Angle θ is an acute angle of a right triangle. Determine the value of cot θ given the value of sin θ below. what is sin θ 15/17?
To determine the value of cot θ, we need to find the cosine of θ. We can use the Pythagorean identity to find the cosine:
cos²θ = 1 - sin²θ
Since sinθ = 15/17, we can substitute this value in:
cos²θ = 1 - (15/17)²
cos²θ = 1 - 225/289
cos²θ = 289/289 - 225/289
cos²θ = 64/289
Taking the square root of both sides, we get:
cosθ = ±8/17
Since θ is an acute angle, cosine is positive, so:
cosθ = 8/17
Finally, we can find the cotangent:
cotθ = cosθ / sinθ
cotθ = (8/17) / (15/17)
cotθ = 8/15
Therefore, cotθ = 8/15.
To determine the value of cot θ, we first need to find the value of sin θ which is given as 15/17.
In a right triangle, sin θ is defined as the ratio of the length of the side opposite angle θ to the length of the hypotenuse.
Given sin θ = 15/17, we can let the side opposite θ be 15 and the hypotenuse be 17 (since 15/17 is in simplified form).
Now, using the Pythagorean Theorem, we can find the length of the adjacent side of the right triangle. The Pythagorean Theorem states that the sum of the squares of the lengths of the two legs of a right triangle is equal to the square of the length of the hypotenuse.
Let's denote the length of the adjacent side as a. Using the Pythagorean Theorem:
a^2 + 15^2 = 17^2
a^2 + 225 = 289
a^2 = 289 - 225
a^2 = 64
a = √64
a = 8
Now, we have the values of the adjacent side and the opposite side: a = 8 and opposite = 15.
Finally, cot θ is the ratio of the adjacent side to the opposite side:
cot θ = adjacent/opposite
cot θ = 8/15
Therefore, the value of cot θ, given sin θ = 15/17, is 8/15.