Q1A. Determine the amplitude A for the sinusoidal voltage function V1 = A \sin(0.5t) using the graph at t = 5 seconds.

Background

Topic: Sinusoidal Functions & Amplitude

This question tests your ability to interpret a graph of a sinusoidal function and use it to solve for the amplitude parameter in the function's equation.

Key Terms and Formulas:

• Amplitude (A): The maximum value of the sinusoidal function.

• Sinusoidal function:

• At a specific time , can be read from the graph.

Step-by-Step Guidance

1. Identify the voltage value at seconds from the graph. This is your at that time.

2. Write the equation: , substituting .

3. Substitute the voltage value from the graph and solve for algebraically, making sure to use radians for the sine function.

4. Set up the equation for but do not compute the final value yet. Prepare to solve for $A$ by isolating it.

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Q1B. Expand the equation using the compound angle formula.

Background

Topic: Trigonometric Identities (Compound Angle Formula)

This question tests your ability to expand a sine function with a sum of angles using the sine addition formula.

Key Terms and Formulas:

• Compound angle formula:

• Amplitude: The coefficient in front of the sine function.

Step-by-Step Guidance

1. Identify and in the expression .

2. Apply the compound angle formula to expand .

3. Distribute the amplitude (36) across the expanded terms.

4. Rearrange the expanded equation to group terms involving and .

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Q1C. Write the fully expanded formula for as the sum of and , then determine values for and .

Background

Topic: Solving Simultaneous Trigonometric Equations

This question tests your ability to equate and solve for unknowns in trigonometric expressions using values from previous parts.

Key Terms and Formulas:

Step-by-Step Guidance

1. Set and equate the expanded forms.

2. Set to simplify the equations and solve for .

3. Substitute back into the equation and use a known value for at to solve for .

4. Prepare to solve for using the cosine inverse function, but do not compute the final value yet.

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Q1D. Plot , , and on the same graph and verify that $V_3$ equals the sum of $V_1$ and $V_2$ at a chosen point.

Background

Topic: Graphical Verification of Trigonometric Sums

This question tests your ability to use graphing software to visually confirm the sum of two sinusoidal functions.

Key Terms and Formulas:

• Graphing: Plotting , , and on the same axes.

• Verification: Comparing values at a specific .

Step-by-Step Guidance

1. Use graphing software to plot , , and .

2. Choose a value of and draw a vertical line at that point.

3. Read the values of and at the intersection with the vertical line.

4. Compare the sum to the value of at the same .

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Q2A. Solve algebraically for the first two values of when for , where .

Background

Topic: Solving Trigonometric Equations

This question tests your ability to solve a trigonometric equation for using algebraic methods and identities.

Key Terms and Formulas:

• Trigonometric identities:

• Quadratic equations in terms of

Step-by-Step Guidance

1. Substitute into the equation and set .

2. Use trigonometric identities to rewrite the equation in terms of .

3. Express the equation as a quadratic in .

4. Factor the quadratic equation and prepare to solve for .

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Q2B. Confirm your solutions for with an annotated graph showing all angles between 0 and 1 radians.

Background

Topic: Graphical Solution Verification

This question tests your ability to use graphing software to visually confirm algebraic solutions to trigonometric equations.

Key Terms and Formulas:

• Graphing: Plotting versus and identifying zeros.

• Annotated graph: Marking solution points.

Step-by-Step Guidance

1. Plot the function for between 0 and 1 radians.

2. Mark the points where on the graph.

3. Compare these points to your algebraic solutions for .

4. Check that the graph visually confirms the calculated values.

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Q2C. Determine how many rows of seats the train should have to ensure each row is photographed, using the BJT voltage threshold and the graph.

Background

Topic: Periodic Functions & Threshold Analysis

This question tests your ability to analyze a periodic voltage function and relate it to a physical threshold for a switching device.

Key Terms and Formulas:

• BJT threshold: Minimum voltage required to turn on the switch.

• Period: Duration of one cycle of the voltage function.

• Number of rows:

Step-by-Step Guidance

1. Identify the voltage threshold (2.5V) and the period during which the voltage is above this threshold.

2. Determine the duration of one cycle from the graph.

3. Calculate the number of cycles (rows) within the total time (1 second).

4. Set up the division to find the number of rows, but do not compute the final value yet.

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Q3A. Write the polar form of each voltage signal and determine the resultant magnitude and angle of using a vector diagram.

Background

Topic: Phasor Addition & Polar Form

This question tests your ability to convert sinusoidal signals to polar form and add them as vectors to find the resultant.

Key Terms and Formulas:

• Polar form:

• Rectangular coordinates: ,

• Resultant magnitude:

• Resultant angle:

Step-by-Step Guidance

1. Convert each voltage signal to polar form, identifying magnitude and phase angle.

2. Convert each signal to rectangular coordinates ().

3. Add the real and imaginary components to find the total and .

4. Set up the formulas for resultant magnitude and angle, but do not compute the final values yet.

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