if P(-4,1) is a point on the terminal side of angle (theta), in standard position, then what is the exact value of csc (theta)?
To find the exact value of csc(theta), we need to determine the value of theta first.
Since point P(-4, 1) lies on the terminal side of angle theta, we can use the Pythagorean theorem to find the length of the hypotenuse (r) of the right triangle formed.
Using the coordinates of the point P, we have:
r = √((-4)^2 + 1^2) = √(16 + 1) = √17
Next, we can determine the value of the sine function (sin(theta)) using the y-coordinate of point P:
sin(theta) = y/r = 1/√17
Finally, we can find the value of csc(theta) by taking the reciprocal of sin(theta):
csc(theta) = 1/sin(theta) = √17/1 = √17
Therefore, the exact value of csc(theta) is √17.
To determine the exact value of csc(theta), we need to find the reciprocal of the sine function.
Given that (-4,1) is a point on the terminal side of angle (theta) in standard position, we can determine the values of the trigonometric functions using the coordinates of the point.
Let's label the coordinates of the point (-4,1) as (x, y).
From the Pythagorean theorem, we have:
x^2 + y^2 = r^2, where r represents the radius of the circle (in this case, the distance from the origin to the point (-4, 1)).
Plugging in the values:
(-4)^2 + (1)^2 = r^2
16 + 1 = r^2
17 = r^2
Taking the square root of both sides gives us:
r = √17
Now, we can determine the sine of theta, which is y/r:
sin(theta) = y/r = 1/√17
Finally, the reciprocal of the sine function is the cosecant function:
csc(theta) = 1/sin(theta) = 1/(1/√17)
csc(theta) = √17
Therefore, the exact value of csc(theta) is √17.
Did you make a sketch?
from (-4,1), x = -4, y = 1, hypotenuse = √17
csc(theta) = 1/sin(theta)
simple from here.