[Image1]
Hey it's a me again @drifter1!
Today we continue with the small series about Trigonometry.
I suggest reading the first article before getting into this one.
Today we will cover Trigonometric Functions and how Right-Angled Triangles can be solved using them...
So, without further ado, let's dive straight into it!
[Custom Figure using draw.io]
Having a right-angled triangle in mind, for any angle θ the three main trigonometric functions are defined as "soh-cah-toa":
The size of the sides doesn't matter, as only the angle changes the ratio.
Similarly, we can also define the less-used cosecant (csc), secant (sec) and cotangent (cot):
A unit circle is a circle with radius 1 and center at O (0, 0) in the Cartesian Space:
[Custom Figure using GeoGebra]
Because the radius is 1, we can directly measure the sine, cosine and tangent, as well as all the other trigonometric functions.
An article about Trigonometry wouldn't be complete without Pythagoras's Theorem:
For a right-angled triangle, the square of the long side (hypotenuse) equals the sum of the squares of the other two sides.
In the case of the unit circle:
And since x = cos θ and y = sin θ, we easily derive the following useful identity:
which is true for any angle θ.
The values of the sine, cosine and tangent of the 30°, 45° and 60° can be easily remembered:
In the two-dimensional Cartesian Coordinate System we define the position of a point on a graph by how far along (x) and how far up (y) it is in respect to the center O (0, 0).
Including negative values for x and y, we easily divide this space into 4 pieces called Quadrants (counter-clockwise direction):
Trigonometric Functions are periodic functions, meaning that two different angles return the same result for sin, cos and tan.
Simple relations to remember are:
Let's now get into inverse trigonometric functions.
In the inverse function we insert the value of the trigonometric function and it returns the angle θ.
Thus, we define the inverse sine, inverse cosine and inverse tangent as:
Of course there are infinite answers, two per 360° degree circle to be exact, and repeating after that.
Mathematically the solution of x = sinθ or θ = arctan(x) can be written:
Similarly, the solution of x = cosθ or θ = arccos(x) is:
And finally the solution of x = tanθ or θ = arctan(x) is:
In these last equations I used radians instead of degrees.
Converting from degrees to radians is: 180 degrees = π radians.
The simplest triangles to solve are those with a right angle.
There are two types of unknown values:
To calculate an unknown angle in a right-angled triangle we have to know the lengths of at least two of its sides.
To find an unknown side in a right-angled triangle we have to know:
Let's consider the following Triangle:
[Custom Figure using GeoGebra]
We only know one angle (excluding the right-angle) and one side. Let's find all the remaining angles and sides!
Let's start with the Hypotenuse. We know the opposite side of the angle of 45° and so to calculate the Hypotenuse we have to use the Sinus function:
Using Pythagoras's Theorem we can now calculate the adjacent side AC:
And because the angles sum up to 180° in all triangles we can also calculate the remaining angle:
Let's consider the following Triangle:
[Custom Figure using GeoGebra]
We only know the lengths of two sides. Let's find the remaining side and all the angles!
Using Pythagoras's Theroem we can easily calculate the Hypotenuse BC:
The angles can be calculated using inverse trigonometric functions.
Let's first calculate the bottom-left angle and then use the sum to 180° to find the upper angle.
Using the inverse tangent (arctan) we calculate (using calculator) the bottom-left angle to be:
Therefore the upper angle is:
Mathematical equations used in this article, where made using quicklatex.
