<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/1/11/Academ_Base_of_trigonometry.svg/600px-Academ_Base_of_trigonometry.svg.png">
<p>[<a href="https://commons.wikimedia.org/wiki/File:Academ_Base_of_trigonometry.svg">Image1</a>]</p>
<h2>Introduction</h2>
<p>
Hey it's a me again <a href="https://peakd.com/@drifter1">@drifter1</a>!<br><br>
Today we continue with the small series about Trigonometry.
I suggest reading the <a href="https://peakd.com/hive-196387/@drifter1/mathematics-all-about-trigonometry-part-1">first article</a> before getting into this one.
Today we will cover Trigonometric Functions and how Right-Angled Triangles can be solved using them...<br><br>
So, without further ado, let's dive straight into it!
</p>
<hr>
<h2>Trigonometric Functions</h2>
<h3>Based on a Right-Angled Triangle</h3>
<p>
<img src="https://i.ibb.co/2qfGTLj/right-angled-triangle.jpg"><br><br>
[Custom Figure using <a href="https://app.diagrams.net/">draw.io</a>]
</p>
<br>
<p>
Having a right-angled triangle in mind, for any angle <em>θ</em> the three main trigonometric functions are defined as "soh-cah-toa":<br><br>
<img src="https://quicklatex.com/cache3/e4/ql_186a532886243c8a295b6a46b0614ae4_l3.png"><br><br>
The size of the sides doesn't matter, as only the angle changes the ratio.
</p>
<br>
<p>
Similarly, we can also define the less-used cosecant (csc), secant (sec) and cotangent (cot):<br><br>
<img src="https://quicklatex.com/cache3/a0/ql_001de4b1d69376c25f3d14834ee657a0_l3.png"><br><br>
</p>
<h3>Unit Circle</h3>
<p>
A unit circle is a circle with radius 1 and center at <em>O</em> (0, 0) in the Cartesian Space:<br><br>
<img src="https://i.ibb.co/P9vhv3y/unit-circle.jpg"><br><br>
[Custom Figure using <a href="https://www.geogebra.org/">GeoGebra</a>]<br><br>
Because the radius is 1, we can directly measure the sine, cosine and tangent, as well as all the other trigonometric functions.
</p>
<h4>Pythagoras's Theorem</h4>
<p>
An article about Trigonometry wouldn't be complete without Pythagoras's Theorem:
<pre>For a right-angled triangle, the square of the long side (hypotenuse) equals the sum of the squares of the other two sides.</pre>
</p>
<p>
In the case of the unit circle:<br><br>
<img src="https://quicklatex.com/cache3/4b/ql_474af1eae454baf7e6d31eba65844b4b_l3.png"><br><br>
And since <em>x = cos θ</em> and <em>y = sin θ</em>, we easily derive the following useful identity:<br><br>
<img src="https://quicklatex.com/cache3/10/ql_81564c667ab77aa4de9299bee16a6110_l3.png"><br><br>
which is true for any angle <em>θ</em>.
</p>
<h3>30°, 45° and 60° Angles</h3>
<p>
The values of the sine, cosine and tangent of the 30°, 45° and 60° can be easily remembered:
<ul>
  <li>sinus goes √1, √2, √3, divided by 2</li>
  <li>cosinus goes √3, √2, √1 divided by 2</li>
  <li>tangent is the division of sinus and cosinus (tan = sin/cos)</li>
</ul>
In tabular form:<br><br>
<img src="https://quicklatex.com/cache3/b1/ql_2bfa2bec27eb251225402182d29529b1_l3.png">
</p>
<h3>Values in the Four Quadrants</h3>
<h4>Cartesian System and Quadrants</h4>
<p>
In the two-dimensional <strong>Cartesian Coordinate System</strong> 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 <em>O</em> (0, 0).<br><br>
Including negative values for x and y, we easily divide this space into 4 pieces called Quadrants (counter-clockwise direction):
<ul>
  <li><em>Quadrant I</em> - both x and y are positive</li>
  <li><em>Quadrant II</em> - x is negative and y is positive</li>
  <li><em>Quadrant III</em> - both x and y are negative</li>
  <li><em>Quadrant IV</em> - x is positive and y is negative</li>
</ul>
Visually:<br>
<img src="https://i.ibb.co/jD12bvb/quadrants.jpg"><br><br>
[Custom Figure using <a href="https://www.geogebra.org/">GeoGebra</a>]
<ul>
<li>A (3, 9) is in Quadrant I</li>
<li>B (-7, 6) is in Quadrant II</li>
<li>C (-4, -5) is in Quadrant III</li>
<li>D (8, -2) is in Quadrant IV</li>
</ul>
</p>
<h4>Sinus, Cosine and Tangent in the Four Quadrants</h4>
<p>
<ul>
<li>Quadrant I - <strong>a</strong>ll trigonometric functions are positive</li>
<li>Quadrant II - only <strong>s</strong>inus positive</li>
<li>Quadrant III - only <strong>t</strong>angent positive</li>
<li>Quadrant IV - only <strong>c</strong>osinus positive</li>
</ul>
So, there is a pattern: "A-S-T-C", that is easy to remember.
