Trigonometry

      <p>Trigonometry is based on the Right-Angled Triangle. Remember <strong>PBH</strong> (Perpendicular combined with Base and Hypotenuse).</p>
      <ul class='list-disc pl-6'>
        <li>$\sin \theta = P/H$</li>
        <li>$\cos \theta = B/H$</li>
        <li>$\tan \theta = P/B$</li>
      </ul>
      <p class='mt-2'><strong>The Master Table (0 to 90):</strong> Memorize values of Sin. Cos is reverse of Sin. Tan = Sin/Cos.</p>
      <div class="overflow-x-auto mt-2">
        <table style="width: 100%; border-collapse: collapse; border: 1px solid #d1d5db; text-align: center; font-size: 0.875rem;">
          <thead style="background-color: #eff6ff;">
            <tr>
              <th class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">Angle</th>
              <th class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">0°</th>
              <th class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">30°</th>
              <th class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">45°</th>
              <th class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">60°</th>
              <th class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">90°</th>
            </tr>
          </thead>
          <tbody>
            <tr><td style="border: 1px solid #d1d5db; padding: 8px; font-weight: bold;">sin</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">0</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">1/2</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">1/$\sqrt{2}$</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">$\sqrt{3}$/2</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">1</td></tr>
            <tr><td style="border: 1px solid #d1d5db; padding: 8px; font-weight: bold;">cos</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">1</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">$\sqrt{3}$/2</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">1/$\sqrt{2}$</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">1/2</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">0</td></tr>
            <tr><td style="border: 1px solid #d1d5db; padding: 8px; font-weight: bold;">tan</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">0</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">1/$\sqrt{3}$</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">1</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">$\sqrt{3}$</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">Inf</td></tr>
            <tr><td style="border: 1px solid #d1d5db; padding: 8px; font-weight: bold;">cot</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">Inf</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">$\sqrt{3}$</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">1</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">1/$\sqrt{3}$</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">0</td></tr>
            <tr><td style="border: 1px solid #d1d5db; padding: 8px; font-weight: bold;">sec</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">1</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">2/$\sqrt{3}$</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">$\sqrt{2}$</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">2</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">Inf</td></tr>
            <tr><td style="border: 1px solid #d1d5db; padding: 8px; font-weight: bold;">cosec</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">Inf</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">2</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">$\sqrt{2}$</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">2/$\sqrt{3}$</td><td class="border border-gray-200 dark:border-slate-700/50 p-2 sm:p-3 text-gray-700 dark:text-gray-300">1</td></tr>
          </tbody>
        </table>
      </div>
    

      <p>This rule decides if a value is Positive or Negative depending on the Quadrant.</p>
      <div class="grid grid-cols-2 gap-2 mt-2 text-sm">
        <div class="border p-2 bg-primary-50">
          <strong>1st Quad (0-90)</strong><br/>
          ALL Positive (A)
        </div>
        <div class="border p-2 bg-yellow-50">
          <strong>2nd Quad (90-180)</strong><br/>
          Sin/Cosec Positive (S)
        </div>
        <div class="border p-2 bg-red-50">
          <strong>3rd Quad (180-270)</strong><br/>
          Tan/Cot Positive (T)
        </div>
        <div class="border p-2 bg-primary-50">
          <strong>4th Quad (270-360)</strong><br/>
          Cos/Sec Positive (C)
        </div>
      </div>
      <p class='mt-2'><strong>Transformation Rule:</strong></p>
      <ul class='list-disc pl-6'>
        <li><strong>90/270 (Odd Multiples):</strong> Change Function (Sin $\leftrightarrow$ Cos, Tan $\leftrightarrow$ Cot).</li>
        <li><strong>180/360 (Even Multiples):</strong> No Change.</li>
        <li><strong>Example:</strong> $\sin(90+\theta) = \cos \theta$ (2nd Quad, Sin is +ve).</li>
      </ul>
    

      <p>These are the tools to simplify complex expressions.</p>
      <ul class='list-disc pl-6 space-y-1'>
        <li>$\sin^2\theta + \cos^2\theta = 1$</li>
        <li>$1 + \tan^2\theta = \sec^2\theta$ (Note: $\sec^2 - \tan^2 = 1 \Rightarrow (\sec-\tan)(\sec+\tan)=1$)</li>
        <li>$1 + \cot^2\theta = \text{cosec}^2\theta$</li>
      </ul>
      <p class='mt-2'><strong>Pythagorean Triplets (Shortcut for H/D):</strong></p>
      <div class="grid grid-cols-2 gap-2 mt-2">
        <div class="border p-1 text-center bg-gray-50">3, 4, 5</div>
        <div class="border p-1 text-center bg-gray-50">5, 12, 13</div>
        <div class="border p-1 text-center bg-gray-50">8, 15, 17</div>
        <div class="border p-1 text-center bg-gray-50">7, 24, 25</div>
        <div class="border p-1 text-center bg-gray-50">20, 21, 29</div>
        <div class="border p-1 text-center bg-gray-50">$a^2-b^2, 2ab, a^2+b^2$</div>
      </div>
    

      <p>Convert hard questions into 5-second arithmetic.</p>
      <ul class='list-disc pl-6'>
        <li><strong>Form involves sin/cos only:</strong> Put $\theta = 0^\circ$ or $90^\circ$.</li>
        <li><strong>Form involves tan/cot/sec/cosec:</strong> Put $\theta = 45^\circ$ (Avoid undefined terms like $\tan 90$).</li>
        <li><strong>If options are numerical:</strong> Puttings works 100%.</li>
        <li><strong>Condition:</strong> Denominator must basic Identity not become zero.</li>
      </ul>
    

      <p>Use Ratios instead of Tan formulas.</p>
      <ul class='list-disc pl-6'>
        <li><strong>30-60-90 Triangle:</strong> Side opposite to $30^\circ$ is $1$, $60^\circ$ is $\sqrt{3}$, $90^\circ$ is $2$.</li>
        <li><strong>45-45-90 Triangle:</strong> Side opposite to $45^\circ$ is $1$, Hypotenuse is $\sqrt{2}$.</li>
        <li><strong>Broken Tree:</strong> Total height = Base + Hypotenuse.</li>
        <li><strong>Shadow Ratio:</strong> Height : Shadow = $1 : \sqrt{3}$ means Angle is $30^\circ$.</li>
      </ul>