Question

The height \(h\) (in feet) of a swing above the ground can be modeled by the function \(h=-8 \cos \theta+10\), where the pivot is 10 feet above the ground, the rope is 8 feet long, and \(\theta\) is the angle that the rope makes with the vertical. Graph the function. What is the height of the swing when \(\theta\) is \(45^{\circ}\) ?

Short answer

The height of the swing when \(\theta = 45^{\circ}\) is approximately 5.66 feet.

Step-by-step solution

Convert degrees to radians

To convert degrees to radians, use the relationship that \(180^{\circ} = \pi\) radians. Therefore, \(45^{\circ} = \frac{\pi}{4}\) radians.

Evaluate function

Substitute \(\theta = \frac{\pi}{4}\) into the function \(h = -8 \cos \theta + 10\) and evaluate. This will determine the height of the swing above the ground when \(\theta = 45^{\circ}\).

Graph the function

The function \(h = -8 \cos \theta + 10\) can be graphed by mapping the angle against the height. The circular shape of the cosine function will be apparent in the graph, with the minimum height of 2 (at \(\cos \theta = 1\)) and the maximum height of 18 (at \(\cos \theta = -1\)).

Key concepts

Converting Degrees to Radians

To accurately graph trigonometric functions, converting degrees to radians is essential, as many mathematical operations in trigonometry are commonly performed in radians. The conversion is based on the fact that a full circle is 360 degrees, which is equivalent to 2π radians. The relationship between degrees and radians is rooted in the formula
\( \frac{{degrees}}{{360}} = \frac{{radians}}{{2\pi}} \). Therefore, to convert degrees to radians, you multiply the degree measure by \( \frac{{\pi}}{{180}} \).
For example, to convert 45 degrees to radians, you would compute \( 45^\circ \times \frac{{\pi}}{{180}} = \frac{{\pi}}{{4}} \).
Understanding this conversion allows students to transition between degrees and radians fluently, crucial for the further steps of evaluating trigonometric functions or graphing them.

Evaluating Cosine Functions

To evaluate a cosine function, like \( h = -8 \cos \theta + 10 \), you need to understand that the cosine function describes the horizontal coordinate of a point on the unit circle. When the angle \( \theta \) is given, finding the cosine value involves looking at the circle with radius 1 centered at the origin.
The cosine value corresponding to that angle \( \theta \) represents a point on the circumference of the circle. The value fluctuates between -1 and 1. If \( \theta = \frac{{\pi}}{{4}} \) radians, we find the cosine value and use it to determine the height \( h \).
In our specific example, \( h = -8 \cos \frac{{\pi}}{{4}} + 10 \) evaluates the height of the swing at that particular angle. The process involves understanding the properties of the cosine function, including its period, amplitude, and its behavior with different angles.

Amplitude and Vertical Shift

In a trigonometric function like \( h = -8 \cos \theta + 10 \), the coefficient in front of the cosine represents the amplitude, and the constant term represents the vertical shift.
In this context, the amplitude, 8, indicates the maximum deviation from the central axis or the vertical shift, 10, of the swing. The minus sign indicates that the graph of this cosine function is a reflection across the horizontal axis running through its vertical shift — the height of the pivot.
The vertical shift moves the entire function up or down on the graph. In our function, the +10 lifts the basic cosine curve up, so that the midline of the function is at 10 feet instead of at 0 feet (ground level). The graph thereby starts at its highest point of 18 feet when \( \theta = 0 \) and oscillates down to 2 feet and back, rather than beginning at the axis and oscillating between your feet length of 8 feet above and below that axis.