Point P is located at the intersection of a circle with a radius of r and the terminal side of an angle q. Find the coordinates of p to the nearest hundreth.
q = -60° , r = 7
Assuming you are using angle rotation convention, that would put point P into the 4th quadrant.
After drawing the triangle with hypotenuse of 7 and angle of 60º, we can see that
sin -60 = y/7 ----> y = 7(sin -60º) = -6.06
cos -60º = x/7 = 7cos -60º = 3.5
so P is (3.5,-6.06)
or
we could have used the ratios of sides of the 30-60-90 triangle which are 1:√3:2
then x:y:7 = 1:√3:2
x/1 = 7/2 ---> x = 3.5
y/√3 = 7/2 ---> y = 7√3/2 = 6.06
Realizing where P is located,
P is (3.5,-6.06)
Well, well, well, we've got ourselves a point P playing hide and seek. Let's go on a wild journey to find its coordinates!
First, we'll need to remember our trigonometry. The coordinates of a point on a circle can be found using some good old sine and cosine action.
Given q = -60° and r = 7, we can unleash our trigonometric powers.
The x-coordinate of P can be found by using the cosine function: x = r * cos(q).
Plugging in our values, we get x = 7 * cos(-60°).
Now, here's the tricky part. Cos(-60°) is the same as cos(300°). Why? Because life is full of surprises and negative angles are just as fun as positive angles. So, let's rewrite our equation:
x = 7 * cos(300°).
Calculating cos(300°) gives us a value of -0.5, which means:
x = 7 * (-0.5) = -3.5.
For the y-coordinate of P, we use the sine function: y = r * sin(q).
Plugging in our values again, we get y = 7 * sin(-60°) = 7 * sin(300°).
And guess what? Sin(-60°) is equal to sin(300°)! So, let's simplify the equation further:
y = 7 * sin(300°).
Calculating sin(300°) gives us a value of -0.866.
y = 7 * (-0.866) = -6.06 (rounded to two decimal places).
So, the coordinates of point P are approximately (-3.5, -6.06) to the nearest hundredth.
And there you have it! Point P is no longer hiding from us. Keep those trigonometry skills sharp, my friend!
To find the coordinates of point P, we need to use trigonometry. The given angle q is in standard position, and since it is negative, we need to rotate clockwise by 60 degrees.
The x-coordinate of P can be found using the formula: x = r * cos(q)
The y-coordinate of P can be found using the formula: y = r * sin(q)
Substituting the given values, we have:
x = 7 * cos(-60°)
y = 7 * sin(-60°)
To calculate the values, we need to convert the angle to radians because trigonometric functions in most programming languages use radians instead of degrees.
To convert degrees to radians, we use the formula: radians = degrees * (pi / 180)
Converting -60° to radians:
radians = -60° * (pi / 180) = -pi / 3
Using the trigonometric identities, we have:
cos(-pi / 3) = 1 / 2
sin(-pi / 3) = -√3 / 2
Substituting the values in the formulas:
x = 7 * (1 / 2) = 3.5
y = 7 * (-√3 / 2) ≈ -6.06
Therefore, the coordinates of point P are approximately (3.5, -6.06) to the nearest hundredth.
To find the coordinates of point P, we can use the concept of trigonometry. The coordinates of point P will be given by (x, y), where x is the horizontal coordinate and y is the vertical coordinate.
For an angle of q = -60° in standard position (where the initial side is the positive x-axis and the terminal side rotates counterclockwise), we can determine the coordinates of point P as follows:
1. Find the x-coordinate using cosine (cos) function:
- In this case, we have q = -60°, so the cosine of -60° can be used.
- cos(-60°) = 0.5
- x = r * cos(q) = 7 * 0.5 = 3.5
2. Find the y-coordinate using sine (sin) function:
- In this case, we have q = -60°, so the sine of -60° can be used.
- sin(-60°) = -√3/2
- y = r * sin(q) = 7 * (-√3/2) = -3.5√3
Therefore, the coordinates of point P to the nearest hundredth are approximately (3.50, -3.80).