The point P(k,24) is 25 units from the origin. If P is on the terminal arm of an angle in standard position, find, for each value of k.
a) The sin of the angle
b) The cosine of the angle
by Pythagoras
k^2 + 24^2 = 25^2
k^2 + 576 = 625
k^2 = 49
k = ± 7
if k = +7
sinØ = 24/25
cosØ = 7/25
if k= -7
sinØ = 24/25
sinØ = -7/25
To solve this problem, we need to use the distance formula and the unit circle.
a) The sine of an angle in standard position is the y-coordinate of the corresponding point on the unit circle.
We are given that the point P(k, 24) is 25 units from the origin. By applying the distance formula, we have:
√(k^2 + 24^2) = 25
Simplifying this equation:
k^2 + 576 = 625
k^2 = 625 - 576
k^2 = 49
Taking the square root of both sides:
k = ±√49
k = ±7
So, for each value of k, the possible points would be (7, 24) and (-7, 24).
Now, looking at the unit circle, we can determine the y-coordinate for each point:
For (7, 24):
The y-coordinate is 24, so the sine of the angle would be 24/25.
For (-7, 24):
The y-coordinate is 24, so the sine of the angle would also be 24/25.
Therefore, for each value of k, the sine of the angle is 24/25.
b) The cosine of an angle in standard position is the x-coordinate of the corresponding point on the unit circle.
Again, referring to the unit circle, we can determine the x-coordinate for each point:
For (7, 24):
The x-coordinate is 7, so the cosine of the angle would be 7/25.
For (-7, 24):
The x-coordinate is -7, so the cosine of the angle would be -7/25.
Therefore, for each value of k, the cosine of the angle is 7/25 for (7, 24) and -7/25 for (-7, 24).
To solve this problem, we can use the Pythagorean theorem to find the value of k. Since the point P(k, 24) is 25 units from the origin, we have the following equation:
sqrt(k^2 + 24^2) = 25
Simplifying the equation, we get:
k^2 + 24^2 = 25^2
k^2 + 576 = 625
k^2 = 625 - 576
k^2 = 49
k = sqrt(49)
k = 7
So, the value of k is 7.
Now, let's calculate the sine (sin) and cosine (cos) of the angle:
a) The sine of the angle is given by the formula:
sin = opposite/hypotenuse
In our case, the opposite side is 24 and the hypotenuse is 25. Therefore, the sine of the angle is:
sin = 24/25
b) The cosine of the angle is given by the formula:
cos = adjacent/hypotenuse
In our case, the adjacent side is k (which is 7) and the hypotenuse is 25. Therefore, the cosine of the angle is:
cos = 7/25
So, for the value of k = 7, the sin of the angle is 24/25 and the cosine of the angle is 7/25.