If the angles a b c of a triangle are in a.P then find the value of cosb
Answers
Answer:
1/2
Given:
Angles of ΔABC are in AP.
To find:
Value of cos B.
Assumption:
⇒ ∠A = a
⇒ ∠B = a + d
⇒ ∠C = a + 2d
(a = first term; d = common difference)
⇒ ∠C = 90° (For finding cos B)
Concepts used:
⇒ Sum of angles of a triangle is 180°.
⇒ Ratio of sides of a triangle of angles 30°, 60° and 90° opposite to respective angles is 1 : √3 : 2.
Step-by-step explanation:
⇒ As the sum of angles of any triangle is 180°, then so is ∠A + ∠B + ∠C. Add these angles in the terms of 'a' and 'd'.
⇒ We get that ∠B = 60°. Consider the triangle of angles ∠A = 30°, ∠B = 60°, ∠C = 90°.
⇒ According to the ratio, as ∠A = 30° and ∠C = 90°, BC and AB becomes 1 and 2. So find the value of cos 60°. The adjacent side of 60° is opposite side of 30°, i.e., 1.
⇒ Thus the answer is 1/2.
-----------------------------------------------------------------
-----------------------------------------------------------------
Method:
⇒ ∠A + ∠B + ∠C = 180°
⇒ a + a + d + a + 2d = 180°
⇒ 3a + 3d = 180°
⇒ 3(a + d) = 180°
⇒ a + d = 180° / 3
⇒ a + d = 60°
⇒ ∠B = 60°
-----------------------------------------------------------------
⇒ ∠A = 180° - (∠B + ∠C)
⇒ ∠A = 180° - (60° + 90°)
⇒ ∠A = 180° - 150°
⇒ ∠A = 30°
-----------------------------------------------------------------
⇒ BC : AC : AB = 1 : √3 : 2
⇒ BC = 1; AC = √3; AB = 2
-----------------------------------------------------------------
⇒ cos 60° = BC / AB
⇒ cos 60° = 1/2
-----------------------------------------------------------------