Each point on this coordinate gird makes a right triangle with the origin, A, and the x-axis. determine the indicated trigonometric ratio in each triangle.
A= (0,0)
P= (-4,3)
1) Sin A and point P
For any angle θ in standard position recall that
r^2 = x^2 + y^2
sinθ = y/r
cosθ = x/r
tanθ = y/x
To determine the indicated trigonometric ratio (sin A) in the right triangle formed by point A and point P, we need to find the length of the side opposite to angle A and the hypotenuse.
1) Find the length of the side opposite angle A:
We have point A at (0, 0) and point P at (-4, 3). To find the length of the side opposite angle A, we need to find the difference in the y-coordinates between the two points.
Length of the side opposite A = difference in the y-coordinates = 3 - 0 = 3.
2) Find the length of the hypotenuse:
To find the length of the hypotenuse, we need to find the distance between point A and point P. The distance between two points can be found using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-4 - 0)^2 + (3 - 0)^2)
= sqrt((-4)^2 + (3)^2)
= sqrt(16 + 9)
= sqrt(25)
= 5.
3) Calculate sin A:
sin A = opposite/hypotenuse
= 3/5.
Therefore, the value of sin A in the right triangle formed by point A and point P is 3/5.