AP EAMCET · Maths · Trigonometric Ratios & Identities

The sides of a triangle inscribed in a given circle subtend angles $α, β, γ$ at the center. The minimum value of the $A . M .$ of $\cos (α + \frac{π}{2}), \cos (β + \frac{π}{2})$ and $\cos (γ + \frac{π}{2})$ is equal to

A $\frac{\sqrt{3}}{2}$
B $\frac{- \sqrt{3}}{2}$
C $\frac{- 2}{\sqrt{3}}$
D $\sqrt{2}$

Verified Solution

Answer & Solution

Correct Answer

(B) $\frac{- \sqrt{3}}{2}$

Step-by-step Solution

Detailed explanation

Let the triangle is ABC, Since the angle made by a chord at the center of a circle is twice the angle made by the chord at the circumference, we get ∠A=α2,∠B=β2 ,∠C=γ2⇒α+β+γ=2π…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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The sides of a triangle inscribed in a given circle subtend angles α,β,γ at the center. The minimum value of the A.M. of cosα+π2,cosβ+π2 and cosγ+π2 is equal to

Step-by-step Solution

The sides of a triangle inscribed in a given circle subtend angles $α, β, γ$ at the center. The minimum value of the $A . M .$ of $\cos (α + \frac{π}{2}), \cos (β + \frac{π}{2})$ and $\cos (γ + \frac{π}{2})$ is equal to