<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</a>, <a href="https://peakd.com/hive-196387/@drifter1/mathematics-all-about-trigonometry-part-2">second</a> and <a href="https://peakd.com/hive-196387/@drifter1/mathematics-all-about-trigonometry-part-3">third</a> part before getting into this one.
Today we will cover Trigonometric Identies and Equations!<br><br>
So, without further ado, let's get straight into it!
</p>
<hr>
<h2>Trigonometric Identities</h2>
<p>
After covering basic Trigonometry, and solving right and general (non-right) triangles, it's time to get even more advanced!
First off, let's cover Trigonometric Identities, which are equations that contain only Trigonometric Functions.
Trigonometric Identities are useful for simplifing expressions, so that they contain only sine and cosine ratios.
</p>
<h3>Quotient Identity</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>
<p>
For a right triangle with an angle <em>θ</em>, we defined that the <strong>main trigonometric functions</strong> are:<br><br>
<img src="https://quicklatex.com/cache3/e4/ql_186a532886243c8a295b6a46b0614ae4_l3.png">
</p>
<p>
Dividing Sine by Cosine we get:<br><br>
<img src="https://quicklatex.com/cache3/9a/ql_5fc7e7dc0087962b1d014a5e7d5d959a_l3.png">
</p>
<p>
And so the first <strong>identity</strong> is:<br><br>
<img src="https://quicklatex.com/cache3/3b/ql_1a2f9d5eb3e09a9013a2d91658c91b3b_l3.png">
</p>
<p>
Something similar is also true for the cosecant (csc), secant (sec) and cotangent (cot) functions:<br><br>
<img src="https://quicklatex.com/cache3/30/ql_33808db7e0745f8d869405335753cb30_l3.png">
</p>
<h3>Square Identity</h3>
<p>
From Pythagoras's Theorem we can also derive an useful identity, that we already covered in a previous article.
There we used the Unit circle and the values of the axes, but the same identiy can also be proven using a different approach.
</p>
<p>
In a right-triangle with Hypotenuse <em>c</em>, dividing <em>a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></em> by <em>c<sup>2</sup></em> gives us:<br><br>
<img src="https://quicklatex.com/cache3/bc/ql_6b68b737f5f9c37776607578d1eefebc_l3.png">
</p>
<p>
Similarly, it's also possible to define the following Identities:<br><br>
<img src="https://quicklatex.com/cache3/91/ql_b60b962b5b9c9d5d509e828138e86e91_l3.png">
</p>
<h3>Opposite Angle Idenitites</h3>
<p>
In a previous article we covered how the values of the trigonometric functions change as we move from Quadrant to Quadrant.
The values follow the pattern: "A-S-T-C" or "All-Sine-Tangent-Cosine", giving the following identities as a result:<br><br>
<img src="https://quicklatex.com/cache3/5e/ql_7957bee93c77ff1d689f308505606c5e_l3.png">
</p>
<h3>Double Angle Identities</h3>
<p>
If we double the angle <em>θ</em> that is "fed into" each trigonometric function the result can also be written in respect to <em>θ</em> as follows:<br><br>
<img src="https://quicklatex.com/cache3/5c/ql_efbe3b933649fce54db3eababe33e05c_l3.png">
</p>
<h3>Half Angle Identities</h3>
<p>
Similarly, it's also possible to write half the angle <em>θ</em> in respect to trigonometric functions containinig <em>θ</em>:<br><br>
<img src="https://quicklatex.com/cache3/40/ql_f67b1e7220e14ad5d666ab96f5a20740_l3.png">
</p>
<h3>Angle Sum and Difference Identities</h3>
<p>
The result of an trigonometric function that takes in a sum of two angles <em>A</em> and <em>B</em> can be written using trigonometric functions that contain only <em>A</em> and <em>B</em> respectively:<br><br>
<img src="https://quicklatex.com/cache3/2c/ql_62080c1bc6c03f4056a72dcc7aab502c_l3.png"><br><br>
In the case of the sine function, that same sign has to be used in both sides!
