If x = 6sinθ, use trigonometric substitution to write square root 36-x^2 as a trigonometric function of θ, where where 0<θ< pi/2
36-x^2
= 36-36sin^2θ
= 36(1-sin^2θ)
= 36cos^2θ
so, √(36-x^2) = 6cosθ
To solve this problem, we can use the trigonometric identity for a right triangle:
sin^2(θ) + cos^2(θ) = 1
Starting with x = 6sinθ, let's solve for sinθ:
x = 6sinθ
Dividing both sides by 6:
sinθ = x/6
Now, we can substitute this value for sinθ in the trigonometric identity:
(x/6)^2 + cos^2(θ) = 1
Expanding the left side:
x^2/36 + cos^2(θ) = 1
Subtracting x^2/36 from both sides:
cos^2(θ) = 1 - x^2/36
Taking the square root of both sides:
cosθ = √(1 - x^2/36)
But we want to express square root (36 - x^2) as a trigonometric function of θ. So, we'll multiply both sides by √36 to get rid of the denominator:
√36 * cosθ = √36 * √(1 - x^2/36)
Simplifying:
6cosθ = √(36 - x^2)
Therefore, the square root of (36 - x^2) can be written as a trigonometric function of θ as:
√(36 - x^2) = 6cosθ
To write the expression √(36 - x^2) as a trigonometric function of θ, we need to substitute x with its equivalent expression in terms of θ, which is x = 6sinθ.
First, let's find the value of x^2 using the given expression:
x^2 = (6sinθ)^2 = 36sin^2θ
Now, let's substitute this value back into the original expression:
√(36 - x^2) = √(36 - 36sin^2θ)
Using a trigonometric identity, we can rewrite sin^2θ as (1 - cos^2θ):
√(36 - 36sin^2θ) = √(36 - 36(1 - cos^2θ))
Simplifying further:
√(36 - 36sin^2θ) = √(36 - 36 + 36cos^2θ)
√(36 - 36sin^2θ) = √(36cos^2θ)
Finally, taking the square root of 36cos^2θ, we get:
√(36 - 36sin^2θ) = 6cosθ
Therefore, √(36 - x^2) is equivalent to 6cosθ when 0 < θ < π/2.