of trigonometric functions

Imagine that you're taking a cable car to the top of a mountain. You know the length of the route to the top ($\text{BA}$) and the elevation angle ($\angle A$), but you don't know the exact height of the mountain ($\text{BC}$). How to compute it?

Let's assume that the length of the route is $1200$ m and the angle of elevation is $30 \degree$. In this right triangle:

$\text{BC} = \text{BA}\cdot\sin(\angle A) = 1200 \cdot \frac{1}{2} = 600 \text{ m}$

Here is an example from physics. A ball is thrown at an angle $\alpha$ to the horizon and its speed is equal to $\overline{V}$ at time $t = 0$. You can easily find $\overline{V_x}$ and $\overline {V_y}$ using the right triangle:

$\overline{V_x} = \overline{V} \cdot \cos(\alpha) \\ \overline{V_y} = \overline{V} \cdot \sin(\alpha)$

Since you've found $\overline{V_x}$ and $\overline {V_y}$, you are able to describe the motion of the ball by using some formulas from physics. We won't do it, because we focus on trigonometric functions and not on physics.

Conclusion

• You can determine the sine, cosine, tangent, and cotangent trigonometric functions for any right triangle using the following formulas:

  $$\sin(\alpha) = \frac{\text{opposite leg}}{\text{hypotenuse}}$$

  $$\cos(\alpha) = \frac{\text{adjacent leg}}{\text{hypotenuse}}$$

  $$\tan(\alpha) = \frac{\text{opposite leg}}{\text{adjacent leg}}$$

  $$\cot(\alpha) = \frac{\text{adjacent leg}}{\text{opposite leg}}$$

• The most essential angles in trigonometry are $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi.$ Remember the values of their trigonometric functions. You can find values of trigonometric functions of all other angles using calculators or special tables.

• Sine and cosine are periodical functions: their values repeat with a period of $2\pi$. There's also a Pythagorean identity $\sin^2(\alpha) + \cos^2(\alpha) = 1$, so you can easily find the value of sine up to a character if you know the value of cosine and vice versa.

• There are also some more important trigonometric identities that you can use to solve different problems:

  $$\sin(\frac{\pi}{2} \! - \! \alpha) = \cos(\alpha)$$

  $$\cos(\frac{\pi}{2} \!- \!\alpha) = \sin(\alpha)$$

  $$\tan(\frac{\pi}{2} \! - \!\alpha) = \cot(\alpha)$$

  $$\cot(\frac{\pi}{2} \!- \!\alpha) = \tan(\alpha)$$

• Two more Pythagorean identities can be used to find values of cosine and sine up to a character if values of tangent and cotangent are known and vice versa:

  $$1 + \tan^2(x) = \frac{1}{\cos^2(x)}$$

  $$1 + \cot^2(x) = \frac{1}{\sin^2(x)}$$