To find the value of tan(-284°24'), we can use the fact that the tangent function has period π, which means that:
tan(x) = tan(x + nπ)
where n is any integer. We can use this fact to convert the angle -284°24' to an equivalent angle between 0° and 360°:
-284°24' = -360° + 75°36' = 75°36'
Now, we need to find the reference angle, which is the acute angle between the terminal side of the angle and the x-axis. Since 75°36' is in the second quadrant (where the tangent function is positive), the reference angle is:
75°36' - 180° = -104°24'
Finally, we can use the identity:
tan(-θ) = -tan(θ)
to find the value of tan(-104°24'):
tan(-104°24') = -tan(104°24')
We can use a calculator to find that:
tan(104°24') ≈ 2.3835
Therefore:
tan(-284°24') ≈ -2.3835 (rounded to the nearest ten-thousandth)
So, the value of tan(-284°24') to the nearest ten-thousandth is -2.3835.