In the angle of the shadow of a 30 m long pool is 10√3m, then the angle elevation of the sun is
In the angle of the shadow of a 30 m long pool is 10√3m, then the angle elevation of the sun is
60°
70°
80°
90°
The Correct Option is A
Solution and Explanation
To determine the angle of elevation of the sun, we can use trigonometric principles. In this scenario, we have a right triangle formed by the height of the pool, its shadow, and the hypotenuse (line from the top of the pool to the end of the shadow). The shadow length given is 10√3 meters, and the pool's height is 30 meters.
To find the angle of elevation θ (the angle between the shadow and the line stretching from the pool top to shadow tip), we use the tangent function:
tan(θ) = Opposite/Adjacent
tan(θ) = Height of pool / Length of shadow = 30 / (10√3)
Simplify the expression:
tan(θ) = 30 / (10√3) = 3/√3
Further simplification yields:
tan(θ) = √3
From trigonometric tables, we know that:
tan(60°) = √3
Therefore, the angle of elevation θ is 60°.
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Concepts Used:
Trigonometric Functions
The relationship between the sides and angles of a right-angle triangle is described by trigonometry functions, sometimes known as circular functions. These trigonometric functions derive the relationship between the angles and sides of a triangle. In trigonometry, there are three primary functions of sine (sin), cosine (cos), tangent (tan). The other three main functions can be derived from the primary functions as cotangent (cot), secant (sec), and cosecant (cosec).
Six Basic Trigonometric Functions:
- Sine Function: The ratio between the length of the opposite side of the triangle to the length of the hypotenuse of the triangle.
sin x = a/h
- Cosine Function: The ratio between the length of the adjacent side of the triangle to the length of the hypotenuse of the triangle.
cos x = b/h
- Tangent Function: The ratio between the length of the opposite side of the triangle to the adjacent side length.
tan x = a/b
Tan x can also be represented as sin x/cos x
- Secant Function: The reciprocal of the cosine function.
sec x = 1/cosx = h/b
- Cosecant Function: The reciprocal of the sine function.
cosec x = 1/sinx = h/a
- Cotangent Function: The reciprocal of the tangent function.
cot x = 1/tan x = b/a
Formulas of Trigonometric Functions:
