16. Let W(ć ) be a point in Quadrant I on the unit circle with center O. W(ć )B is perpendicular to
the x-axis at B, OB = 0.6. and W(ƒá )B ƒ 0.8. Find:
Why was the unit circle always so confident?
Because it knew its center was always "O-K"!
To solve this problem, we'll use the Pythagorean theorem and trigonometric identities. Let's start step by step:
Step 1: Draw a diagram
Draw a diagram representing the situation described in the problem. Label the point on the unit circle as W(ć ), the center of the unit circle as O, and the point on the x-axis as B.
Step 2: Use the Pythagorean theorem
Since OB is perpendicular to the x-axis, we can create a right triangle with OB as the hypotenuse and the coordinates of B as its legs. Let's call the coordinates of B as (x, 0), where x represents the distance of B from the origin.
According to the Pythagorean theorem, we have:
OB^2 = x^2 + 0^2
Simplifying, we get:
(0.6)^2 = x^2
0.36 = x^2
x = ± √0.36
Since W(ć ) is in Quadrant I, we take the positive square root:
x = √0.36
x ≈ 0.6
Step 3: Calculate the y-coordinate of W(ć )
Since W(ć )B is perpendicular to the x-axis, the y-coordinate of W(ć ) is equal to the length of W(ć )B. Given that W(ć )B = 0.8, we can determine the y-coordinate of W(ć ) as follows:
W(ć ) = W(ć )O - OB
W(ć ) = 1 - 0.6
W(ć ) Š0.4
Step 4: Write the coordinates of W(ć )
The coordinates of W(ć ) are (0.6, 0.4)
Step 5: Find the angle ć
To find the angle ć, we can use trigonometric identities. Since W(ć ) lies on the unit circle, we know that the sine of the angle ć is equal to the y-coordinate of the point. Therefore:
sin(ć ) = 0.4
To find the value of ć , we can use the inverse sine function (also known as arcsine):
ć = arcsin(0.4)
ć Š0.4115 radians (approximately)
So, the coordinates of W(ć ) are approximately (0.6, 0.4), and the angle ć is approximately 0.4115 radians.
To find the coordinates of point W(ć), we need to use the trigonometric ratios sine and cosine.
Let's break down the information given:
1. Point W(ć) is in Quadrant I, which means its coordinates will have positive x and y values.
2. Point B is the intersection of the line W(ć)B with the x-axis, and OB = 0.6.
3. The line W(ć)B is perpendicular to the x-axis at B, which means it makes a right angle with the x-axis.
4. The length of W(ć)B is 0.8.
Since OB is the same as the x-coordinate of W(ć), we can conclude that the x-coordinate of W(ć) is 0.6.
Now, we can use the Pythagorean theorem to find the length of the line segment W(ć)O (i.e., the distance from W(ć) to the origin O).
Using the Pythagorean theorem:
(W(ć)B)^2 + (OB)^2 = (W(ć)O)^2
(0.8)^2 + (0.6)^2 = (W(ć)O)^2
0.64 + 0.36 = (W(ć)O)^2
1 = (W(ć)O)^2
Taking the square root of both sides, we find:
W(ć)O = 1
Since W(ć)O is the same as the radius of the unit circle, we know that point W(ć) lies on the unit circle.
Therefore, the coordinates of W(ć) are (0.6, 0.8).