Find y if the point (7, y) is on the terminal side of θ and cos(θ) = 7/25
y^2 + 7^2 = 25^2
y^2 = 625 - 49 = 576
y = 24
so y^2 + 7^2 = 25^2
y^2 = 576
y = ± √576 = ± 24
To find the value of y if the point (7, y) is on the terminal side of angle θ and cos(θ) = 7/25, we can use the Pythagorean identity for cosine:
cos^2(θ) + sin^2(θ) = 1
Since cos(θ) = 7/25, we can substitute this value into the equation:
(7/25)^2 + sin^2(θ) = 1
Simplifying this equation, we get:
49/625 + sin^2(θ) = 1
To find sin^2(θ), we subtract 49/625 from both sides:
sin^2(θ) = 1 - 49/625
sin^2(θ) = 625/625 - 49/625
sin^2(θ) = 576/625
Taking the square root of both sides, we get:
sin(θ) = ±√(576/625)
sin(θ) = ±(24/25)
Since the point (7, y) is on the terminal side, y represents the y-coordinate. Since sin(θ) = y/r, we can find y by multiplying sin(θ) by the radius (r). In this case, the radius is 25.
So, y = ±(24/25) * 25
y = ±24
Therefore, the possible values of y are 24 and -24.
To find the value of y, we need to use the given information that (7, y) lies on the terminal side of an angle θ, and the cosine of θ is 7/25.
Since the point (7, y) lies on the terminal side of θ, we can find the value of y by determining the y-coordinate when x = 7 in the coordinate system.
To find y, we can use the unit circle and the relationship between the cosine and the x-coordinate on the unit circle. The unit circle is a circle with a radius of 1, centered at the origin (0,0). The x-coordinate of a point on the unit circle is equal to the cosine of the angle, and the y-coordinate is equal to the sine of the angle.
Since we are given that the cosine of θ is 7/25, we know that the x-coordinate of the corresponding point on the unit circle is 7/25.
Next, we need to find the y-coordinate of this point. We can use the Pythagorean theorem to do this. The Pythagorean theorem states that for any point (x, y) on the unit circle, the sum of the squares of the x and y coordinates is equal to 1.
In this case, the x-coordinate is 7/25, and the y-coordinate is what we are trying to find. So we have:
(7/25)^2 + y^2 = 1
Simplifying, we get:
49/625 + y^2 = 1
Now, we can solve for y:
y^2 = 1 - 49/625
y^2 = 576/625
Taking the square root of both sides, we get:
y = ±√(576/625)
Since (7, y) lies on the terminal side of the angle, y must be positive. So the value of y is:
y = √(576/625)
y = 24/25
Therefore, the value of y is 24/25 when the point (7, y) is on the terminal side of the angle θ, and cos(θ) = 7/25.