</p>
<h4>Angles with same Trigonometric Function Result</h4>
<p>
Trigonometric Functions are periodic functions, meaning that two different angles return the same result for sin, cos and tan.
Simple relations to remember are:<br><br>
<img src="https://quicklatex.com/cache3/32/ql_cebadd75d0e8576d7c5d0fa29ec5c132_l3.png"><br><br>
</p>
<h4>Inverse Trigonometric Functions</h4>
<p>
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 <em>θ</em>.<br><br>
Thus, we define the inverse sine, inverse cosine and inverse tangent as:<br><br>
<img src="https://quicklatex.com/cache3/ff/ql_862704a0b400a862a81d57eb0e73a2ff_l3.png">
</p>
<h4>Infinite Solutions</h4>
<p>
Of course there are infinite answers, two per 360° degree circle to be exact, and repeating after that.<br><br>
Mathematically the solution of <em>x = sinθ</em> or <em>θ = arctan(x)</em> can be written:<br><br>
<img src="https://quicklatex.com/cache3/e6/ql_7b16a4b8eb3478ad79aa03321c2b7ce6_l3.png"><br><br>
Similarly, the solution of <em>x = cosθ</em> or <em>θ = arccos(x)</em> is:<br><br>
<img src="https://quicklatex.com/cache3/ef/ql_66cb4fa3e73f9835f666c52ff2ac9fef_l3.png"><br><br>
And finally the solution of <em>x = tanθ</em> or <em>θ = arctan(x)</em> is:<br><br>
<img src="https://quicklatex.com/cache3/06/ql_2a7d6e92e8c95f4b2eacec2d19879606_l3.png"><br><br>
In these last equations I used radians instead of degrees.
Converting from degrees to radians is: <em>180 degrees = π radians</em>.
</p>
<hr>
<h2>Solving Right Triangles</h2>
<p>
The simplest triangles to solve are those with a right angle.<br><br>
There are two types of unknown values:<br>
<ol>
  <li>Unknown Angles</li>
  <li>Unknown Sides</li>
</ol>
</p>
<h3>Finding an unknown angle</h3>
<p>
To calculate an unknown angle in a right-angled triangle we have to know the lengths of at least two of its sides.
<ul>
  <li>If all sides are known then we can use any of the trigonometric functions (sin, cos, tan)</li>
  <li>If only two are known then we have to identify which of the trigonometric functions has to be used:
    <ul>
      <li>Opposite and Hypotenuse known &rarr; Sinus</li>
      <li>Adjacent and Hypotenuse known &rarr; Cosinus</li>
      <li>Adjacent and Opposite known &rarr; Tangent</li>
    </ul>
  </li>
</ul>
</p>
<h3>Finding an unknown side</h3>
<p>
To find an unknown side in a right-angled triangle we have to know:
<ul>
<li>the length of at least one side</li>
<li>one angle (not including the right-angle!)</li>
</ul>
If two sides are known then we can directly apply Pythagoras's Theroem to calculate the remaining side!<br><br>
If not, then we of course have to choose the correct trigonometric function to calculate the side that we want to find.
The choice depends on which angle we know and the relation of the known side to that angle.<br><br>
For example if the side that we know is opposite to the known angle, and the side that we want to find is the Hypotenuse, then we have to use the Sinus Function.
