In some building especially in industries the roof is inclined this in...
Application of Trigonometry in Inclined Roofs
Given Information:
- The roof of the industry is inclined at angles alpha and beta with the horizontal line.
- The value of cosec alpha is √2.
- The value of cot beta is 1.
- Both alpha and beta are acute angles.
Solution:
Drawing:
First, draw a rough diagram of the inclined roof with the given angles α and β as shown below:
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Trigonometric Functions:
We know that:
- cosec α = 1/sin α
- cot β = 1/tan β
Using the Given Information:
From the given information, we have:
Finding sin α and sin β:
We can find sin α using the value of cosec α:
- cosec α = 1/sin α
- √2 = 1/sin α
- sin α = 1/√2 = √2/2
We can find sin β using the value of cot β:
- cot β = 1/tan β
- 1 = cos β/sin β
- sin β = cos β/1 = cos β
Finding sin (α + β):
Using the formula for sin (α + β), we get:
- sin (α + β) = sin α cos β + cos α sin β
- sin (α + β) = (√2/2) * 1 + (1/√2) * cos α
Finding cos α:
To find cos α, we can use the identity:
Solving for cos α, we get:
- sin² α + cos² α = 1
- (√2/2)² + cos² α = 1
- 1/2 + cos² α = 1
- cos² α = 1/2
- cos α = ±√(1/2)
Since alpha is an acute angle, cos α is positive. Therefore, cos α = √(1/2) = √2/2.
Substituting Values:
Substituting the values of sin α, cos α, and sin β into the formula for sin (α + β), we get: