좌표평면 위의 점으로 삼각함수 값 계산하기

문제

좌표평면에서 원점 $\mathrm{O}$와 점 $\mathrm{P}(-12, 5)$를 잇는 선분 $\mathrm{OP}$가 나타내는 각을 $\theta$라고 할 때, $13 \sin \theta + 5 \tan \theta$의 값은?

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<!-- Coordinate System Setup -->
<!-- Math origin (0,0) maps to SVG (200, 150) -->
<!-- Scale: 1 math unit = 10 SVG pixels -->

<!-- SVG Coordinates for key points: -->
<!-- O (0,0) -> (200, 150) -->
<!-- P (-12, 5) -> (200 + (-12)*10, 150 - 5*10) = (80, 100) -->
<!-- H (-12, 0) -> (200 + (-12)*10, 150) = (80, 150) -->

<!-- X-axis -->
<line x1="50" y1="150" x2="350" y2="150" class="main-line"/>
<text x="340" y="165" class="label">x</text>
<!-- Y-axis -->
<line x1="200" y1="50" x2="200" y2="250" class="main-line"/>
<text x="215" y="55" class="label">y</text>

<!-- Ticks & Labels on Axes -->
<!-- X-axis ticks -->
<line x1="80" y1="148" x2="80" y2="152" class="main-line"/> <text x="80" y="165" class="label small">-12</text>
<line x1="100" y1="148" x2="100" y2="152" class="main-line"/> <text x="100" y="165" class="label small">-10</text>
<line x1="150" y1="148" x2="150" y2="152" class="main-line"/> <text x="150" y="165" class="label small">-5</text>
<line x1="250" y1="148" x2="250" y2="152" class="main-line"/> <text x="250" y="165" class="label small">5</text>
<!-- Y-axis ticks -->
<line x1="198" y1="100" x2="202" y2="100" class="main-line"/> <text x="185" y="105" class="label small right-align">5</text>
<line x1="198" y1="200" x2="202" y2="200" class="main-line"/> <text x="185" y="205" class="label small right-align">-5</text>

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<text x="208" y="165" class="label">O</text>

<!-- Point P(-12, 5) -->
<circle cx="80" cy="100" r="3" class="point"/>
<text x="80" y="90" class="label">P(-12, 5)</text>

<!-- Line segment OP (highlighted as it's key to the angle) -->
<line x1="200" y1="150" x2="80" y2="100" class="secondary-line"/>

<!-- Projection from P to x-axis to form a right triangle -->
<line x1="80" y1="100" x2="80" y2="150" class="projection-line"/> <!-- Line PH -->
<!-- Label for point H (projection) -->
<circle cx="80" cy="150" r="2" class="point" fill="#6C63FF"/>
<text x="80" y="170" class="label small">H</text>

<!-- Angle theta -->
<!-- Arc radius `r_arc = 20`. -->
<!-- End point of arc on OP: x_end_arc = 200 + 20*(-12/13) = 181.54, y_end_arc = 150 - 20*(5/13) = 142.31 -->
<!-- Start point of arc on positive x-axis: (200 + 20, 150) = (220, 150) -->
<!-- large-arc-flag = 0 (angle < 180), sweep-flag = 1 (counter-clockwise) -->
<path class="secondary-line"
      d="M 220 150 A 20 20 0 0 1 181.54 142.31"/>
<text x="180" y="130" class="label highlight">θ</text> <!-- Label for theta -->

<!-- Labels for lengths of the right triangle -->
<!-- Length OH = |-12| = 12 (horizontal distance) -->
<text x="140" y="165" class="label highlight">12</text>
<!-- Length PH = |5| = 5 (vertical distance) -->
<text x="70" y="125" class="label highlight">5</text>
<!-- Length OP = R = 13 (hypotenuse) -->
<text x="140" y="115" class="label highlight">13</text>