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Conditional Trigonometric Identities Worksheet


(1)  If A + B + C = 180°, show that 

(i) sin 2A + sin2B + sin2C = 4 sin A sin B sin C

Answer

(ii)  cos A + cos B − cos C = −1 + 4cos(A/2)cos(B/2)sin(C/2)

Answer

(iii) sin2 A + sin2 B + sin2 C = 2 + 2cosAcosB cosC

Answer

(iv)  sin2 A + sin2 B − sin2 C = 2 sin A sin B cos C

Answer

(v)  tan A/2 tan B/2 + tan B/2 tan C/2 + tan C/2 tan A/2 = 1

Answer

(vi)  sinA + sinB + sinC = 4cos A/2 cos B/2 cos C/2

Answer

(vii) sin(B + C − A) + sin(C + A − B) + sin(A + B − C) = 4sinAsinB sinC.                Answer

(2)  If A + B + C = 2s, then show that sin(s − A) sin(s − B) + sins sin(s − C) = sin A sin B.          Answer

(3)  If x + y + z = xyz, then show that

(2x/1 − x2) + (2y/1 − y2) + (2z/1 − z2) = (2x/1 − x2) (2y/1 − y2) (2z/1 − z2)    Answer

(4)  If A + B + C = π/2, show the next

(i) sin 2A + sin2B + sin2C = 4cosAcosB cosC      Answer

(ii) cos 2A + cos2B + cos2C = 1 + 4sinAsinB cosC 

Answer 

(5)  If triangle ABC is a proper triangle and if ∠A = π/2, then show that

(i) cos2 B + cos2 C = 1

(ii) sin2 B + sin2 C = 1

(iii) cosB − cosC = −1 + 2 √2 cos B/2 sin C/2      Answer


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