</p>
<hr>
<h2>Full-on Examples</h2>
<h3>One side and one angle known</h3>
<p>
Let's consider the following Triangle:<br>
<img src="https://i.ibb.co/4T978Gc/Created-with-GIMP.jpg"><br>
[Custom Figure using <a href="https://www.geogebra.org/">GeoGebra</a>]<br><br><br>
We only know one angle (excluding the right-angle) and one side. Let's find all the remaining angles and sides!<br><br><br>
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:<br><br>
<img src="https://quicklatex.com/cache3/c5/ql_a65e6753c5b1962ecd3f00ed4b6e74c5_l3.png"><br><br><br>
Using Pythagoras's Theorem we can now calculate the adjacent side <em>AC</em>:<br><br>
<img src="https://quicklatex.com/cache3/21/ql_e16fe2b684600d355c5244c41497d521_l3.png"><br><br><br>
And because the angles sum up to 180° in all triangles we can also calculate the remaining angle:<br><br>
<img src="https://quicklatex.com/cache3/8e/ql_8210e35c9f04341d48fb3632deb5128e_l3.png">
</p>
<h3>Two sides known</h3>
<p>
Let's consider the following Triangle:<br>
<img src="https://i.ibb.co/TcN4Dd9/example-2.jpg"><br>
[Custom Figure using <a href="https://www.geogebra.org/">GeoGebra</a>]<br><br><br>
We only know the lengths of two sides. Let's find the remaining side and all the angles!<br><br><br>
Using Pythagoras's Theroem we can easily calculate the Hypotenuse <em>BC</em>:<br><br>
<img src="https://quicklatex.com/cache3/3f/ql_187af90cd9f6d46a99c2657961369d3f_l3.png"><br><br><br>
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.<br><br>
Using the inverse tangent (arctan) we calculate (using calculator) the bottom-left angle to be:<br><br>
<img src="https://quicklatex.com/cache3/14/ql_2de53afbac06c3927ffa75ac9e63f714_l3.png"><br><br>
Therefore the upper angle is:<br><br>
<img src="https://quicklatex.com/cache3/ca/ql_6272400224c0e8b656da92d5b03334ca_l3.png">
</p>
<hr>
<h2>RESOURCES:</h2>
<h3>References</h3>
<ol>
<li><a href="https://www.britannica.com/science/trigonometry">https://www.britannica.com/science/trigonometry</a></li>
<li><a href="https://www.mathsisfun.com/algebra/trigonometry-index.html">https://www.mathsisfun.com/algebra/trigonometry-index.html</a></li>
<li><a href="https://www.khanacademy.org/math/trigonometry">https://www.khanacademy.org/math/trigonometry</a></li>
<li><a href="https://www.math24.net/basic-trigonometric-equations/">https://www.math24.net/basic-trigonometric-equations/</a></li>
<li><a href="http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-trigeqn-2009-1.pdf">http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-trigeqn-2009-1.pdf</a></li>
<li><a href="https://www.mathportal.org/algebra/trigonometry/index.php">https://www.mathportal.org/algebra/trigonometry/index.php</a></li>
<li><a href="https://intl.siyavula.com/read/maths/grade-11/trigonometry">https://intl.siyavula.com/read/maths/grade-11/trigonometry</a></li>
<li><a href="https://betterexplained.com/articles/intuitive-trigonometry/">https://betterexplained.com/articles/intuitive-trigonometry/</a></li>
</ol>
<h3>Images</h3>
<ol>
<li><a href="https://commons.wikimedia.org/wiki/File:Academ_Base_of_trigonometry.svg">https://commons.wikimedia.org/wiki/File:Academ_Base_of_trigonometry.svg</a></li>
</ol>
<p>Mathematical equations used in this article, where made using <a href="http://quicklatex.com/">quicklatex</a>.</p>
<hr>
<h2>Final words | Next up</h2>
And this is actually it for today's post!<br><br>
Next time we will cover how we solve any Triangle (non-right-angled), as well as Trigonometric identities, equations etc.<br><br>
Also, currently my ideas for "All About" articles include:
<ul>
  <li>Geometry</li>
  <li>Polynomial Arithmetic</li>
  <li>Exponentials and Logarithms</li>
  <li>Rational Expressions</li>
</ul>
Basically High-School Math Refreshers for Students in Universities or maybe even Middle-Age people that have problems helping their kids out in Math at school!<br><br>
See ya!
<p><img src="https://steemitimages.com/0x0/https://media.giphy.com/media/ybITzMzIyabIs/giphy.gif" width="500" height="333"/></p>
Keep on drifting!