**Law of Cosines and Sines Worksheet :**

Here we are going to see some practice questions on law of cosines and sines.

(1) In a triangle ABC, if sin A/sin C = sin(A − B)/sin(B − C), prove that a^{2}, b^{2}, c^{2} are in Arithmetic Progression.

(2) The angles of a triangle ABC, are in Arithmetic Progression and if b : c = √3 : √2, find ∠A.

(3) In a triangle ABC, if cos C = sin A / 2 sin B, show that the triangle is isosceles. Solution

(4) In a triangle ABC, prove that sin B/sinC = (c − a cosB)/(b − a cosC) Solution

(5) In a triangle ABC, prove that a cosA + b cosB + c cosC = 2a sinB sinC. Solution

(6) In a triangle ABC, ∠A = 60°. Prove that b + c = 2a cos (B − C)/2 Solution

In a triangle ABC, prove the following

(i) a sin (A/2 + B) = (b + c) sin A/2 Solution

(ii) a(cos B + cos C) = 2(b + c) sin^{2} A/2 Solution

(iii) (a^{2} − c^{2}) / b^{2} = sin(A − C) / sin(A + C) Solution

(iv) a sin(B − C)/(b^{2} − c^{2}) = b sin(C − A)/c^{2} − a^{2} = c sin(A − B)/(a^{2} − b^{2}) Solution

(v) (a + b)/(a − b) = tan (A + B)/2 cot (A − B)/2 Solution

(8) In a triangle ABC, prove that (a^{2} − b^{2} + c^{2}) tanB = (a^{2} + b^{2} − c^{2}) tanC. Solution

(9) An Engineer has to develop a triangular shaped park with a perimeter 120 m in a village. The park to be developed must be of maximum area. Find out the dimensions of the park. Solution

(10) A rope of length 12 m is given. Find the largest area of the triangle formed by this rope and find the dimensions of the triangle so formed Solution

(11) Derive Projection formula from (i) Law of sines, (ii) Law of cosines. Solution

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