Solution of triangles

Key takeaways

Solution of triangles belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Solution of triangles to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Solution of triangles from memory before moving on to harder problems.

Reference excerpt

Solution of triangles (Latin: solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.

Solving plane triangles

A general form triangle has six main characteristics (see picture): three linear (side lengths a, b, c) and three angular (α, β, γ). The classical plane trigonometry problem is to specify three of the six characteristics and determine the other three. A triangle can be uniquely determined in this sense when given any of the following:

Three sides (SSS) Two sides and the included angle (SAS, side-angle-side) Two sides and an angle not included between them (SSA), if the side length adjacent to the angle is shorter than the other side length. A side and the two angles adjacent to it (ASA) A side, the angle opposite to it and an angle adjacent to it (AAS). For all cases in the plane, at least one of the side lengths must be specified. If only the angles are given, the side lengths cannot be determined, because any similar triangle is a solution.

Trigonomic relations

The standard method of solving the problem is to use fundamental relations.

Law of cosines a 2 = b 2 + c 2 − 2 b c cos ⁡ α b 2 = a 2 + c 2 − 2 a c cos ⁡ β c 2 = a 2 + b 2 − 2 a b cos ⁡ γ {\displaystyle {\begin{aligned}a^{2}&=b^{2}+c^{2}-2bc\cos \alpha \\b^{2}&=a^{2}+c^{2}-2ac\cos \beta \\c^{2}&=a^{2}+b^{2}-2ab\cos \gamma \end{aligned}}}

Law of sines a sin ⁡ α = b sin ⁡ β = c sin ⁡ γ {\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}}

Sum of angles

α + β + γ = 180 ∘ {\displaystyle \alpha +\beta +\gamma =180^{\circ }}

Law of tangents a − b a + b = tan ⁡ 1 2 ( α − β ) tan ⁡ 1 2 ( α + β ) . {\displaystyle {\frac {a-b}{a+b}}={\frac {\tan {\frac {1}{2}}(\alpha -\beta )}{\tan {\tfrac {1}{2}}(\alpha +\beta )}}.}

There are other (sometimes practically useful) universal relations: the law of cotangents and Mollweide's formula.

Notes To find an unknown angle, the law of cosines is safer than the law of sines. The reason is that the value of sine for the angle of the triangle does not uniquely determine this angle. For example, if sin β = 0.5, the angle β can equal either 30° or 150°. Using the law of cosines avoids this problem: within the interval from 0° to 180° the cosine value unambiguously determines its angle. On the other hand, if the angle is small (or close to 180°), then it is more robust numerically to determine it from its sine than its cosine because the arc-cosine function has a divergent derivative at 1 (or −1). We assume that the relative position of specified characteristics is known. If not, the mirror reflection of the triangle will also be a solution. For example, three side lengths uniquely define either a triangle or its reflection.

Three sides given (SSS)

Let three side lengths a, b, c be specified. To find the angles α, β, the law of cosines can be used:

… excerpt ends here. Continue reading the full article.

Illustrations

Solution of triangles: Three sides given — Three sides given

Solution of triangles: Two sides and the included angle given — Two sides and the included angle given

Solution of triangles: Two sides and a non-included angle given — Two sides and a non-included angle given

Solution of triangles: Two solutions for the triangle — Two solutions for the triangle

Worked examples

Example 1 — a first encounter with Solution of triangles

Start with the simplest possible case. Write down what Solution of triangles claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Solution of triangles before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Solution of triangles ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Solution of triangles

In research: Solution of triangles appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Solution of triangles in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Solution of triangles is common in secondary-school and first-year university syllabi. It links to neighbouring topics Spherical trigonometry, Triangle problems, Trigonometry, so understanding it makes those chapters shorter.
In everyday life: Look for Solution of triangles outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Solution of triangles in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Solution of triangles means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Solution of triangles out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Solution of triangles in simple terms?

Solution of triangles (Latin: solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere.

Why does Solution of triangles matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Solution of triangles?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Solution of triangles.