- Identify the right triangle and the given dimensions: a side length of 4 inches and an angle of 80 degrees.
- Use the cosine definition: \(\cos(80^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}}\).
- Calculate the hypotenuse: \(c = \frac{4}{\cos(80^\circ)} \approx 23.0\) inches.
- The correct answer is \(\boxed{23.0 \text{ in.}}\) (option c).
Explanation
- Analyze the right triangle properties.
In a right triangle, we are given one side that measures \(4\) in. Let this side be \(a = 4\) in. We are also told there is an angle of \(80^\circ\) in the triangle. To determine the hypotenuse \(c\), we examine the relationship between the side and the angle. If we assume the side of \(4\) in is adjacent to the \(80^\circ\) angle, we use the cosine function: \(\cos(80^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{c}\). Rearranging this gives \(c = \frac{4}{\cos(80^\circ)}\).
- Calculate the hypotenuse length.
Using the cosine relationship, we calculate the hypotenuse \(c\):
\[c = \frac{4}{\cos(80^\circ)}\]
The value of \(\cos(80^\circ)\) is approximately \(0.17365\).
Thus, the hypotenuse is:
\[c = \frac{4}{0.17365} \approx 23.035\]
Rounding to the nearest tenth, we get \(23.0\) in. This corresponds to option (c).
Examples
In architecture, right triangles are used to determine the lengths of support beams or rafters. If a roof has a slope of \(80^\circ\) and the horizontal run of the rafter is \(4\) feet, you can use the cosine function to calculate the required length of the rafter (the hypotenuse). By calculating \(4 / \cos(80^\circ) \approx 23.0\) feet, you ensure that the materials ordered for the roof construction are accurate and structural integrity is maintained.