Problem 37
Question
GEOGRAPHY For Exercises 37 and 38 , use the following information. A Mercator projection map uses a flat projection of Earth in which the distance between the lines of latitude increases with their distance from the equator. The calculation of the location of a point on this projection uses the expression tan \(\left(45^{\circ}+\frac{L}{2}\right)\) where \(L\) is the latitude of the point. Write this expression in terms of a trigonometric function of \(L .\)
Step-by-Step Solution
Verified Answer
The expression in terms of a trigonometric function of \( L \) is \( \frac{1 + \tan \frac{L}{2}}{1 - \tan \frac{L}{2}} \).
1Step 1: Understanding the Expression
The given expression is \( \tan \left(45^{\circ} + \frac{L}{2}\right) \), where \( L \) is the latitude. Our goal is to rewrite this expression in terms of a trigonometric function of \( L \).
2Step 2: Applying Trigonometric Identities
We use the tangent angle addition formula: \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \). Let \( A = 45^{\circ} \) and \( B = \frac{L}{2} \).
3Step 3: Plugging in Values for the Angle Addition Formula
Since \( \tan 45^{\circ} = 1 \), the tangent formula becomes:\[ \tan \left(45^{\circ} + \frac{L}{2}\right) = \frac{1 + \tan \frac{L}{2}}{1 - \tan \frac{L}{2}} \]
4Step 4: Simplifying the Expression
The expression \( \frac{1 + \tan \frac{L}{2}}{1 - \tan \frac{L}{2}} \) directly uses the tangent trigonometric function. This cannot be simplified further without additional information on \( L \).
5Step 5: Conclusion
The expression \( \tan(45^{\circ} + \frac{L}{2}) \) can be rewritten as \( \frac{1 + \tan \frac{L}{2}}{1 - \tan \frac{L}{2}} \), which is a trigonometric function of \( L \).
Key Concepts
Mercator ProjectionLatitudeTangent FunctionAngle Addition Formula
Mercator Projection
The Mercator projection is a well-known map-making technique used to represent the round surface of the Earth on a flat plane. This method was created by Gerardus Mercator in 1569. It is particularly useful for navigation, as it allows for straight-lines to represent constant compass bearings.
A major feature of the Mercator projection is how it represents lines of latitude, also known as parallels. On a Mercator map, these lines spread out as they approach the poles. This is because the distance representing one degree of latitude increases the further you move from the equator. This can distort land masses, making regions near the poles look much larger than they actually are.
While valuable for maritime purposes, the Mercator projection's distortion makes it less ideal for representing global land sizes accurately. However, its navigational benefits keep it in use.
A major feature of the Mercator projection is how it represents lines of latitude, also known as parallels. On a Mercator map, these lines spread out as they approach the poles. This is because the distance representing one degree of latitude increases the further you move from the equator. This can distort land masses, making regions near the poles look much larger than they actually are.
While valuable for maritime purposes, the Mercator projection's distortion makes it less ideal for representing global land sizes accurately. However, its navigational benefits keep it in use.
Latitude
Latitude measures how far north or south a point is from the equator. It is expressed in degrees, starting from 0° at the equator to 90° at the poles. Knowing the latitude of a location allows us to determine various geographical characteristics, such as climate and daylight hours.
Latitude lines, or parallels, wrap around the Earth horizontally. They remain equidistant from one another on the spherical Earth, but appear spaced out on a Mercator projection. For instance, cities closer to the equator have negligible distortion, while those located farther toward the poles appear stretched. Understanding latitude is essential in geographical mapping, navigation, and even climate studies. It's a fundamental geographical concept that we often encounter in various fields.
Latitude lines, or parallels, wrap around the Earth horizontally. They remain equidistant from one another on the spherical Earth, but appear spaced out on a Mercator projection. For instance, cities closer to the equator have negligible distortion, while those located farther toward the poles appear stretched. Understanding latitude is essential in geographical mapping, navigation, and even climate studies. It's a fundamental geographical concept that we often encounter in various fields.
Tangent Function
The tangent function is a fundamental trigonometric function. It's part of the six primary trigonometric functions. Simply put, in a right-angled triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side.
The tangent function is periodic, with a period of 180°, and has a range from negative to positive infinity. It is important in mathematics, particularly in trigonometry, calculus, and when working with periodic functions. The function can also be represented on a unit circle. As the angle increases, the value of the tangent can become undefined, such as at 90° and 270°. Beyond right triangles, the tangent function has practical applications in various contexts, like physics and engineering, where wave patterns and oscillations occur.
The tangent function is periodic, with a period of 180°, and has a range from negative to positive infinity. It is important in mathematics, particularly in trigonometry, calculus, and when working with periodic functions. The function can also be represented on a unit circle. As the angle increases, the value of the tangent can become undefined, such as at 90° and 270°. Beyond right triangles, the tangent function has practical applications in various contexts, like physics and engineering, where wave patterns and oscillations occur.
Angle Addition Formula
The angle addition formula for tangent helps us find the tangent of a sum of two angles. The formula is stated as:
Given our initial problem with a Mercator projection, understanding how to apply the angle addition formula has helped us transform the expression \( \tan (45^{\circ} + \frac{L}{2}) \) into a form where the tangent of a latitude-related angle \( L \) is directly involved.This concept is a powerful tool in trigonometry, often used in various real-world applications and complex mathematical problems.
- For tangent: \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \)
Given our initial problem with a Mercator projection, understanding how to apply the angle addition formula has helped us transform the expression \( \tan (45^{\circ} + \frac{L}{2}) \) into a form where the tangent of a latitude-related angle \( L \) is directly involved.This concept is a powerful tool in trigonometry, often used in various real-world applications and complex mathematical problems.
Other exercises in this chapter
Problem 36
For Exercises \(36-38,\) use the following information. The population of predators and prey in a closed ecological system tends to vary periodically over time.
View solution Problem 37
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Which One Does Belong? Identify the equation that does not belong with the other three. Explain your reasoning. $$ \begin{array}{ll}{\sin ^{2} \theta+\cos ^{2}
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Verify that each of the following is an identity. \(\sin \left(60^{\circ}+\theta\right)+\sin \left(60^{\circ}-\theta\right)=\sqrt{3} \cos \theta\)
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