Problem 49 - Algebra 2

Question

sKYCOASTING Mikhail and Anya visit a local amusement park to ride a skycoaster. After the first several swings, the angle the skycoaster makes with the vertical is modeled by \(\theta=0.2 \mathrm{cos} \pi t,\) with \(\theta\) measured in radians and \(t\) measured in seconds. Determine the measure of the angle for \(t=0,\) \(0.5,1,1.5,2,2.5,\) and 3 in both radians and degrees.

Step-by-Step Solution

Answer

At \( t = 0, 0.5, 1, 1.5, 2, 2.5, 3 \), the angles are: \( 0.2, 0, -0.2, 0, 0.2, 0, -0.2 \) radians, or \( 11.46, 0, -11.46, 0, 11.46, 0, -11.46 \) degrees respectively.

Step 1: Substitute the Time Value

For each given value of time \( t \), substitute it into the function \( \theta = 0.2 \cos(\pi t) \) to find the angle in radians. Start with \( t = 0 \) and continue through \( t = 3 \).

Step 2: Calculate Radians Using the Cosine Function

Use a calculator or a cosine table to find the cosine of \( \pi t \) and then multiply by 0.2. For example, at \( t = 0 \), \( \cos(\pi \cdot 0) = \cos(0) = 1 \), so \( \theta = 0.2 \cdot 1 = 0.2 \) radians.

Step 3: Convert Radians to Degrees

To convert radians to degrees, use the formula \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \). For example, at \( t = 0 \), \( 0.2 \text{ radians} \times \frac{180}{\pi} \approx 11.46 \text{ degrees} \).

Step 4: Compile and List the Solutions

Repeat the calculations for each \( t \) value. The values are:- \( t = 0 \): \( 0.2 \) radians, \( 11.46 \) degrees- \( t = 0.5 \): \( 0 \) radians, \( 0 \) degrees- \( t = 1 \): \( -0.2 \) radians, \( -11.46 \) degrees- \( t = 1.5 \): \( 0 \) radians, \( 0 \) degrees- \( t = 2 \): \( 0.2 \) radians, \( 11.46 \) degrees- \( t = 2.5 \): \( 0 \) radians, \( 0 \) degrees- \( t = 3 \): \( -0.2 \) radians, \( -11.46 \) degrees

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that expresses the relationship between the sides and angles of a right triangle. It's denoted as \( \cos \theta \), where \( \theta \) is the angle. When graphing, the cosine function behaves in a wave-like pattern, alternating between a maximum of 1 and a minimum of -1.

For any angle \( \theta \), the cosine value is the ratio of the adjacent side to the hypotenuse in a right triangle. In our skycoaster problem, the cosine function \( \cos(\pi t) \) is used to model the oscillation of the angle over time. It helps determine how the angle changes as the swing progresses back and forth.

Key properties of the cosine function include:

• Periodicity: It repeats every \( 2\pi \) radians.
• Range: Values range from -1 to 1.
• Even Function: Symmetric about the y-axis \( \cos(-x) = \cos(x) \).

Understanding these properties allows us to predict and calculate the angle for any given point in time.

Angle Conversion

Converting angles between radians and degrees is a common task in trigonometry. This step is crucial when applications require angles in a specific unit, such as the skycoaster exercise.

The conversion formula between radians and degrees is:

• Degrees = Radians \( \times \frac{180}{\pi} \)
• Radians = Degrees \( \times \frac{\pi}{180} \)

Using this conversion, one can translate complex trigonometric calculations, like those seen with radian measures, to more familiar units such as degrees.

In the context of the exercise, once we find \( \theta \) in radians using the cosine function, we apply the formula to convert it into degrees. This allows the swing angles to be interpreted in a range most people find easier to visualize, providing clarity to the results in both scientific and everyday contexts.

Radians and Degrees

Radians and degrees are two units of measuring angles. Understanding their relationship is crucial in trigonometry as it influences how we calculate and interpret angles.

Radians are based on the radius of a circle. One radian is the angle made when the arc length of a circle is equal to the radius of the circle. It is a natural way of measuring angles and is heavily utilized in advanced mathematics and engineering.

Degrees subdivide the circle into 360 equal parts, making it intuitive and common in practical scenarios.
Important to remember:

• \( 2\pi \) radians equal 360 degrees.
• \( \pi \) radians equal 180 degrees.
• 1 radian is approximately 57.3 degrees.

In our exercise, using radians enables precise modeling of the swinging angle with the cosine function. Yet, when communicating findings, degrees help in sharing understandable results for everyone. Hence, both units play essential roles in the full interpretation of angle measurements.