Triangle solver

<p class="paragraph-header">Match the known information</p>
<p>In triangle \(ABC\), sides \(a,b,c\) are opposite angles \(A,B,C\), respectively.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Information in a triangle</strong></th><th style="text-align: center; --cell-justify: center"><strong>Useful relation</strong></th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">Right angle and two relevant sides or an acute angle</td><td style="text-align: left; --cell-justify: flex-start">\(\sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">A known side–opposite-angle pair</td><td style="text-align: left; --cell-justify: flex-start">\(\dfrac a{\sin A}=\dfrac b{\sin B}\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">Two sides and their included angle</td><td style="text-align: left; --cell-justify: flex-start">\(c^2=a^2+b^2-2ab\cos C\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">Area from two sides and their included angle</td><td style="text-align: left; --cell-justify: flex-start">\(\text{Area}=\dfrac12ab\sin C\)</td></tr></tbody></table></div></div>
<p>The shortest distance from a point to a straight line is the perpendicular distance. If the question refers to a path segment, check that the perpendicular foot lies on that segment.</p>
<p>For a practical decision, distinguish the calculated geometric length from any allowance for fittings, waste or clearance. Compare unrounded quantities with the stated limit; a rounded answer can hide a small shortfall.</p>
<p>Label a sketch before choosing a formula. Work in consistent units and state the assumptions that make a right angle or horizontal plane available. A diagram in perspective does not display true spatial angles.</p>

In triangle \(ABC\), sides \(a,b,c\) are opposite angles \(A,B,C\), respectively.

Right angle and two relevant sides or an acute angle

\(\sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}\)

Area from two sides and their included angle

The shortest distance from a point to a straight line is the perpendicular distance. If the question refers to a path segment, check that the perpendicular foot lies on that segment.

For a practical decision, distinguish the calculated geometric length from any allowance for fittings, waste or clearance. Compare unrounded quantities with the stated limit; a rounded answer can hide a small shortfall.

Label a sketch before choosing a formula. Work in consistent units and state the assumptions that make a right angle or horizontal plane available. A diagram in perspective does not display true spatial angles.

Selecting a method and interpreting a result

Selecting a method and interpreting a result

Information in a triangle	Useful relation
Right angle and two relevant sides or an acute angle	\(\sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}\)
A known side–opposite-angle pair	\(\dfrac a{\sin A}=\dfrac b{\sin B}\)
Two sides and their included angle	\(c^2=a^2+b^2-2ab\cos C\)
Area from two sides and their included angle	\(\text{Area}=\dfrac12ab\sin C\)