Problem 22 - College Trigonometry

Question

A naturalist sets off on a hike from a lodge on a bearing of \(\mathrm{S} 80^{\circ} \mathrm{W}\). After \(1.5\) miles, she changes her bearing to \(\mathrm{S} 17^{\circ} \mathrm{W}\) and continues hiking for 3 miles. Find her distance from the lodge at this point. Round your answer to the nearest hundredth of a mile. What bearing should she follow to return to the lodge? Round your angle to the nearest degree.

Step-by-Step Solution

Answer

The naturalist is approximately 2.68 miles from the lodge and should return on a bearing of \( \mathrm{S} 13^{\circ} \mathrm{E} \).

Step 1: Draw the Path

Draw a diagram representing the naturalist's path. First, mark a point for the lodge and draw the initial path 1.5 miles long on a bearing of \( \mathrm{S} 80^{\circ} \mathrm{W} \). Then, draw the second leg of the path for 3 miles on a bearing of \( \mathrm{S} 17^{\circ} \mathrm{W} \). This gives two line segments with an included angle.

Step 2: Convert Bearings to Angles

The bearing \( \mathrm{S} 80^{\circ} \mathrm{W} \) converts to an angle of \( 260^{\circ} \) clockwise from north. Similarly, \( \mathrm{S} 17^{\circ} \mathrm{W} \) converts to \( 197^{\circ} \). The included angle between the paths is the absolute difference between these directions, which is \( |197^{\circ} - 260^{\circ}| = 63^{\circ} \).

Step 3: Apply the Law of Cosines

Using the Law of Cosines to find the distance from the lodge, let \( c \) be the unknown distance, and \( a = 1.5 \) miles and \( b = 3 \) miles be the two sides, with the included angle \( \theta = 63^{\circ} \). The formula is:\[ c^2 = a^2 + b^2 - 2ab \cos(\theta) \]Insert the values:\[ c^2 = 1.5^2 + 3^2 - 2 \times 1.5 \times 3 \times \cos(63^{\circ}) \]

Step 4: Calculate the Unknown Distance

Calculate numerically:\[ c^2 = 2.25 + 9 - 9 \times 0.454 \] (using \( \cos(63^{\circ}) \approx 0.454 \))\[ c^2 = 11.25 - 4.086 \]\[ c^2 = 7.164 \]\[ c = \sqrt{7.164} \approx 2.68 \]Thus, the distance from the lodge is approximately 2.68 miles.

Step 5: Determine the Return Bearing

Using the Law of Sines to find the angle opposite to the first path:\[ \frac{\sin(A)}{1.5} = \frac{\sin(63^{\circ})}{2.68} \]\[ \sin(A) = \frac{1.5 \times \sin(63^{\circ})}{2.68} \]Calculate \( A \) using \( \sin(63^{\circ}) \approx 0.891 \):\[ \sin(A) = \frac{1.5 \times 0.891}{2.68} = 0.499 \]\[ A \approx 30^{\circ} \]The naturalist should head at a bearing of \(197^{\circ} - 30^{\circ} = 167^{\circ} \) from the south, which is \( \mathrm{S} 13^{\circ} \mathrm{E} \).

Key Concepts

Bearing Calculation

Bearings are a crucial part of navigation and help us determine direction. They are usually measured in degrees from a reference direction, often north. In this context, a bearing helps to trace the path of the hiker on her journey from the lodge.

• Bearing is denoted in the form 'N40°E', 'S80°W' etc., where 'N' stands for north, 'S' for south, and the letter after the angle specifies the direction to which you turn.
• For example, a bearing of 'S80°W' means starting facing south and turning 80 degrees towards the west.

Converting these bearings into standard angles involves measuring clockwise from the north. For instance:

• 'S80°W' converts to 260° clockwise from north.
• Similarly, 'S17°W' converts to 197°.

Subtract these to find the angle between the two bearings, which is 63°. The understanding of bearings is essential in calculating the correct path and ensuring accurate distance and direction measurements.

Trigonometry Applications

Trigonometry is more than just triangles and angles. It has practical applications in real-world scenarios including navigation and surveying. In this exercise, trigonometry helps to calculate unknown distances and angles using known measurement paths.

• We use the Law of Cosines to find the unknown side of a triangle when we know two sides and the included angle.
• Similarly, the Law of Sines is helpful in finding unknown angles when two sides and an angle are known.

By calculating the unknown distance using the Law of Cosines, the hiker can determine how far off course she is from her starting point. Additionally, calculating her return bearing becomes simpler as we apply the Law of Sines to determine necessary angles.
Applications like these highlight how trigonometry is embedded in everyday problem-solving.

Law of Sines

The Law of Sines is a powerful tool in trigonometry when dealing with non-right triangles, as in this exercise. It states that the ratio of the length of a side of a triangle to the sine of the angle opposite is constant for all three sides of the triangle.

• Formula: \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \)

In this example, it helps us determine the angle that dictates the return journey of the hiker:

• By knowing two sides of the triangle and an angle (63°) between them, we can find unknown angles.
• Here, we find angle \( A \), which helps us in determining the bearing she needs to take to return to the lodge.

The calculation involves equating \( \frac{\sin(A)}{1.5} = \frac{\sin(63°)}{2.68} \). Solving this helps us find angle \( A \), which aids in charting an optimal return path.
The Law of Sines is instrumental in ensuring accurate angular estimations, leading to informed navigation decisions.