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Let x = 33/65 be the x-coordinate of the point P(x, y), where the terminal side of angle θ (in standard position) meets the unit circle. If P is in Quadrant IV,
what is sin (θ)?
Since P is in Quadrant IV, the x-coordinate is positive and the y-coordinate is negative. To find the y-coordinate, we can use the Pythagorean identity:
sin(θ) = y/r
Since the unit circle has a radius of 1, we can rewrite the equation as:
sin(θ) = y
To find the y-coordinate, we can use the Pythagorean theorem:
x^2 + y^2 = r^2
Substituting the given x-coordinate and solving for y:
(33/65)^2 + y^2 = 1
(1089/4225) + y^2 = 1
y^2 = 1 - 1089/4225
y^2 = 4225/4225 - 1089/4225
y^2 = 3136/4225
Taking the square root of both sides to solve for y:
y = ± √(3136/4225)
Since P is in Quadrant IV, the y-coordinate is negative:
y = -√(3136/4225)
Now we can substitute the value of y into the equation for sin(θ):
sin(θ) = -√(3136/4225)
Therefore, sin(θ) is approximately -0.7857.