Create an identity and find the formula for cos (3θ) in terms of cos (θ) and sin (θ) using de Moivre’s Theorem and test the identity with real values to confirm its validity.

Scenario:

You are working as an Engineering Research Technician in a research and development laboratory where good analytical skills and understanding of mathematics is essential for everyday work. Your line manager has informed you that you need to improve and update your present Mathematical skills to work effectively & contribute to the research team.

The essential skills include Number theory, Matrices, graphical and numerical methods, Models of

engineering systems using ordinary differential equations .

To test your suitability for this position in research, the following tasks have been compiled for you to complete.

Unit Learning Outcomes

LO1 Apply instances of number theory in practical construction situations.

Assignment Brief and Guidance

Task 1

a. Convert each number into denary,
• 11001.01
• 4D
b) calculate the following in both binary and denary
• 1101+1001

Task 2
Apply de’Moivre’s theorem or otherwise to solve for Zo and C from these expressions given
below :
Z0=Z/Y and C=Z*Y
Where:
• Z is a complex number.
• Y is also a complex number.
• Re (Z0) >0 and Re (C) >0
Find Z0 and C when:
Z = 1 + 5 j,Y = 1 − 3 j

Task 3

a. Simplify the following equation:

G = 1× e j2π × 2 × e j0.5 × 0.5 × e j0.75

b. Express the following expression in complex exponential form:
v=20sin (1000t-30°)

Task 4

Create an identity and find the formula for cos (3θ) in terms of cos (θ) and sin (θ) using de Moivre’s Theorem and test the identity with real values to confirm its validity.

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