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Application of Eulers formula Prove the following identities. c.pdf

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Application of Euler\'s formula: Prove the following identities. cos(A + B) = cosAcosB - sinAsinB and sin(A + B) = sinAcosB + cosAsinB (Hint: consider z1 = Ai and z2 = Bi where A and B > 0.) Solution eAi = cosA + i sinA eBi = cosB + i sinB eAi eBi = (cosA + i sinA)(cosB + i sinB) e(A + B)i = cosAcosB + i2 sinAsinB + i(sinAcosB + cosAsinB) cos(A + B) + i sin(A + B)= cosAcosB - sinAsinB + i (sinAcosB + cosAsinB) Real part: cos(A + B) = cosAcosB - sinAsinB Imaginary part: sin(A + B)= sinAcosB + cosAsinB.

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Application of Euler's formula: Prove the following identities. cos(A + B) = cosAcosB - sinAsinB and sin(A + B) = sinAcosB + cosAsinB (Hint: consider z1 = Ai and z2 = Bi where A and B > 0.) Solution eAi = cosA + i sinA eBi = cosB + i sinB eAi eBi = (cosA + i sinA)(cosB + i sinB) e(A + B)i = cosAcosB + i2 sinAsinB + i(sinAcosB + cosAsinB) cos(A + B) + i sin(A + B)= cosAcosB - sinAsinB + i (sinAcosB + cosAsinB) Real part: cos(A + B) = cosAcosB - sinAsinB Imaginary part: sin(A + B)= sinAcosB + cosAsinB

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Application of Eulers formula Prove the following identities. c.pdf

  • 1. Application of Euler's formula: Prove the following identities. cos(A + B) = cosAcosB - sinAsinB and sin(A + B) = sinAcosB + cosAsinB (Hint: consider z1 = Ai and z2 = Bi where A and B > 0.) Solution eAi = cosA + i sinA eBi = cosB + i sinB eAi eBi = (cosA + i sinA)(cosB + i sinB) e(A + B)i = cosAcosB + i2 sinAsinB + i(sinAcosB + cosAsinB) cos(A + B) + i sin(A + B)= cosAcosB - sinAsinB + i (sinAcosB + cosAsinB) Real part: cos(A + B) = cosAcosB - sinAsinB Imaginary part: sin(A + B)= sinAcosB + cosAsinB