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).
draw the triangle!
adjacent/opposite = 4/1, so the hypotenuse is √17
now you can find sin, cos, and 2*sin*cos
To find the values of sinθ and cos(2θ), we can start by using the given information: cot θ = 4.
To recall the relationship between trigonometric functions, we can use the identity cot θ = 1/tan θ. Therefore, cot θ = 4 is equivalent to tan θ = 1/4.
Now, let's find sinθ:
1. Start with the equation tan θ = 1/4.
2. Recall that tan θ = sin θ / cos θ.
3. Substitute sin θ / cos θ for tan θ: sin θ / cos θ = 1/4.
4. Cross multiply: sin θ = cos θ / 4.
Next, let's find cos(2θ):
1. Recall the double-angle identity for cosine: cos(2θ) = cos^2 θ - sin^2 θ.
2. Substitute the value of sin θ we found earlier: sin θ = cos θ / 4.
3. Square both sides of the equation: sin^2 θ = (cos θ)^2 / 16.
4. Substitute the expression for sin^2 θ in the double-angle identity: cos(2θ) = cos^2 θ - (cos θ)^2 / 16.
5. Simplify the expression: cos(2θ) = (16cos^2 θ - cos^2 θ) / 16.
6. Combine like terms: cos(2θ) = 15cos^2 θ / 16.
Therefore, the exact values are:
A. sin θ = cos θ / 4.
B. cos(2θ) = 15cos^2 θ / 16.
Note that we do not have enough information to determine the exact values of sin θ and cos θ separately because cot θ only provides the ratio between the adjacent and opposite sides of a right triangle, not their actual values.