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In this section, students learn how to determine the height of a tower based on the distance from the tower and the angle of elevation using basic trigonometric ratios, specifically the tangent function. The problem is solved step-by-step, showcasing the application of tan(60°).
In Example 1, we are tasked with finding the height of a vertical tower from a given point on the ground, situated 15 meters away from the base. An angle of elevation of 60° is observed towards the top of the tower, prompting us to apply trigonometric ratios. The key concept here is the tangent function in a right-angled triangle, which helps relate the angle of elevation to the opposite side (height of the tower, denoted as AB) and the adjacent side (distance from the point to the tower, denoted as BC). Utilizing the ratio tan(60°) = AB / BC, we substitute the value of BC (15m) and perform the necessary calculations to derive the height of the tower (15√3 m), reinforcing the significance of understanding angles and right triangles in practical applications.
Angle of Elevation: The angle from the observer's horizontal line of sight up to an object.
Right Triangle: A triangle with one angle equal to 90°.
Tangent: A trigonometric function relating opposite and adjacent sides in a right triangle.
If you want to find the height, with a glance up to the light, use tan to find the ratio right!
Imagine standing far from a tall tower, looking up as its peak begins to flower, using angles to measure height like a shower.
To remember tan, think 'Tall And Now' for the opposite over adjacent!
Example of measuring the height of a building or tree using its angle of elevation from a certain distance.
Finding the height of a ladder leaned against a wall by forming a right triangle.
Term: Angle of Elevation
Definition: The angle formed by a horizontal line to an observer's line of sight upward to an object.
The angle formed by a horizontal line to an observer's line of sight upward to an object.
Term: Trigonometric Ratio
Definition: A ratio of two sides of a right triangle, commonly used in trigonometry, such as sine, cosine, or tangent.
A ratio of two sides of a right triangle, commonly used in trigonometry, such as sine, cosine, or tangent.
Term: Tangent Function (tan)
Definition: A function that relates the angle θ in a right triangle to the ratio of the length of the opposite side to the length of the adjacent side.
A function that relates the angle θ in a right triangle to the ratio of the length of the opposite side to the length of the adjacent side.