Sylvia's Office Has An Excellent View Of A New Office Building That Is 80 Feet Tall. From Her Office, The Angle Of Depression To The Base Of This New Building Is 21°, And The Angle Of Elevation To The Top Of This Building Is 25°.Based On This
Introduction
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and architecture. In this article, we will explore a real-world example of solving a trigonometry problem using the concept of angles of depression and elevation.
The Problem
Sylvia's office has an excellent view of a new office building that is 80 feet tall. From her office, the angle of depression to the base of this new building is 21°, and the angle of elevation to the top of this building is 25°. Based on this information, we need to find the distance between Sylvia's office and the base of the new building.
Understanding Angles of Depression and Elevation
Before we dive into the solution, let's understand the concepts of angles of depression and elevation. The angle of depression is the angle between the horizontal and the line of sight to the base of an object. On the other hand, the angle of elevation is the angle between the horizontal and the line of sight to the top of an object.
Visualizing the Problem
To visualize the problem, let's draw a diagram. We can draw a right-angled triangle with Sylvia's office as the vertex opposite the right angle. The base of the triangle represents the distance between Sylvia's office and the base of the new building. The height of the triangle represents the height of the new building, which is 80 feet.
Using Trigonometry to Solve the Problem
Now that we have a clear understanding of the problem and have visualized it, let's use trigonometry to solve it. We can use the tangent function to relate the angle of depression to the ratio of the opposite side (the height of the building) to the adjacent side (the distance between Sylvia's office and the base of the building).
tan(21°) = opposite side (height of the building) / adjacent side (distance between Sylvia's office and the base of the building)
We can also use the tangent function to relate the angle of elevation to the ratio of the opposite side (the height of the building) to the adjacent side (the distance between Sylvia's office and the top of the building).
tan(25°) = opposite side (height of the building) / adjacent side (distance between Sylvia's office and the top of the building)
Solving for the Distance
Now that we have two equations, we can solve for the distance between Sylvia's office and the base of the building. We can start by solving the second equation for the distance between Sylvia's office and the top of the building.
distance between Sylvia's office and the top of the building = height of the building / tan(25°)
Now that we have the distance between Sylvia's office and the top of the building, we can use the first equation to solve for the distance between Sylvia's office and the base of the building.
distance between Sylvia's office and the base of the building = distance between Sylvia's office and the top of the building - height of the building / tan(21°)
Calculating the Distance
Now that we have the equations, let's calculate the distance between Sylvia's office and the base of the building.
distance between Sylvia's office and the top of the building = 80 feet / tan(25°) = 80 feet / 0.4663 = 171.6 feet
distance between Sylvia's office and the base of the building = 171.6 feet - 80 feet / tan(21°) = 171.6 feet - 80 feet / 0.3839 = 171.6 feet - 207.7 feet = -36.1 feet
However, the distance cannot be negative, so we need to re-evaluate our calculation.
distance between Sylvia's office and the base of the building = 80 feet / tan(21°) = 80 feet / 0.3839 = 207.7 feet
Conclusion
In this article, we have used trigonometry to solve a real-world problem involving angles of depression and elevation. We have visualized the problem, used the tangent function to relate the angles to the ratios of the sides, and solved for the distance between Sylvia's office and the base of the building. The final answer is 207.7 feet.
Real-World Applications
Trigonometry has numerous real-world applications, including:
- Physics: Trigonometry is used to describe the motion of objects in terms of position, velocity, and acceleration.
- Engineering: Trigonometry is used to design and analyze the structural integrity of buildings, bridges, and other infrastructure.
- Architecture: Trigonometry is used to design and plan the layout of buildings and other structures.
- Navigation: Trigonometry is used to determine the position and course of ships and aircraft.
Final Thoughts
Q: What is the difference between an angle of depression and an angle of elevation?
A: An angle of depression is the angle between the horizontal and the line of sight to the base of an object, while an angle of elevation is the angle between the horizontal and the line of sight to the top of an object.
Q: How do I use trigonometry to solve problems involving angles of depression and elevation?
A: To use trigonometry to solve problems involving angles of depression and elevation, you can use the tangent function to relate the angle to the ratio of the opposite side (the height of the object) to the adjacent side (the distance between the observer and the object).
Q: What is the formula for calculating the distance between an observer and an object using the tangent function?
A: The formula for calculating the distance between an observer and an object using the tangent function is:
distance = height / tan(angle)
Q: How do I calculate the height of an object using the tangent function?
A: To calculate the height of an object using the tangent function, you can rearrange the formula to solve for the height:
height = distance x tan(angle)
Q: What is the difference between a right-angled triangle and a non-right-angled triangle?
A: A right-angled triangle is a triangle with one angle that is 90 degrees, while a non-right-angled triangle is a triangle with no angles that are 90 degrees.
Q: How do I use trigonometry to solve problems involving right-angled triangles?
A: To use trigonometry to solve problems involving right-angled triangles, you can use the sine, cosine, and tangent functions to relate the angles to the ratios of the sides.
Q: What is the sine function?
A: The sine function is a trigonometric function that relates the angle of a right-angled triangle to the ratio of the opposite side (the height of the triangle) to the hypotenuse (the longest side of the triangle).
Q: What is the cosine function?
A: The cosine function is a trigonometric function that relates the angle of a right-angled triangle to the ratio of the adjacent side (the base of the triangle) to the hypotenuse (the longest side of the triangle).
Q: What is the tangent function?
A: The tangent function is a trigonometric function that relates the angle of a right-angled triangle to the ratio of the opposite side (the height of the triangle) to the adjacent side (the base of the triangle).
Q: How do I use the sine, cosine, and tangent functions to solve problems involving right-angled triangles?
A: To use the sine, cosine, and tangent functions to solve problems involving right-angled triangles, you can use the following formulas:
- sin(angle) = opposite side / hypotenuse
- cos(angle) = adjacent side / hypotenuse
- tan(angle) = opposite side / adjacent side
Q: What are some real-world applications of trigonometry?
A: Some real-world applications of trigonometry include:
- Physics: Trigonometry is used to describe the motion of objects in terms of position, velocity, and acceleration.
- Engineering: Trigonometry is used to design and analyze the structural integrity of buildings, bridges, and other infrastructure.
- Architecture: Trigonometry is used to design and plan the layout of buildings and other structures.
- Navigation: Trigonometry is used to determine the position and course of ships and aircraft.
Q: How can I practice trigonometry?
A: You can practice trigonometry by:
- Solving problems: Practice solving problems involving right-angled triangles and trigonometric functions.
- Using online resources: Use online resources such as calculators and interactive simulations to practice trigonometry.
- Working with a tutor: Work with a tutor or teacher to practice trigonometry and get feedback on your work.