Textbook Question
Solve each problem. See Examples 1–4. Altitude of a Triangle Find the altitude of an isosceles triangle having base 184.2 cm if the angle opposite the base is 68°44'.
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2. Trigonometric Functions on Right Triangles
Solving Right Triangles
3:38 minutesProblem 57
Textbook Question
Solve each problem.See Examples 3 and 4. Angle of Elevation of the Sun The length of the shadow of a building 34.09 m tall is 37.62 m. Find the angle of elevation of the sun to the nearest hundredth of a degree.
Verified step by step guidance1
Identify the right triangle formed by the building, its shadow, and the line of sight to the sun. The building height is the opposite side, the shadow length is the adjacent side, and the angle of elevation is the angle between the ground and the line of sight to the sun.
Label the known values: opposite side (height) = 34.09 m, adjacent side (shadow length) = 37.62 m, and the angle of elevation as \( \theta \).
Use the tangent function, which relates the opposite side and adjacent side in a right triangle: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). Substitute the known values: \( \tan(\theta) = \frac{34.09}{37.62} \).
To find the angle \( \theta \), take the inverse tangent (arctangent) of the ratio: \( \theta = \tan^{-1} \left( \frac{34.09}{37.62} \right) \).
Use a calculator to evaluate the inverse tangent and round the result to the nearest hundredth of a degree to find the angle of elevation.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Angle of Elevation
The angle of elevation is the angle formed between the horizontal ground and the line of sight to an object above the horizontal. In this problem, it is the angle between the ground and the sun’s rays as they hit the top of the building.
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Right Triangle Trigonometry
The building, its shadow, and the line from the top of the building to the tip of the shadow form a right triangle. Understanding how to apply trigonometric ratios like tangent, which relates opposite and adjacent sides, is essential to find the angle.
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Tangent Function
The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side. Here, tangent(angle) = height of building / length of shadow. Using the inverse tangent function allows calculation of the angle of elevation.
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