If an angle, x, is in standard position, and P(-3, 4) is a point on the terminal side of the angle, what is the value of sin x?
sinx=4/5
To find the value of sin x, we need to use the coordinates of point P (-3, 4) to determine the length of the side opposite to the angle x and the length of the hypotenuse.
In this case, the x-coordinate of point P is -3 and the y-coordinate is 4. This means that the point P is 3 units to the left of the origin (on the x-axis) and 4 units above the origin (on the y-axis).
To create a right-angle triangle, we draw a line from the origin (0,0) to point P (-3,4) and another line from point P to the x-axis.
The length of the side opposite to the angle x is the y-coordinate of point P, which is 4.
The hypotenuse can be found by using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the length of the hypotenuse is the distance between the origin (0,0) and point P (-3,4), which can be calculated using the distance formula.
Using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Distance = sqrt((-3 - 0)^2 + (4 - 0)^2)
Distance = sqrt((-3)^2 + 4^2)
Distance = sqrt(9 + 16)
Distance = sqrt(25)
Distance = 5
So, the length of the hypotenuse is 5.
Now, we can use the defined values to find the value of sin x. In trigonometry, sin x is defined as the ratio of the side opposite to the angle x to the hypotenuse. Therefore:
sin x = opposite/hypotenuse
sin x = 4/5
Hence, the value of sin x is 4/5.