📐 Similarity & Trigonometry
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Laws of Sines & Cosines

Solve non-right triangles using the Law of Sines and Law of Cosines.

When triangles are not right triangles, we need more powerful tools. The Law of Sines relates sides and opposite angles; the Law of Cosines generalizes the Pythagorean theorem. Together, they allow us to solve any triangle given sufficient information.

Law of Sines: asin⁡A=bsin⁡B=csin⁡C\text{Law of Sines: } \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
Law of Cosines: c2=a2+b2−2abcos⁡C\text{Law of Cosines: } c^2 = a^2 + b^2 - 2ab\cos C

Use Law of Sines when given:

AAS: two angles and any sideASA: two angles and the included sideSSA: two sides and a non-included angle (ambiguous!)

Use Law of Cosines when given:

SAS: two sides and the included angleSSS: all three sides

Law of Cosines avoids the ambiguous case of SSA.

📐Applying the Law of Cosines
In △ABC with a = 8, b = 5, C = 60°. Find c. c² = 64 + 25 − 2(8)(5)cos60° = 89 − 80(0.5) = 89 − 40 = 49. So c = 7.
c2=82+52−2(8)(5)cos⁡60∘=64+25−40=49  ⟹  c=7c^2 = 8^2 + 5^2 - 2(8)(5)\cos 60^\circ = 64 + 25 - 40 = 49 \implies c = 7
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Remember This!

After finding all three sides with the Law of Cosines, switch to the Law of Sines to find remaining angles — the arithmetic is simpler.

✏️ Try It!

In a triangle with angles A = 45°, B = 75°, and side a = 10, what is the value of b using the Law of Sines?

Practice with CalcVerse
🔷Oblique Triangle Solver🥧Arc Length & Sector Area
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