</p>
<h3>Product to Sum Identities</h3>
<p>
A product between trigonometric functions of possibly different angles, <em>x</em> and <em>y</em>, can be turned into a sum:<br><br>
<img src="https://quicklatex.com/cache3/f7/ql_f0aefaa2d330a56aca679bc1161911f7_l3.png">
</p>
<h3>Sum to Product Identities</h3>
<p>
A sum between trigonometric functions of possible different angles, <em>x</em> and <em>y</em>, can be turned into a product:<br><br>
<img src="https://quicklatex.com/cache3/f9/ql_76cf0583fd42cbcc586adf85c892f0f9_l3.png">
</p>
<h3>Laws of General Triangles</h3>
<p>
Of course the Law of Sines and Law of Cosines can also be used as Idenitites.
And there is also something called the Law of Tangents, that we didn't cover before.
</p>
<p>
Mathematically:<br><br>
<img src="https://quicklatex.com/cache3/4e/ql_47e3797b00f0384ab07e933760031d4e_l3.png">
</p>
<hr>
<h2>Solving Trigonometric Equations</h2>
<h3>Equal Sines, Cosines and Tangents</h3>
<p>
Two angles <em>θ</em> and <em>φ</em>, which are both less than 360°, give us the same result for the trigonometric functions.
How are those two angles related?
</p>
<h4>Sine</h4>
<p>
<img src="https://quicklatex.com/cache3/c6/ql_ba53316a2ff4febc1c2d16f6762d49c6_l3.png"><br><br>
Sine has the same value in Quadrants A-B (positive) and C-D (negative), meaning that <em>θ = 180° - φ</em> or <em>φ = 180° - θ</em>.
</p>
<h4>Cosine</h4>
<p>
<img src="https://quicklatex.com/cache3/09/ql_6778f2c72957662055b952d69bc13409_l3.png"><br><br>
Cosine has the same value in Quadrants A-D (positive) and B-C (negative), meaning that <em>θ = 360° - φ</em> or <em>φ = 360° - θ</em> 
</p>
<h4>Tangent</h4>
<p>
<img src="https://quicklatex.com/cache3/8c/ql_e524dbf1db077a5fc4cce21f9d656f8c_l3.png"><br><br>
Tangent has the same value in Quadrants A-C (positive) and B-D (negative), meaning that <em>θ = φ - 180°</em> or <em>φ = θ - 180°</em> 
</p>
<h3>Solutions of a simple Trigonometric Equation (Example of Reference 5)</h3>
<p>
Consider the following equation:<br><br>
<img src="https://quicklatex.com/cache3/89/ql_02630e6964916f309693aa78a804fb89_l3.png"><br><br>
What is the value of <em>x</em>?
</p>
<h4>Solution</h4>
<p>
The Cosinus function gives <em>1 / 2</em> in the case of 60° degrees.
The same is also true for <em>360° - 60° = 300°</em>, because Quadrant A and D have the same value.
Using the idenitity of the opposite angle (<em>cos(-θ) = cos(θ)</em>), we also get <em>-60°</em> and <em>-300°</em>.
</p>
<p>
Thus:<br><br>
<img src="https://quicklatex.com/cache3/82/ql_b462cf01be2c13c6f46ee120a7851482_l3.png">
</p>
<p>
Using Trigonometric Identities this way, any equation is solveable or simplifiable!
</p>
<h3>General Solutions of a Trigonometric Equation (Example 17 of Reference 7)</h3>
<p>
Consider the following equation:<br><br>
<img src="https://quicklatex.com/cache3/77/ql_780de2e273cfa7b103bc8c4c9504c377_l3.png"><br><br>
Let's find the general solution for <em>a</em>.
</p>
<h4>Solution</h4>
<p>
For an angle <em>x = 2a - 10°</em>, the tangent <em>tan(x) = 2.5</em> and so one solution is:<br><br>
<img src="https://quicklatex.com/cache3/bf/ql_03e492eef35dcf9cc547c04b01c16abf_l3.png"><br><br>
Solutions in the first quadrant:<br><br>
<img src="https://quicklatex.com/cache3/f6/ql_800a842b7987d1992325ade0de556df6_l3.png"><br><br>
Solutions in the third quadrant:<br><br>
<img src="https://quicklatex.com/cache3/53/ql_d2a8a13959090c0af23a8eafd8dbcb53_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>
</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 and this small series about Trigonometry!<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!