MP Board · Class 10 · Mathematics · TrianglesState and prove the Sine Rule and Cosine Rule for any triangle. Explain, with clear headings and bullet points, how each rule is derived and discuss at least two different types of problems where these rules are applied in the study of triangles.

1. Sine Rule (Law of Sines)

• Statement: For any triangle (\triangle ABC) with sides (a, b, c) opposite to angles (A, B, C) respectively, [\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R] where (R) is the radius of the circum‑circle.
• Proof:
  1. Draw the circum‑circle of (\triangle ABC) and denote its centre as (O).
  2. Join (O) to the vertices (A, B, C). Each side becomes a chord of the circle.
  3. Using the definition of a chord, the length of a chord is (2R\sin\theta) where (\theta) is the subtended angle at the centre.
  4. Hence, (a = 2R\sin A), (b = 2R\sin B), (c = 2R\sin C).
  5. Dividing each equation by the corresponding sine gives the required ratio.

2. Cosine Rule (Law of Cosines)

• Statement: For the same triangle, [a^{2}=b^{2}+c^{2}-2bc\cos A] and cyclic permutations for (b^{2}) and (c^{2}).
• Proof:
  1. Drop a perpendicular from (A) to side (BC) meeting it at (D). Let (AD = h) and (BD = x). Then (DC = b - x).
  2. From right‑angled triangles (\triangle ABD) and (\triangle ADC):
     • (a^{2}=h^{2}+x^{2})
     • (c^{2}=h^{2}+(b-x)^{2})
  3. Subtract the first equation from the second and simplify to obtain (c^{2}=a^{2}+b^{2}-2ab\cos C). Similar steps give the other forms.

3. Applications

• Finding an unknown side: When two sides and the included angle are known, the cosine rule directly yields the third side.
• Solving ambiguous case: In SSA (two sides and a non‑included angle) the sine rule helps determine possible angles, and the cosine rule checks feasibility.
• Area calculation: Combining the sine rule with (\text{Area}=\frac{1}{2}ab\sin C) provides a method to compute area when only angles and one side are given.
• Real‑world problems: Navigation, surveying, and physics problems involving vectors often rely on these rules to resolve components.

Summary: Both rules are extensions of the basic trigonometric relationships in right‑angled triangles, allowing us to work with any triangle. Mastery of their derivations reinforces understanding of circle geometry and the interplay between sides and angles.