if the terminal side of an angle , theta, passes through the point (3,-5). find sin(theta)
sin(theta) = y/r
just to remind you,
r^2 = x^2+y^2
To find sin(theta), we need to determine the value of theta in a right triangle.
We are given that the terminal side of the angle theta passes through the point (3, -5). From this information, we can find the length of the side opposite to the angle (vertical distance) and the length of the hypotenuse of the right triangle.
To find the length of the opposite side, we use the y-coordinate of the given point, which is -5. The length of the opposite side is equal to the absolute value of the y-coordinate, so the length of the opposite side is 5.
To find the length of the hypotenuse, we use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Using the formula, c^2 = a^2 + b^2, where c is the hypotenuse, and a and b are the other two sides, we substitute the values we know:
c^2 = 3^2 + (-5)^2
c^2 = 9 + 25
c^2 = 34
Now, we take the square root of both sides to find the length of the hypotenuse:
c = √34
Now that we know the length of the opposite side (5) and the length of the hypotenuse (√34), we can determine the value of sin(theta) by dividing the opposite side by the hypotenuse:
sin(theta) = opposite/hypotenuse
sin(theta) = 5/√34
Note: The value of sin(theta) can be further simplified by rationalizing the denominator (multiplying the numerator and denominator by √34) to remove the radical sign if necessary.