Q1. Convert 37.2° to degrees, minutes, and seconds (DMS).

Background

Topic: Angle Measurement Conversion

This question tests your ability to convert decimal degrees into the DMS (degrees, minutes, seconds) format, which is commonly used in navigation and geometry.

Key Terms and Formulas

• 1 degree () = 60 minutes ()

• 1 minute () = 60 seconds ()

Step-by-Step Guidance

1. Take the decimal part of the degree and multiply by 60 to get the minutes.

2. Take the decimal part of the minutes and multiply by 60 to get the seconds.

3. Combine the whole number degrees, minutes, and seconds for the DMS format.

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Q2. Convert 42°6’36” to decimal degrees.

Background

Topic: Angle Measurement Conversion

This question tests your ability to convert an angle given in DMS format to decimal degrees, which is often used in calculations.

Key Terms and Formulas

• Decimal degrees = degrees + (minutes/60) + (seconds/3600)

Step-by-Step Guidance

1. Divide the minutes by 60 to convert to degrees.

2. Divide the seconds by 3600 to convert to degrees.

3. Add all parts together to get the decimal degree value.

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Q3. Find the arc length of a circle with a radius of 3 cm intercepted by an angle of radians.

Background

Topic: Arc Length in Circles

This question tests your understanding of how to calculate the arc length of a circle given the radius and the central angle in radians.

Key Terms and Formulas

• Arc length formula:

• Where is arc length, is radius, is angle in radians.

Step-by-Step Guidance

1. Identify the radius cm and angle radians.

2. Plug these values into the arc length formula: .

3. Multiply the radius by the angle in radians to set up the calculation.

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Q4a. Find a trigonometric equation for the height of the water as a function of time (hours since midnight).

Background

Topic: Trigonometric Modeling of Periodic Phenomena

This question tests your ability to model real-world periodic events (like tides) using trigonometric functions.

Key Terms and Formulas

• General cosine function:

• Amplitude (), period (), phase shift (), vertical shift ()

Step-by-Step Guidance

1. Determine the amplitude: .

2. Find the midline: .

3. Calculate the period: .

4. Set up the cosine function using amplitude, period, phase shift, and midline.

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Q4b. When does the next high tide occur according to your model?

Background

Topic: Application of Trigonometric Models

This question tests your ability to interpret a trigonometric model to predict future events.

Key Terms and Formulas

• High tide corresponds to the maximum value of the cosine function.

• Use the model equation to solve for when is at its maximum.

Step-by-Step Guidance

1. Set the cosine function equal to its maximum value.

2. Solve for to find the time of the next high tide.

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Q5A. Determine possible coordinates for the five points: F, G, J, K, and P.

Background

Topic: Graphing Periodic Functions

This question tests your ability to interpret and assign coordinates to points on a graph of a periodic function, such as a sine wave.

Key Terms and Formulas

• Periodic function: models the distance as a function of time.

• Maximum, minimum, and midline values correspond to specific points on the graph.

Step-by-Step Guidance

1. Identify the maximum, minimum, and midline values from the graph.

2. Assign coordinates to each labeled point based on their position relative to the cycle.

3. Use the period and amplitude to estimate the -values for each point.

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Q5B. Write the function in the form . Find values of , , , and .

Background

Topic: Sinusoidal Function Parameters

This question tests your ability to write a sine function to model a periodic situation and determine its parameters.

Key Terms and Formulas

• Amplitude ():

• Vertical shift ():

• Period:

• Phase shift (): determined by the starting point of the cycle

Step-by-Step Guidance

1. Calculate amplitude and vertical shift using the maximum and minimum values.

2. Determine the period from the time it takes for one full cycle.

3. Find the phase shift based on the starting position.

4. Write the function in the required form using these values.

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Q5C. On the interval , which of the following is true about ?

Background

Topic: Behavior of Periodic Functions

This question tests your understanding of the increasing/decreasing behavior of a sine function over a given interval.

Key Terms and Formulas

• Positive/negative values: above/below the midline

• Increasing/decreasing: determined by the slope of the function

Step-by-Step Guidance

1. Examine the graph to determine whether is above or below the midline in the interval.

2. Check the direction of the curve (up or down) to decide if it is increasing or decreasing.

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Q5D. Describe how the rate of change of is changing over the interval .

Background

Topic: Concavity and Rate of Change

This question tests your understanding of how the rate of change (derivative) of a periodic function behaves over an interval, including concepts of concavity.

Key Terms and Formulas

• Concave up/down: relates to whether the rate of change is increasing or decreasing

• Second derivative: indicates concavity

Step-by-Step Guidance

1. Determine the concavity of the function in the interval by examining the graph.

2. Relate concavity to the behavior of the rate of change (is it increasing or decreasing?).

Try solving on your own before revealing the answer!