Find the exact values requested. No decimal approximations.

Given that cot θ = 4 for an acute angle θ,
A. Find sinθ.
B. Find cos(2θ).

cot Ø = 4

then tan Ø = 1/4, and Ø is acute, thus in quadrant I

Did you make a sketch of the triangle?
recall that tanØ = y/x
so y = 1, and x = 4
can you find r ?
How is sinØ defined in terms of x, y, and r ?
Did you know that
cos 2Ø = cos^2 Ø - sin^2 Ø ?

To find the exact values of sinθ and cos(2θ) when cotθ = 4, we can use the trigonometric identities and the given information to solve for these values step by step.

A. Find sinθ:
We know that cotθ is the reciprocal of tanθ, so we can use the identity:
cotθ = 1 / tanθ

Substituting the given value cotθ = 4:
4 = 1 / tanθ

To find tanθ, take the reciprocal of both sides of the equation:
1/4 = tanθ

Now, we can use the Pythagorean identity for the tangent:
tanθ = sinθ / cosθ

Substituting the given equation:
1/4 = sinθ / cosθ

To solve for sinθ, multiply both sides of the equation by cosθ:
sinθ = (1/4) * cosθ

At this point, we still need to determine the exact value of cosθ in order to find sinθ. Since θ is an acute angle, we can use the Pythagorean identity for sine and cosine:
sin²θ + cos²θ = 1

Substituting the value of cotθ = 4, we can determine cos²θ as follows:
cot²θ + 1 = csc²θ
(4)² + 1 = csc²θ
16 + 1 = csc²θ
17 = csc²θ

Taking the square root of both sides of the equation, we find:
cscθ = √17

Since cscθ is the reciprocal of sinθ, we have:
1/sinθ = √17

Multiply both sides of the equation by sinθ to solve for sinθ:
sinθ = 1 / √17

To rationalize the denominator, multiply both the numerator and the denominator by √17:
sinθ = √17 / (√17 * √17)
sinθ = √17 / 17

Therefore, the exact value of sinθ is √17 / 17.

B. Find cos(2θ):
To find cos(2θ), we can use the double angle identity:
cos(2θ) = cos²θ - sin²θ

Using the values we found earlier, substitute sinθ = √17 / 17 and cosθ = √(1 - sin²θ):
cos(2θ) = (√(1 - (√17 / 17)²))² - (√17 / 17)²

Simplifying the equation:
cos(2θ) = (√(1 - 17/289))² - 17/289
cos(2θ) = (√(272/289))² - 17/289
cos(2θ) = (16/17)² - 17/289
cos(2θ) = 256/289 - 17/289
cos(2θ) = 239/289

Therefore, the exact value of cos(2θ) is 239/289.