How to Apply Trigonometry to General Triangles?
TL;DR: Stuck on a triangle that doesn’t have a right angle? You’ve got two tools built for exactly this. The Law of Sines says the ratio of each side to the sine of its opposite angle stays constant across the triangle. The Law of Cosines is a souped-up Pythagorean theorem that handles any included angle. Which one you reach for depends on what you’re given — SSS or SAS calls for cosines, while ASA and AAS belong to sines. Pick the right tool and the triangle solves itself.
Key takeaways:
- Law of Sines: \(\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}\).
- Law of Cosines: \(c^2 = a^2 + b^2 – 2ab\cos C\).
- Use Law of Sines for ASA, AAS, and SSA (ambiguous case).
- Use Law of Cosines for SSS and SAS.
- Angles in any triangle add to \(180^\circ\) — use this to find a missing angle quickly.
\(cos B =\frac {c^2+a^2-b^2}{2ca}\)
\(cos β = \frac {19^2+11^2-23^2}{2(19)(11)}=\frac{361+121-529}{418}=\frac{-47}{418}=-0.112\)
\( β ≅ 96.45^\circ\)
Then, use the law of sines to find the size of the smallest angle \((α)\):
\(\frac {sin A}{a}=\frac {sin B}{b}=\frac {sin C}{c}\)
\(\frac {sin\ α }{11}=\frac {sin\ 96.45}{23}\)
\( sin\ α =\frac{11× sin\ 96.45}{23}=\frac {11 × 0.99}{23}=\frac {10.89}{23}=0.47\)
\( α ≅ 28.37^\circ\)
Now, the difference between the largest and smallest angle is:
\( 96.45^\circ\ – 28.37^\circ=68.08 ^\circ\ \)
Solving General Triangles – Example 2:
(SAS) Find the length of the third side of the triangle.
To find side \(c\) use the law of cosines: \(c^2=a^2+b^2-2ab.\cos C\)
\(c^2=31^2+31^2-2(31)(31)(cos 17)=961+961-(1922 × cos 17)=1,922-(1,922 × 0.96)=1,922-1845.12=76.88\)
\(c^2=76.88→ c=\sqrt{76.88}= 8.77\)
Exercises for Solving General Triangles
- (AAS) Find the value of \(x\) in the triangle.
- (ASA) Find the value of \(d\) in the triangle.
- \(\color{blue}{83.32}\)
- \(\color{blue}{311.95}\)
- \(\color{blue}{19.44}\)
The Best Books to Ace the Trigonometry Test
Recommended EffortlessMath Books
For a trigonometry workbook that covers Law of Sines, Law of Cosines, and identities with worked examples, Trigonometry for Beginners walks through every topic from right triangles to general triangles. For a wider algebra-through-precalculus review that includes trig, The Ultimate Algebra Bundle covers the prerequisites you need first.
Frequently Asked Questions
When do I use the Law of Sines vs the Law of Cosines?
Law of Sines works when you have an angle and its opposite side, plus one other piece (ASA, AAS, SSA). Law of Cosines works when you have all three sides (SSS) or two sides and the angle between them (SAS).
What is the Law of Sines?
\(\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}\). Each angle and its opposite side form a ratio, and all three ratios are equal. Useful any time you know an angle-side pair plus one more piece.
What is the Law of Cosines?
\(c^2 = a^2 + b^2 – 2ab\cos C\). Solve for the side opposite the angle you know. You can also rearrange to find an angle: \(\cos C = \dfrac{a^2 + b^2 – c^2}{2ab}\).
What’s the ambiguous case (SSA)?
When you’re given two sides and a non-included angle, there can be zero, one, or two triangles that fit. After using the Law of Sines, check whether the supplementary angle (\(180^\circ – \) your angle) also gives a valid triangle whose angles add to less than \(180^\circ\). If yes, both triangles are valid.
Do these laws work for right triangles too?
Yes, but it’s overkill. For right triangles, basic SOHCAHTOA and the Pythagorean theorem are faster. Save the Law of Sines and Cosines for triangles without a \(90^\circ\) angle.
How do I find the area of a triangle without a right angle?
Use \(\text{Area} = \tfrac{1}{2}ab\sin C\), where \(a\) and \(b\) are two sides and \(C\) is the angle between them. If you only have the three sides, use Heron’s formula instead.
What’s Heron’s formula?
\(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\), where \(s = \tfrac{a+b+c}{2}\) is the semi-perimeter. Useful when you have all three sides and no angle.
How precise should my answers be?
Carry at least four decimal places through the work and round only at the end. Most tests want angles to the nearest tenth of a degree and sides to two decimal places. Rounding mid-problem stacks small errors into big ones.
My calculator gives the wrong angle. What went wrong?
The \(\sin^{-1}\) function only returns angles between \(-90^\circ\) and \(90^\circ\). In the SSA case, the actual angle might be the supplement (\(180^\circ – \) the calculator’s value). Always check both possibilities.
Where do general triangle problems show up on tests?
SAT (rarely), ACT, Precalculus and Trigonometry final exams, AP Precalculus, ALEKS placement tests, and almost every college calculus prerequisite test. Expect at least one Law of Sines or Cosines problem on any trig-heavy test.
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