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59 lines
2.8 KiB
HTML
59 lines
2.8 KiB
HTML
<div class="book-prose">
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<h2 id="history" class="book-prose__heading">History</h2>
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<p>
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Trigonometry grew from astronomy, navigation, and surveying. Mathematicians in ancient Greece,
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India, and the Islamic Golden Age developed angle and chord/sine tables for practical measurement.
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</p>
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<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
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<h3 id="similarity-idea" class="book-prose__heading">Why trig ratios depend only on angle</h3>
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<p>
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Any two right triangles with the same acute angle are similar, so corresponding side ratios are
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equal. That is why \(\sin\theta,\cos\theta,\tan\theta\) are functions of \(\theta\) only.
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</p>
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<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
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<details class="book-question">
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<summary class="book-question__prompt">Find \(\sin30^\circ\) and \(\cos60^\circ\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> Both are \(\frac12\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">In a right triangle, opposite to \(\theta\) is 9 and hypotenuse is 15. Find \(\sin\theta\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(\sin\theta=\frac{9}{15}=\frac35\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Adjacent is 7, hypotenuse is 25. Find \(\cos\theta\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(\cos\theta=\frac{7}{25}\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">If \(\tan\theta=0.75\), find \(\theta\) (to 1 d.p.).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(\theta=\tan^{-1}(0.75)\approx 36.9^\circ\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">A ladder leans against a wall. Foot is 4 m from wall and ladder length is 6 m. Find angle with ground.</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(\cos\theta=\frac{4}{6}=\frac23\Rightarrow \theta\approx 48.2^\circ\).</p>
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</div>
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</details>
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<h2 id="applications" class="book-prose__heading">Applications</h2>
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<ul>
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<li><strong>Surveying:</strong> height and distance estimation without direct measurement.</li>
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<li><strong>Navigation:</strong> direction and angle-based positioning.</li>
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<li><strong>Engineering:</strong> force components and incline analysis.</li>
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</ul>
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<h2 id="references" class="book-prose__heading">References</h2>
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<ul>
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<li>Standard high-school trigonometry curricula (right-triangle ratios).</li>
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<li>Geometry and trigonometry texts on SOH-CAH-TOA applications.</li>
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</ul>
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</div>
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