Problem 38 - Contemporary Precalculus

Question

Solve the triangle. The Law of Cosines may be needed. A straight path makes an angle of \(6^{\circ}\) with the horizontal. A statue at the higher end of the path casts a 6.5 -meter-long shadow straight down the path. The angle of elevation from the end of the shadow to the top of the statue is \(32^{\circ} .\) How tall is the statue?

Step-by-Step Solution

Answer

Answer: The height of the statue is approximately 0.68 meters.

Step 1: Determine the angles of the triangle

Since the angle between the path and the horizontal is \(6^{\circ}\), we can consider the angle at the base of the statue (where the path meets the ground) as \(6^{\circ}\). We are given the angle of elevation (from the end of the shadow) to the top of the statue as \(32^{\circ}\). Let's denote this angle as \(\angle A\), the angle at the base of the statue as \(\angle B\), and the angle at the top of the statue as \(\angle C\). Thus, we have: \(\angle A = 32^{\circ}\) \(\angle B = 6^{\circ}\) \(\angle C = 180^{\circ} - (\angle A + \angle B) = 180^{\circ} - (32^{\circ} + 6^{\circ}) = 142^{\circ}\)

Step 2: Calculate the hypotenuse using the Law of Sines

Let the length of the shadow (from the base of the statue to the end of the shadow) be side \(a\), the distance from the top of the statue to the end of the shadow be side \(b\), and the height of the statue be side \(c\). We can use the Law of Sines as follows: \(\frac{a}{\sin \angle A} = \frac{b}{\sin \angle B}\) We are given that \(a=6.5\). From step 1, we know that \(\angle A = 32^{\circ}\) and \(\angle B = 6^{\circ}\). Therefore, we can calculate side \(b\): \(b = \frac{a\sin \angle B}{\sin \angle A} = \frac{6.5\sin 6^{\circ}}{\sin 32^{\circ}}\)

Step 3: Calculate side b

Using the formula derived in step 2, we can now find the length of side b: \(b = \frac{6.5\sin 6^{\circ}}{\sin 32^{\circ}} \approx 3.73\) meters

Step 4: Calculate the height of the statue using trigonometry

Now, we can use the angle at the base of the statue, \(\angle B\), to calculate the height of the statue, side \(c\). Since we have side a and angle B, we can use the following relationship: \(c = a\sin(\angle B) = 6.5 \sin 6^{\circ}\)

Step 5: Calculate side c (height of the statue)

Using the formula derived in step 4, we can now find the height of the statue: \(c = 6.5\sin 6^{\circ} \approx 0.68\) meters So, the height of the statue is approximately 0.68 meters.

Key Concepts

Law of Cosines

The Law of Cosines is a powerful tool in trigonometry, especially when dealing with oblique triangles (triangles that do not contain a right angle). It allows us to find unknown sides or angles of a triangle when we know:

• Two sides and the included angle, or
• All three sides

This law states that for any triangle with sides \( a, b, c \) opposite to angles \( A, B, C \) respectively, the squared length of any side equals the sum of the squares of the other two sides minus twice the product of those two sides and the cosine of their included angle:\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]Here, it wasn’t directly used in our solution, but understanding it can help solve more complex triangle problems, where the Law of Sines may fall short. Understanding the Law of Cosines deeply enriches your trigonometric toolkit and makes solving triangles—even without a right angle—manageable.

Law of Sines

The Law of Sines relates the ratios of each side length of a triangle to the sine of its opposite angle. For a triangle with sides \( a, b, c \) and respective opposite angles \( A, B, C \), the law states:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]This relationship becomes incredibly useful when you need to find an unknown side or angle, especially when:

• An angle and its opposite side are known, and at least one other side or angle is given.
• There is no right angle, which is frequently the case in applied problems.

The solution utilized the Law of Sines to calculate side \( b \), the distance from the top of the statue to the end of the shadow. This law is particularly useful because it allows us to connect angles and sides directly, simplifying complex trigonometric problems into manageable calculations.

Angle of Elevation

The angle of elevation is the angle between the horizontal line from the observer's eye to an object and the line of sight to the object. It is commonly used during situations where we need to find the height of an object or the distance from the observer to the object when only limited direct measurements are available. In this exercise:

• The angle of elevation is \(32^{\circ}\) from the end of the shadow to the top of the statue.
• This angle helps determine the vertical height of the statue using trigonometric relationships.

By knowing this angle, along with the length of the shadow, we applied trigonometric methods to find the height of the statue. Understanding the concept of angles of elevation is essential for solving problems involving heights and distances where direct measurement isn't possible.

Triangle Geometry

Triangle geometry is the study of the properties and relations of triangles, which include angles, sides, and specific points. In the context of trigonometry, problems often involve applying geometric concepts to determine unknown values, as seen in this exercise about the statue:

• We are aware of three angles, each contributing to understanding how to deconstruct the triangular relationships.
• The relationships between the angles help us apply trigonometric laws accurately.

Understanding triangle geometry ensures the correct application of laws and uncovering unknown measurements from known values. It's the fundamental background that enables anyone working with triangles to translate measurements into real-world understanding, or vice versa. This foundational knowledge aids in solving a variety of different geometric and real-world problems effectively and accurately.