Problem 73 - Algebra and Trigonometry Real Mathematics, Real People

Question

True or False Determine whether the statement is true or false. Justify your answer. Writing Find two bearings perpendicular to \(\mathrm{N} 32^{\circ} \mathrm{E}\) and explain how you found them.

Step-by-Step Solution

Answer

The two bearings perpendicular to \(N 32^{\circ} E\) are \(S 58^{\circ} E\) and \(N 58^{\circ} W\).

Step 1: Determining Perpendicular Directions Considering Compass

Given the bearing \(N 32^{\circ} E\), to find the two bearings that are perpendicular to this, consider the compass. From North (start of the bearing), go 90 degrees clockwise and anti-clockwise. This would give the two directions which are perpendicular to \(N 32^{\circ} E\).

Step 2: Calculation of Bearings for Clockwise Direction

Start from \(N 32^{\circ} E\) and go 90 degrees clockwise. Since moving from North to East is positive, the bearing in the clockwise direction would be the given bearing plus 90 degrees which gives \(N 32^{\circ} E + 90^{\circ} = S 58^{\circ} E\).

Step 3: Calculation of Bearings for Anti-Clockwise Direction

Now, navigate 90 degrees anti-clockwise from the given bearing \(N 32^{\circ} E\). To do this, use the same method as above but subtract 90 degrees instead: \(N 32^{\circ} E - 90^{\circ} = N 58^{\circ} W\).

Key Concepts

Perpendicular Directions

When working with bearings, understanding perpendicular directions is crucial. Bearings describe a specific direction relative to the cardinal points on a compass. When two directions are perpendicular, they form a right angle of 90 degrees with each other. For example, if you have a bearing of North 32° East, finding its perpendicular directions involves rotating 90 degrees both clockwise and counterclockwise from this position.

• Clockwise rotation adds 90 degrees to the initial bearing.
• Counterclockwise rotation subtracts 90 degrees from the initial bearing.

In practical terms, if someone is facing North 32° East, a 90-degree clockwise turn will point them toward South 58° East, while a 90-degree counterclockwise turn will direct them to North 58° West.

Compass Navigation

Compass navigation is essential for determining bearings, which specify directions based on compass points. A compass is marked by the cardinal directions: North, East, South, and West. When navigating using bearings, these directions serve as reference points, meaning each has a corresponding degree: North at 0 or 360 degrees, East at 90 degrees, South at 180 degrees, and West at 270 degrees.
Understanding compass directions helps in navigation by providing clear steps to move from one point to another.

• Bearings describe the direction in terms of the smallest angle a line makes with the reference direction.
• All bearings are typically measured clockwise from the North.

For complex navigation, bearings help you locate positions or navigate efficiently between two distinct points. This is particularly useful in activities like hiking or sailing where precise direction is essential.

Angle Measurement

Calculating bearings involves precise angle measurements. Angles in bearings are typically measured in degrees, starting from the north direction moving clockwise. Accurately measuring and manipulating angles is fundamental when determining bearings.

• The compass is divided into 360 degrees.
• Measurements utilize angles to determine direction shifts.

When you have a specific bearing, such as North 32° East, this represents an angle of 32 degrees measured clockwise from the north. To find perpendicular bearings, adjust the angle by 90 degrees. This approach allows navigators to map their direction precisely and helps ensure accurate navigation across various terrains and circumstances.