The difference of two acute angles in a right angled triangle is 2π/5.find the value of smallest angle?
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their sum is π/2, so if the smaller angle is x, then
x + x+2π/5 = π/2
To find the value of the smallest angle in a right-angled triangle, we need to determine the measures of the other two angles.
Let's assume:
- The smallest angle is denoted by θ.
- The other acute angle is denoted by α.
Given that the difference between these two acute angles is 2π/5, we have the equation:
α - θ = 2π/5
In a right-angled triangle, one of the angles is always 90 degrees or π/2 radians. Therefore, the sum of the two acute angles in a right-angled triangle is π/2.
We can express this as an equation:
θ + α = π/2
To find the value of the smallest angle, we need to solve this system of equations:
α - θ = 2π/5 (Equation 1)
θ + α = π/2 (Equation 2)
To eliminate α, we can add Equations 1 and 2:
α - θ + θ + α = 2π/5 + π/2
This simplifies to:
2α = 2π/5 + π/2
Now, we can solve for α by dividing both sides of the equation by 2:
α = (2π/5 + π/2)/2
Simplifying further:
α = (4π + 5π)/20
α = 9π/20
To find the value of the smallest angle (θ), substitute the value of α into either Equation 1 or 2:
θ + 9π/20 = π/2
Now, isolate θ by subtracting 9π/20 from both sides of the equation:
θ = π/2 - 9π/20
To simplify:
θ = (10π - 9π)/20
θ = π/20
Therefore, the value of the smallest angle in the right-angled triangle is π/20 radians.