Problem 40 - Contemporary Precalculus

Question

A fence post is located 50 feet from one corner of a building and 40 feet from the adjacent corner. Fences are put up between the post and the building corners to form a triangular garden area. The 40 -foot fence makes a \(58^{\circ}\) angle with the building. How long is the building wall?

Step-by-Step Solution

Answer

Explain your answer. Answer: No, it is not possible to find the length of the building wall with the given information. When applying the sine rule to the triangle, we calculated an invalid value for the sine of angle A, which was greater than 1. This indicates an error in the given problem or inconsistencies in the provided measurements and angles.

Step 1: Understand the given information

To solve this problem, we need to create a triangle in which one of the building's wall acts as a base, and the remaining two sides will be the fences. The angle formed between the 40-foot fence and the building, which is \(58^{\circ}\), will be used to apply the sine rule to the triangle.

Step 2: Apply the sine rule to find the length of the building wall

Let the length of the building wall be 'b'. In our triangle, we can assume the 50-foot side (opposite the \(58^{\circ}\) angle) as 'a', and the 40-foot side (adjacent to the \(58^{\circ}\) angle) as 'c'. We can write the sine rule as \(\frac{a}{\sin{A}} = \frac{b}{\sin{B}} = \frac{c}{\sin{C}}\) We know the values of a, c, and \(\angle C\) and need to find the length of 'b'. We can write two equations using the sine rule to solve for the length of 'b': 1. \(\frac{50}{\sin{A}} = \frac{40}{\sin{58}}\) 2. \(\frac{b}{\sin{B}} = \frac{40}{\sin{58}}\) Using the first equation, we can solve for \(\sin{A}\): \(\sin{A} = \frac{50\cdot\sin{58}}{40} = \frac{50\cdot0.848}{40} = 1.13\) Since \(\sin{A}\) is greater than 1, this cannot be a valid value for sine. Thus, there must be an error in the given problem. Please check the given information and make sure it is correct.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. It is a crucial tool for solving problems in various fields such as engineering, physics, architecture, and even in simpler everyday situations like determining the height of a tree or a building.

The main functions in trigonometry are sine (sin), cosine (cos), and tangent (tan), along with their reciprocals cosecant (csc), secant (sec), and cotangent (cot). These functions relate the angles of a triangle to the lengths of its sides. Trigonometry is especially useful for solving right-angled triangles, where one angle is known to be 90 degrees.

Triangle Problem Solving

Problem-solving with triangles involves finding missing dimensions or angles based on the given information. Key strategies include the use of Pythagorean theorem for right-angled triangles and trigonometric ratios for non-right triangles. An important part of this process is interpreting the question correctly and drawing a clear diagram to visualize the scenario.

When dealing with non-right triangles, as in the fence post problem, we identify which elements of the triangle we know (such as sides and angles), and choose the appropriate law or theorem to apply. Additionally, understanding the range of possible values for trigonometric functions (for example, the sine function only takes values between -1 and +1) is essential to avoid errors and identify contradictory or incomplete information.

Law of Sines

The law of sines, also known as the sine rule, is a powerful trigonometric principle that provides a reliable method to solve for unknown sides and angles in any (not exclusively right-angled) triangle. It states that the ratio of the length of a side to the sine of the angle opposite that side is the same for all sides and angles of a triangle:

\[\frac{a}{\sin{A}} = \frac{b}{\sin{B}} = \frac{c}{\sin{C}}\]
Where 'a', 'b', and 'c' are the lengths of the sides, and \(A\), \(B\), and \(C\) are the corresponding opposite angles.

When applying the law of sines, it is important to remember this rule is valid for any triangle, not just right-angled ones. It is also essential to consider the domain of the sine function, as it cannot exceed a value of 1. This provides a useful check against possible errors in calculations or the initial problem setup.