Find the value of csc Q for angle Q in standard position if the point at (3, –1) lies on its terminal side.
To find the value of csc(Q), we first need to determine the position of angle Q in standard position.
Given that the point (3, -1) lies on the terminal side of angle Q, we can visualize this as a right triangle formed by the coordinates of the point and the origin (0, 0) as follows:
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(0, 0) (3, -1)
We can see that the horizontal leg of the triangle has a length of 3, and the vertical leg has a length of -1 (since the point lies below the x-axis).
Using the Pythagorean theorem, we can find the length of the hypotenuse (r) by applying the formula:
r^2 = x^2 + y^2,
where x is the length of the horizontal leg and y is the length of the vertical leg.
r^2 = 3^2 + (-1)^2
r^2 = 9 + 1
r^2 = 10
r = sqrt(10)
In standard position, the value of csc(Q) is defined as the reciprocal of the sine of angle Q. To find the sine of angle Q, we divide the length of the vertical leg by the length of the hypotenuse:
sin(Q) = y / r
sin(Q) = -1 / sqrt(10)
Finally, we can determine csc(Q) by taking the reciprocal of the sine of angle Q:
csc(Q) = 1 / sin(Q)
csc(Q) = 1 / (-1 / sqrt(10))
csc(Q) = -sqrt(10)
To find the value of csc(Q), we need to first determine the value of the sine of angle Q.
We know that the point (3, -1) lies on the terminal side of angle Q in standard position. Using this information, we can determine the values of the x and y coordinates.
The x-coordinate represents the adjacent side, which is 3.
The y-coordinate represents the opposite side, which is -1.
Now, we can use the formula for sine:
sin(Q) = opposite/hypotenuse
sin(Q) = (-1)/hypotenuse
To find the hypotenuse, we can use the Pythagorean theorem:
hypotenuse^2 = adjacent^2 + opposite^2
hypotenuse^2 = 3^2 + (-1)^2
hypotenuse^2 = 9 + 1
hypotenuse^2 = 10
Taking the square root of both sides, we find:
hypotenuse = √10
Now, we can substitute this value back into the sine equation:
sin(Q) = (-1)/√10
Using the reciprocal property of trigonometric functions, we can find the value of csc(Q):
csc(Q) = 1/sin(Q)
csc(Q) = 1/((-1)/√10)
csc(Q) = -√10
Therefore, the value of csc(Q) is -√10.