Point P(k, 24) is 25 units from the origin. If P lies on the terminal arm of an angle in standard position, determine the measures of the angle(s)
Learn to recognize the easy Pythagorean triples. One of these is 7-24-25
Or, if you want to figure it out, just work with
k^2 + 24^2 = 25^2
Now you can find the angles.
To determine the measures of the angle(s), we need to find the value of k.
Given that point P(k, 24) is 25 units from the origin, we can use the distance formula to calculate the distance between P and the origin. The distance formula is:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
where (x₁, y₁) is the coordinate of the origin and (x₂, y₂) is the coordinate of point P. In this case, (x₁, y₁) = (0, 0) and (x₂, y₂) = (k, 24).
Applying the distance formula, we have:
25 = √((k - 0)² + (24 - 0)²)
25 = √(k² + 576)
25² = k² + 576
625 = k² + 576
k² = 625 - 576
k² = 49
Taking the square root of both sides, we get:
sqrt(k²) = sqrt(49)
k = ±7
Therefore, the possible values of k are 7 and -7.
Now, let's determine the measures of the angles. Since point P lies on the terminal arm of an angle in standard position, the angle's measure can be found using the inverse tangent function.
The formula for finding the angle θ when given the coordinates (x, y) is:
θ = atan(y / x)
In this case, we have a horizontal line with a y-coordinate of 24 (y = 24) and variable x-coordinate (x = k).
So, for k = 7, the corresponding angle θ is:
θ = atan(24 / 7)
And for k = -7, the corresponding angle θ is:
θ = atan(24 / -7)
Using a scientific calculator or trigonometric table, we can find the measures of the angles θ by evaluating the inverse tangent function for each case.