Answer:
sin(t) = -4/7
Explanation:
If the terminal ray of an angle contains point (x, y), the sine of the angle will be ...
sin(t) = y/√(x^2 +y^2)
For your given points, the sine of the angle is ...
sin(t) = (-4/7)/√((-√33/7)^2 +(-4/7)^2) = -4/√(33 +4^2) = -4/√49
sin(t) = -4/7
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The signs of the coordinates tell you this is a 3rd-quadrant angle, where the tangent is positive and the sine is negative. You can use your calculator to find the reference angle:
θ = arctan(y/x) = arctan(-4/-√33) ≈ 38.499°
sin(38.499°) = 0.571428...(6-digit repeat) = 4/7
The sine of the 3rd-quadrant angle will be the opposite of this:
sin(t) = -4/7
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Additional comment
When you use your calculator in this way, do not round the value that is the result of the arctangent function. Use it to full calculator precision. If your calculator can convert the result to a fraction, you should have it do so.