Answer: Half-angle sine formula using s (semi-perimeter).
- A Tangent half-angle formula expressed using the semi-perimeter s
- B Half-angle sine formula using s (semi-perimeter)
- C A variant of the cosine rule rearranged in terms of s
- D Projection formula relating a side to the cosines of the other angles
Correct answer: B. Half-angle sine formula using s (semi-perimeter)
Explanation: Half-angle sine formula using semi-perimeter s=(a+b+c)/2. Derived by combining cosine rule cosA=(b²+c²−a²)/2bc with cos A=1−2sin²(A/2): substituting a²=(b+c)²−4bc·sin²(A/2) and simplifying yields sin(A/2)=√[(s−b)(s−c)/bc].
Unit circle with standard angles 0, 30, 45, 60, 90 degrees; at 45 degrees the point is (cos45, sin45) = (1/sqrt2, 1/sqrt2).
Concept context
Ratios, identities, and applications in triangles