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61 lines
2.4 KiB
HTML
61 lines
2.4 KiB
HTML
<div class="book-prose">
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<p>
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Trigonometry relates angles and side lengths in right triangles. The three core ratios are sine,
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cosine, and tangent.
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</p>
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<h2 id="right-triangle-ratios" class="book-prose__heading">Right-triangle ratios</h2>
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<p>
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For angle \(\theta\) in a right triangle:
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</p>
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<p class="book-formula">\[
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\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\quad
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\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\quad
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\tan\theta=\frac{\text{opposite}}{\text{adjacent}}.
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\]</p>
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<p>
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Remember: opposite/adjacent depend on the chosen angle.
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</p>
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<h2 id="basic-values" class="book-prose__heading">Common exact values</h2>
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<p>
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For \(30^\circ,45^\circ,60^\circ\):
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</p>
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<ul>
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<li>\(\sin30^\circ=\frac12,\ \cos30^\circ=\frac{\sqrt3}{2},\ \tan30^\circ=\frac1{\sqrt3}\)</li>
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<li>\(\sin45^\circ=\cos45^\circ=\frac{\sqrt2}{2},\ \tan45^\circ=1\)</li>
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<li>\(\sin60^\circ=\frac{\sqrt3}{2},\ \cos60^\circ=\frac12,\ \tan60^\circ=\sqrt3\)</li>
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</ul>
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<h2 id="solve-sides" class="book-prose__heading">Finding missing sides</h2>
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<p><strong>Example.</strong> Hypotenuse \(=10\), angle \(=37^\circ\). Find opposite side \(x\):</p>
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<p class="book-formula">\[
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\sin37^\circ=\frac{x}{10}\Rightarrow x=10\sin37^\circ\approx 6.02.
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\]</p>
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<h2 id="solve-angles" class="book-prose__heading">Finding missing angles</h2>
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<p><strong>Example.</strong> Opposite \(=5\), adjacent \(=12\). Find \(\theta\):</p>
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<p class="book-formula">\[
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\tan\theta=\frac{5}{12}\Rightarrow \theta=\tan^{-1}\!\left(\frac{5}{12}\right)\approx 22.6^\circ.
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\]</p>
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<h2 id="elevation-depression" class="book-prose__heading">Angles of elevation and depression</h2>
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<p>
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Angle of elevation looks upward from horizontal; angle of depression looks downward. Draw a clear
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diagram and label horizontal lines before forming trig equations.
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</p>
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<h2 id="pythagoras-link" class="book-prose__heading">Link with Pythagoras</h2>
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<p>
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In right triangles, trigonometry and Pythagoras work together: first find one side/angle, then use
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the other method to complete the triangle.
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</p>
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<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
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<ul>
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<li>Choose ratio from known and required sides (SOH-CAH-TOA).</li>
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<li>Set calculator mode to degrees unless stated otherwise.</li>
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<li>Round only at final step unless instructed.</li>
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<li>Include units and context in word-problem answers.</li>
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</ul>
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</div>
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