Question #1
Suppose that θ is an angle in standard position whose terminal side intersects the unit circle at
−−2029,2129
Find the exact values of
cscθ
cotθ
sinθ
.
She means it intersects at (-20/29,-21/29)
x=-20/29
y=-21/29
To find the exact values of cscθ, cotθ, and sinθ, we need to find the coordinates of the point where the terminal side of the angle θ intersects the unit circle.
The given point is (-2029, 2129), which lies in the second quadrant of the coordinate plane. To find the coordinates of the point on the unit circle, we can use the Pythagorean theorem:
r² = x² + y²,
where r represents the radius of the unit circle (which is 1).
Let's find the length of the hypotenuse of the right triangle formed by the given point and the origin (0, 0):
r² = (-2029)² + 2129²,
r² = 4133641 + 4525841,
r² = 8659482.
Taking the square root of both sides:
r = √8659482.
Since the radius of the unit circle is 1, the coordinates of the point on the unit circle can be found by dividing the coordinates of the given point by the square root of 8659482:
(-2029/√8659482, 2129/√8659482).
Now, we can find the trigonometric values:
1. cscθ (Cosecθ):
cscθ = 1/sinθ.
Since sinθ is one of the values we need to find, we will calculate sinθ first.
sinθ = y/r,
sinθ = 2129/√8659482.
And therefore,
cscθ = 1/(2129/√8659482),
cscθ = √8659482/2129.
2. cotθ (Cotangentθ):
cotθ = x/y,
cotθ = -2029/2129.
3. sinθ (Sineθ):
As already calculated,
sinθ = 2129/√8659482.
So, the exact values of the trigonometric functions are:
cscθ = √8659482/2129,
cotθ = -2029/2129,
sinθ = 2129/√8659482.
To find the exact values of cscθ, cotθ, and sinθ, we first need to determine the coordinates of the point where the terminal side of θ intersects the unit circle.
The point given, (-2029, 2129), represents the (x, y) coordinates of a point on the unit circle. However, these coordinates are outside the range of the unit circle (radius 1). To find the corresponding point on the unit circle, we can normalize these coordinates to obtain new coordinates (x', y').
To normalize the coordinates, we divide both x and y by the distance from the origin to the given point. The distance can be found using the Pythagorean theorem:
distance = √(x^2 + y^2)
distance = √((-2029)^2 + 2129^2)
= √(4112441 + 4525041)
= √(8637482)
= 2941.29 (rounded to 2 decimal places)
Now, we divide both x and y by the distance:
x' = x / distance = -2029 / 2941.29 ≈ -0.689
y' = y / distance = 2129 / 2941.29 ≈ 0.724
So, the normalized coordinates (x', y') ≈ (-0.689, 0.724).
Now that we have the normalized coordinates, we can find the exact values of cscθ, cotθ, and sinθ by referring to the definitions of these trigonometric functions:
cscθ = 1 / sinθ
cotθ = cosθ / sinθ
sinθ = opposite / hypotenuse
In this case, since we have the y-coordinate (opposite) and the radius of the unit circle (hypotenuse = 1), we can directly obtain the value of sinθ:
sinθ = y' ≈ 0.724
Using sinθ, we can then find cscθ and cotθ:
cscθ = 1 / sinθ = 1 / 0.724 ≈ 1.380
cotθ = cosθ / sinθ
To find cosθ, we can use the Pythagorean identity:
cosθ = √(1 - sin^2(θ))
= √(1 - 0.724^2)
= √(1 - 0.525)
= √(0.475)
≈ 0.689
cotθ = cosθ / sinθ ≈ 0.689 / 0.724 ≈ 0.951
Therefore, the exact values of cscθ, cotθ, and sinθ for the given angle in standard position are approximately:
cscθ ≈ 1.380
cotθ ≈ 0.951
sinθ ≈ 0.724
well, you know that
x = -2029
y = 2129
so, r = 2941
So, now you just have to review your basic trig functions for angles in standard position. I was a bit confused by the double minus sign, but if I have it right, you're ready to move along.