If θ is an angle in standard position and its terminal side passes through the point (-20,21), find the exact value of \sin\thetasinθ in simplest radical form.
you have
x = -20
y = 21
r = 29
sinθ = y/r = 21/29
I assume you want to find sinθ?
We know that sinθ = opposite/hypotenuse and that our coordinate point is in Quadrant II. If we construct a right triangle with a height of 21 units and a length of 20 units from right to left, then the side opposite of θ is 21 and the side adjacent to θ is -20. We need to determine the hypotenuse, which can easily be solved with the Pythagorean Theorem:
opp^2 + adj^2 = hyp^2
21^2 + (-20)^2 = hyp^2
441 + 400 = hyp^2
841 = hyp^2
29 = hyp
Therefore, sinθ=21/29
Well, aren't you asking the tough questions! Let's solve this math puzzle one step at a time.
First, let's draw the given point (-20, 21) on the coordinate plane. Hmmm... it seems like we have quite an adventurous angle on our hands!
Now, we need to find the length of the hypotenuse, which is the straight line segment connecting the origin (0,0) to the point (-20, 21). We can use the Pythagorean theorem for that. The hypotenuse's length is given by √((-20)^2 + 21^2). Simplifying that square root might be a bit tricky, but luckily, I'm up for the challenge!
√((-20)^2 + 21^2) = √(400 + 441) = √841 = 29.
Great! Now, we can find sin(θ) by using the definition of sine: sin(θ) = opposite/hypotenuse. The opposite side in this case is 21, and the hypotenuse is 29.
Therefore, sin(θ) = 21/29.
And there you have it, the exact value of sin(θ) in simplest radical form. Now, wasn't that a sin-tillating journey?
To find the exact value of sinθ, we first need to find the values of the hypotenuse and the opposite side of the right triangle formed by the terminal side of the angle θ.
The hypotenuse is the distance from the origin (0,0) to the point (-20,21), which can be found using the Pythagorean theorem:
h = √((-20)^2 + 21^2)
= √(400 + 441)
= √841
= 29
The opposite side is the vertical distance from the point (-20,21) to the x-axis, which is 21.
Therefore, sinθ is the ratio of the opposite side to the hypotenuse:
sinθ = opposite/hypotenuse
= 21/29
So, the exact value of sinθ in simplest radical form is 21/29.
To find the exact value of sin(θ), we need to determine the trigonometric ratio using the given information. Here are the steps to solve the problem:
Step 1: Determine the hypotenuse of the right triangle
Since the point (-20,21) lies on the terminal side of the angle θ, we can use the distance formula to find the length of the hypotenuse. The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Here, (x1, y1) = (0, 0) (origin) and (x2, y2) = (-20, 21). Substituting the values, we get:
d = √((-20 - 0)^2 + (21 - 0)^2)
= √((-20)^2 + 21^2)
= √(400 + 441)
= √841
= 29
So, the length of the hypotenuse (d) is 29.
Step 2: Determine the opposite side of the right triangle
The opposite side of the right triangle corresponds to the y-coordinate of the given point. Since the y-coordinate of the point (-20, 21) is 21, the length of the opposite side is 21.
Step 3: Determine the sin(θ) ratio
The sin of an angle θ is given by the ratio of the opposite side to the hypotenuse:
sin(θ) = opposite/hypotenuse
Substituting the values, we get:
sin(θ) = 21/29
Therefore, the exact value of sin(θ) is 21/29 in simplest radical form.