Angle of Elevation and Depression Word Problems Trigonometry, Finding Sides, Angles, Right Triangles
TLDRThis educational video script offers a detailed guide on solving word problems involving angles of elevation and depression. It begins with a scenario where a man measures a 30-degree angle of elevation to a building 800 feet away, using the tangent function to calculate the building's height as approximately 461.88 feet. The script then explains how to calculate the angle of elevation to a 50-foot tree from a point 20 feet away, resulting in an angle of 68.2 degrees. Lastly, it addresses how to find the distance of a boat from a 100-foot observation tower when the angle of depression is 10 degrees, concluding the boat is about 567 feet away. The video emphasizes the importance of understanding trigonometric functions and the correct application of tangent, sine, and cosine in these types of problems.
Takeaways
- ๐ Solving word problems involving angles of elevation and depression.
- ๐ Example problem: Finding the height of a building using a measured angle of elevation (30 degrees) from 800 feet away.
- ๐ Draw a right triangle to visualize the problem.
- ๐ข Use the tangent trigonometric function for problems involving opposite and adjacent sides.
- ๐ Tangent formula: tan(theta) = opposite / adjacent.
- ๐งฎ Example calculation: Height of the building = 800 * tan(30 degrees), approximately 461.88 feet.
- ๐ Another problem: Finding the angle of elevation to a 50-foot tree 20 feet away.
- ๐ Use the inverse tangent function to find the angle: arctan(2.5) โ 68.2 degrees.
- ๐ข Example problem: Finding the distance from a 100-foot observation tower to a boat with a 10-degree angle of depression.
- ๐ Distance calculation: x = 100 / tan(10 degrees), approximately 567 feet.
- ๐ Remember: Angles of elevation are above the horizontal line, and angles of depression are below it.
Q & A
What is the main focus of the video?
-The video focuses on solving word problems related to the angle of elevation and depression.
What is the scenario described in the first problem of the video?
-The scenario involves a man who measures the angle of elevation between the ground and a building that is 800 feet away to be 30 degrees, and the task is to find out how tall the building is.
Why is the height of the man considered irrelevant to the height of the building in the first problem?
-The height of the man is irrelevant because the problem is concerned with the angle of elevation from the ground to the building, not from the man's eye level to the building.
What trigonometric function is used to solve for the height of the building in the first problem?
-The tangent function is used to solve for the height of the building, as it relates the opposite side (the height of the building) to the adjacent side (the distance from the man to the building).
What is the value of tangent for a 30-degree angle, and why is it used in the first problem?
-The value of tangent for a 30-degree angle is โ3/3. It is used because the problem involves finding the height of the building (opposite side) using the distance (adjacent side) and the angle of elevation.
How is the height of the building calculated in the first problem?
-The height of the building is calculated by multiplying 800 feet (the distance) by the tangent of 30 degrees, which gives 800 * (โ3/3), resulting in approximately 461.88 feet.
What is the purpose of the second problem in the video?
-The second problem demonstrates how to calculate the angle of elevation when given the height of an object (a 50-foot tree) and the distance from the object (20 feet away).
How is the angle of elevation found in the second problem?
-The angle of elevation is found by taking the inverse tangent (arctan) of the ratio of the height of the tree (50 feet) to the distance from the tree (20 feet), which results in an angle of approximately 68.2 degrees.
What is the scenario in the third problem presented in the video?
-The third problem involves a man on a 100-foot observation tower who measures the angle of depression to a boat to be 10 degrees, and the task is to find out how far the boat is from the tower.
How is the distance of the boat from the tower calculated in the third problem?
-The distance is calculated by dividing 100 feet (the height of the tower) by the tangent of the angle of depression (10 degrees), which is approximately 0.17633, resulting in a distance of about 567 feet.
What is the key takeaway from the video regarding solving angle of elevation and depression problems?
-The key takeaway is to remember that for angle of elevation problems, the angle is above the horizontal line, and for angle of depression problems, the angle is below the horizontal line, and to use the appropriate trigonometric function (sine, cosine, or tangent) based on the sides of the right triangle you know.
Outlines
๐ Solving Word Problems Involving Angle of Elevation and Depression
This paragraph introduces the focus of the video on solving word problems related to the angle of elevation and depression. It presents a problem where a man measures the angle of elevation between the ground and a building 800 feet away to be 30 degrees and aims to determine the building's height. The explanation includes drawing a right triangle, understanding the concepts of angle of elevation and depression, and using trigonometric functions (SOHCAHTOA) to find the solution. The tangent ratio is identified as the most suitable for solving the problem, leading to the height of the building being approximately 461.88 feet.
๐งฎ Calculating Angle of Elevation from Ground to a Tree
This paragraph discusses how to calculate the angle of elevation from a point on the ground to a 50-foot tree that is 20 feet away. It explains the process of drawing a right triangle and using the tangent ratio (opposite side divided by the adjacent side) to determine the angle. By simplifying the ratio and using the inverse tangent function, the angle of elevation is calculated to be approximately 68.2 degrees. The explanation emphasizes the importance of using the correct trigonometric function based on the given sides of the triangle.
๐ค Finding Distance Using Angle of Depression
This paragraph explains how to determine the distance of a boat from a 100-foot observation tower by measuring the angle of depression to be 10 degrees. The description includes drawing a right triangle, identifying the angle of depression, and using the tangent ratio to solve for the distance. The height of the tower and the angle of depression are used to find that the boat is approximately 567 feet away from the shoreline. The explanation reinforces the concept of using the tangent ratio for problems involving angles of elevation and depression.
๐ Reviewing Angles of Elevation and Depression
This paragraph provides a review of the concepts of angle of elevation and angle of depression. It highlights the differences between the two, where the angle of elevation is above the horizontal line and the angle of depression is below it. The summary emphasizes the importance of understanding these concepts for solving related word problems. The video concludes by thanking viewers and encouraging them to remember these key points when approaching similar problems.
Mindmap
Keywords
๐กAngle of Elevation
๐กRight Triangle
๐กTrigonometric Functions
๐กSOHCAHTOA
๐กHypotenuse
๐กAdjacent Side
๐กOpposite Side
๐กTangent Ratio
๐กInverse Tangent
๐กAngle of Depression
๐กObservation Tower
Highlights
Introduction to solving word problems involving angle of elevation and depression.
Explanation of a scenario where a man measures the angle of elevation to a building 800 feet away at 30 degrees.
Irrelevance of the man's height to the building's height in the problem.
Drawing a right triangle to represent the problem visually.
Differentiation between angle of elevation and angle of depression in the context of the problem.
Introduction to the trigonometric functions sine, cosine, and tangent.
Selection of the tangent function as the most suitable for solving the building height problem.
Use of the tangent ratio to set up the equation for the building's height.
Calculation of the building's height using the tangent of 30 degrees.
Explanation of the exact value of tangent 30 as root 3 over 3.
Conversion of the exact answer to a decimal value for practical use.
Demonstration of calculating the angle of elevation to a 50-foot tree from a 20-foot distance.
Use of the inverse tangent function to find the angle when given two sides of a triangle.
Problem involving a man on a 100-foot observation tower measuring the angle of depression to a boat.
Assumption that the boat's height is negligible for the problem.
Application of the tangent ratio to find the distance of the boat from the tower.
Final calculation showing the boat is approximately 567 feet away from the tower.
Review and summary of the importance of understanding angles in relation to the horizontal line for solving word problems.
Transcripts
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