Precalculus Study Guide: Angles, Arc Length, and Trigonometric Modeling

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 ($1^\circ$) = 60 minutes ($60'$)

• 1 minute ($1'$) = 60 seconds ($60''$)

Step-by-Step Guidance

1. Start with the decimal degree: $37.2^\circ$.

2. Extract the whole number of degrees: $37^\circ$.

3. Multiply the decimal part ($0.2$) by $60$to convert to minutes: $0.2 \times 60$.

4. Take the whole number of minutes and multiply the remaining decimal by $60$ to get seconds.

Try solving on your own before revealing the answer!

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. Start with the given values: $42^\circ$, $6'$, $36''$.

2. Convert minutes to degrees: $6/60$.

3. Convert seconds to degrees: $36/3600$.

4. Add all three values together to get the decimal degree.

Try solving on your own before revealing the answer!

Q3. Find the arc length of a circle with a radius of 3 cm intercepted by an angle of $\frac{2\pi}{3}$ 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: $s = r\theta$

• $r$ = radius (in cm)

• $\theta$ = central angle (in radians)

Step-by-Step Guidance

1. Identify the radius: $r = 3$ cm.

2. Identify the angle in radians: $\theta = \frac{2\pi}{3}$.

3. Plug these values into the arc length formula: $s = r\theta$.

4. Set up the multiplication: $s = 3 \times \frac{2\pi}{3}$.

Try solving on your own before revealing the answer!

Q4a. Find a trigonometric equation for the height of the water as a function of time $t$ (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, specifically cosine or sine functions.

Key Terms and Formulas:

• General cosine function: $y = a \cos(b(t - c)) + d$

• $a$ = amplitude (half the difference between high and low tide)

• $b$ = frequency (related to the period)

• $c$ = phase shift (horizontal shift)

• $d$ = vertical shift (average value)

Step-by-Step Guidance

1. Calculate the amplitude: $a = \frac{18 - 6}{2}$.

2. Find the midline (vertical shift): $d = \frac{18 + 6}{2}$.

3. Determine the period: Time between two consecutive high tides.

4. Calculate the frequency: $b = \frac{2\pi}{\text{period}}$.

5. Determine the phase shift based on the timing of high tide.

Try solving on your own before revealing the answer!

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, such as the timing of the next high tide.

Key Terms and Formulas:

• Use the model from part (a) to solve for $t$ when the height is at its maximum.

Step-by-Step Guidance

1. Set the trigonometric equation equal to the maximum height (high tide).

2. Solve for $t$ using the properties of the cosine function.

3. Interpret the value of $t$ in terms of hours since midnight.

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Q5A. Determine possible coordinates $(t, d(t))$ for the five points $F, G, J, K, P$ on the graph.

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 or cosine wave.

Key Terms and Formulas:

• Periodic function: $d(t)$ models the distance as a function of time.

• Key points: maximum, minimum, midline crossings.

Step-by-Step Guidance

1. Identify the period and amplitude from the problem description.

2. Use the graph to estimate the $t$-values for each labeled point.

3. Assign the corresponding $d(t)$ values based on the graph's vertical position.

4. Check for symmetry and periodicity to help with coordinate assignment.

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Q5B. Write the function $d(x)$ in the form $d(x) = a \sin(b(t + c)) + d$ and find values for $a, b, c, d$.

Background

Topic: Sinusoidal Function Modeling

This question tests your ability to write a sine function to model a periodic phenomenon, and to determine the amplitude, frequency, phase shift, and vertical shift.

Key Terms and Formulas:

• General sine function: $d(x) = a \sin(b(t + c)) + d$

• $a$ = amplitude

• $b$ = frequency

• $c$ = phase shift

• $d$ = vertical shift

Step-by-Step Guidance

1. Find the amplitude: $a = \frac{\text{max} - \text{min}}{2}$.

2. Find the vertical shift: $d = \frac{\text{max} + \text{min}}{2}$.

3. Determine the period and calculate $b = \frac{2\pi}{\text{period}}$.

4. Find the phase shift $c$ based on the starting point of the cycle.

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Q5C(i). On the interval $(t_1, t_2)$, which statement is true about $d$?

Background

Topic: Behavior of Periodic Functions

This question tests your ability to analyze the sign and monotonicity (increasing/decreasing) of a periodic function over a given interval.

Key Terms and Formulas:

• Positive/negative values: Above/below midline.

• Increasing/decreasing: Slope of the function.

Step-by-Step Guidance

1. Examine the graph between $t_1$ and $t_2$ to determine if $d$ is above or below the midline.

2. Check the direction of the curve (up or down) to determine if $d$ is increasing or decreasing.

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Q5C(ii). Describe how the rate of change of $d$ is changing over the interval $(t_1, t_2)$.

Background

Topic: Concavity and Rate of Change

This question tests your understanding of how the rate of change (derivative) of a function behaves, specifically whether it is increasing or decreasing (concavity).

Key Terms and Formulas:

• Concave up/down: Second derivative test.

• Rate of change: Slope of the tangent line.

Step-by-Step Guidance

1. Look at the curvature of the graph in the interval $(t_1, t_2)$.

2. Determine if the function is concave up or concave down (rate of change increasing or decreasing).

Try solving on your own before revealing the answer!

Final Answers

1. 42°12’ 2. 37.11° 3. $2\pi$ 4. a) $y = 6 \cos \left(\frac{\pi}{6.25}(t-11.25)\right) + 12$ or $y = -6 \cos \left(\frac{\pi}{6.25}(t-5)\right) + 12$ 4. b) 23.75 or 11:45 pm 5. A) F (1.5, 60) G(2.25, 42) J(3, 24) K(3.75, 42) P(4.5, 60) 5. B) $d(x) = 18 \sin \left(\frac{2\pi}{3}(t-3.75)\right) + 42$ 5. C) b 5. D) Since $d(t)$ is concave down in the interval, the rate of change of $d(t)$ is decreasing.

Each answer uses the appropriate formulas and reasoning for the type of question asked.