Find the value of secant theta for angle theta in standard position if a point with coordinates (-3/4) lies on its terminal side
When in STD position, the center i8s at
the origin:
C(h , k),
C(0 , o), P(-3 , 4),
r^2 = (x - h)^2 + (y - k)^2,
r^2 = (-3 - 0)^2 + (4 - 0)^2,
r^2 = 9 + 16 = 25,
r = 5.
rcos(theta) = x -h = -3 -0 = -3,
5cos(theta) = -3,
cos(theta) = -3/5 = -0.6,
theta = 126.9 deg.
Sec(theta) = 1/cos(theta) = 1/-0.6 =
-1.67.
To find the value of secant theta for an angle in standard position, we need to determine the ratio of the hypotenuse to the adjacent side of the right triangle formed by the angle and the coordinate point.
In this case, we are given that a point with coordinates (-3/4) lies on the terminal side of angle theta in standard position. Let's denote the adjacent side as 'x' and the hypotenuse as 'r'. The point (-3/4) lies on the terminal side, and we can use the Pythagorean theorem to determine the value of the opposite side:
opposite side^2 + adjacent side^2 = hypotenuse^2
(-3/4)^2 + x^2 = r^2
9/16 + x^2 = r^2
Now, using the definition of secant:
sec(theta) = r / x
We want to find sec(theta), which means we need to find r and x. To determine these values, we can use the Pythagorean theorem:
r^2 = 9/16 + x^2
Since we know that the point (-3/4) lies on the terminal side, it means that the coordinates (-3/4) represent the ratio of x to r:
-3/4 = x / r
Now we have a system of equations to solve. We can substitute the second equation into the first equation:
r^2 = 9/16 + (-3/4)^2
r^2 = 9/16 + 9/16
r^2 = 18/16
Simplifying, we get:
r^2 = 9/8
Taking the square root of both sides, we find:
r = ± √(9/8)
Since r represents the hypotenuse, which is always positive, we can discard the negative value:
r = √(9/8)
Now we can substitute this value of r back into the second equation:
-3/4 = x / √(9/8)
Multiplying both sides by √(9/8), we have:
-3√(8/9) = x
Simplifying, we get:
-3√(8/9) = x
Finally, we can substitute the values of x and r into the formula for secant:
sec(theta) = r / x
sec(theta) = √(9/8) / (-3√(8/9))
By simplifying the expression in the numerator and denominator, we get:
sec(theta) = -√(9/8) / 3√(8/9)
Rationalizing the denominator, we multiply the numerator and the denominator by the conjugate of the denominator:
sec(theta) = -√(9/8) / 3√(8/9) * (√(8/9) / √(8/9))
Simplifying, we get:
sec(theta) = -√(9/8 * 8/9) / 3 * 8/9
Finally, canceling the common factors, we obtain:
sec(theta) = -√(1) / 3 * 8/9
sec(theta) = -1 / (3 * 8/9)
Multiplying the numerator and denominator, we have:
sec(theta) = -9 / 24
Reducing the fraction, we get:
sec(theta) = -3 / 8
Therefore, the value of secant theta for angle theta in standard position, when a point with coordinates (-3/4) lies on its terminal side, is -3/8.