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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Trigonometry - Deep Study</title>
<style>
* {
margin: 0;
padding: 0;
box-sizing: border-box;
}
body {
font-family: 'Arial', sans-serif;
background-color: #f8f9fa;
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line-height: 1.6;
padding: 20px;
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header {
background-color: #007bff;
color: #fff;
padding: 20px;
text-align: center;
border-radius: 8px;
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header h1 {
font-size: 2.5rem;
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.content {
background-color: #ffffff;
padding: 30px;
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max-width: 1200px;
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</style>
</head>
<body>
<header>
<h1>Trigonometry: An In-depth Study</h1>
</header>
<div class="content">
<h2>1. The Basics of Trigonometry</h2>
<p>Trigonometry primarily deals with right-angled triangles, where one angle is <b>90°</b>. In any right-angled triangle, the sides are categorized as follows:</p>
<ul>
<li><b>Hypotenuse (h):</b> The longest side, opposite the right angle.</li>
<li><b>Opposite side (o):</b> The side opposite to the angle of interest.</li>
<li><b>Adjacent side (a):</b> The side next to the angle of interest, but not the hypotenuse.</li>
</ul>
<h3>Fundamental Trigonometric Ratios</h3>
<p>In a right-angled triangle, the three basic trigonometric functions are defined as:</p>
<ul>
<li><b>Sine (sin):</b> <code>sin(θ) = opposite / hypotenuse</code></li>
<li><b>Cosine (cos):</b> <code>cos(θ) = adjacent / hypotenuse</code></li>
<li><b>Tangent (tan):</b> <code>tan(θ) = opposite / adjacent</code></li>
</ul>
<h3>The Reciprocal Functions</h3>
<ul>
<li><b>Cosecant (csc):</b> <code>csc(θ) = 1 / sin(θ) = hypotenuse / opposite</code></li>
<li><b>Secant (sec):</b> <code>sec(θ) = 1 / cos(θ) = hypotenuse / adjacent</code></li>
<li><b>Cotangent (cot):</b> <code>cot(θ) = 1 / tan(θ) = adjacent / opposite</code></li>
</ul>
<h2>2. Trigonometric Identities</h2>
<h3>Pythagorean Identity</h3>
<p>The most fundamental trigonometric identity is the Pythagorean identity:</p>
<code>sin²(θ) + cos²(θ) = 1</code>
<h3>Other Pythagorean Identities</h3>
<ul>
<li><code>1 + tan²(θ) = sec²(θ)</code></li>
<li><code>1 + cot²(θ) = csc²(θ)</code></li>
</ul>
<h3>Reciprocal Identities</h3>
<ul>
<li><code>csc(θ) = 1 / sin(θ), sec(θ) = 1 / cos(θ), cot(θ) = 1 / tan(θ)</code></li>
</ul>
<h3>Quotient Identities</h3>
<ul>
<li><code>tan(θ) = sin(θ) / cos(θ), cot(θ) = cos(θ) / sin(θ)</code></li>
</ul>
<h2>3. Trigonometric Graphs</h2>
<p>The trigonometric functions can be graphed on a coordinate plane, where the x-axis represents the angle (in degrees or radians), and the y-axis represents the value of the trigonometric function.</p>
<ul>
<li><b>Sine Graph:</b> The sine function is periodic with a period of 360° or 2π radians. It oscillates between -1 and 1.</li>
<li><b>Cosine Graph:</b> Like the sine graph, the cosine graph is also periodic, oscillating between -1 and 1.</li>
<li><b>Tangent Graph:</b> The tangent function has a period of 180° or π radians. It has vertical asymptotes where cos(x) = 0.</li>
<li><b>Cotangent, Secant, and Cosecant:</b> These functions also have periodic graphs and asymptotes at specific points.</li>
</ul>
<h2>4. Applications of Trigonometry</h2>
<ul>
<li><b>Physics:</b> Trigonometry is used to model wave functions, analyze forces, and solve problems in mechanics.</li>
<li><b>Engineering:</b> Trigonometry is critical in the design of buildings, bridges, and roads, especially in structural engineering and the study of oscillations.</li>
<li><b>Astronomy:</b> Trigonometry is used to calculate distances and angles in space, especially in measuring the positions of celestial bodies.</li>
<li><b>Computer Graphics:</b> Trigonometry is used to calculate rotations, translations, and scaling of objects in 2D and 3D graphics.</li>
</ul>
<h2>5. Special Angles and Their Trigonometric Values</h2>
<ul>
<li><b>30° (π/6 radians):</b>
<ul>
<li>sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3</li>
</ul>
</li>
<li><b>45° (π/4 radians):</b>
<ul>
<li>sin(45°) = √2/2, cos(45°) = √2/2, tan(45°) = 1</li>
</ul>
</li>
<li><b>60° (π/3 radians):</b>
<ul>
<li>sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3</li>
</ul>
</li>
<li><b>90° (π/2 radians):</b>
<ul>
<li>sin(90°) = 1, cos(90°) = 0, tan(90°) = undefined</li>
</ul>
</li>
</ul>
</div>
<footer>
<p>© 2025 Yoseph Feyisa Wegi | All Rights Reserved</p>
</footer>
</body>
</html>