Suppose that θ is an angle in standard position whose terminal side intersects the unit circle at , 1161−6061.
Find the exact values of secθ, cotθ, and sinθ.
To find the exact values of secθ, cotθ, and sinθ, we need to use the given coordinates of the point on the unit circle where the terminal side intersects.
Let's label the given point as (x, y) = (1161, -6061).
First, we need to find the radius (r) of the unit circle using the Pythagorean theorem:
r = √(x^2 + y^2)
r = √((1161)^2 + (-6061)^2)
r = √(1350721 + 36723221)
r = √38073942
r ≈ 6169.26
Now, we can find the values of secθ, cotθ, and sinθ.
secθ = r/x
cotθ = x/y
sinθ = y/r
Using the given coordinates:
secθ = 6169.26 / 1161 ≈ 5.316
cotθ = 1161 / (-6061) ≈ -0.1917
sinθ = (-6061) / 6169.26 ≈ -0.9829
Therefore, the exact values of secθ, cotθ, and sinθ are approximately 5.316, -0.1917, and -0.9829, respectively.
To find the values of sec(θ), cot(θ), and sin(θ), we first need to determine the position of the angle θ in the coordinate plane.
Given that the terminal side of θ intersects the unit circle at the point (x, y) = (1161, -6061), we can use the Pythagorean theorem to find the length of the radius (r) of the unit circle. The Pythagorean theorem states that r^2 = x^2 + y^2.
Let's calculate r:
r^2 = (1161)^2 + (-6061)^2
r^2 = 1350641 + 36758521
r^2 = 38109162
r ≈ 6179.07 (approximately)
Now that we have the radius, we can find the values of sec(θ), cot(θ), and sin(θ) using the trigonometric definitions:
1. sec(θ) is the reciprocal of the cosine function, which is equal to sec(θ) = 1 / cos(θ).
To find cos(θ), we can use the coordinates of the point (x, y) on the unit circle.
cos(θ) = x / r = 1161 / 6179.07 ≈ 0.188
Now we can calculate sec(θ):
sec(θ) = 1 / cos(θ) ≈ 1 / 0.188 ≈ 5.319
2. cot(θ) is the reciprocal of the tangent function, which is equal to cot(θ) = 1 / tan(θ).
To find tan(θ), we can use the coordinates of the point (x, y) on the unit circle.
tan(θ) = y / x = -6061 / 1161 ≈ -5.227
Now we can calculate cot(θ):
cot(θ) = 1 / tan(θ) ≈ 1 / -5.227 ≈ -0.191
3. sin(θ) is equal to y / r.
sin(θ) = y / r = -6061 / 6179.07 ≈ -0.981
Therefore, the exact values of sec(θ), cot(θ), and sin(θ) are approximately:
sec(θ) ≈ 5.319
cot(θ) ≈ -0.191
sin(θ) ≈ -0.981
recall that
sinθ = y/r
secθ = r/x
cotθ = x/y