When trying to solve
arctan(sqrt 567.2 / -459.6) my answer was -2.96. My teacher said that i handled the radical wrong. She said I have to square both numbers, then add them, and then find the square root. I am confused because you cannot take the square root of -459.6. Please explain.
Im not sure by your equation why you are to follow that process, but if you are to first square the two numbers then you get rid of the -ve
ie:
567.2^2 and (-459.6)^2
i hope this helps
arctan(sqrt 567.2 / -459.6)
= arctan(23.81596/-459.6)
= arctan(-.051819)
= -2.97°
most calculators have been programmed that for any arc(trig function) they will give you the smallest value of rotation from 0,
so -2.97 would be in the 4th quadrant, coterminal with 360-2.97 or 357.03°
the tangent is also negative in the 2nd quadrant, so
the smallest positive answer is 180-2.97 = 177.03°
check: tan 357.03 = tan 177.03 = .05188
√567.2/-459.6 = .0518189 , close enough,
The reason you cannot take the square root of a negative number like -459.6 is because it leads to what is known as an imaginary number. The square root of a negative number is not a real number.
In the given expression, arctan(sqrt(567.2 / -459.6)), you are taking the square root of both 567.2 and -459.6. However, since -459.6 is a negative number, you can't find its square root directly.
To handle this situation, you need to remember that the square root of a quotient is equal to the quotient of the square roots. In other words, sqrt(a/b) = sqrt(a) / sqrt(b), as long as a and b are positive numbers.
So to evaluate arctan(sqrt(567.2 / -459.6)), you should first evaluate sqrt(567.2) and sqrt(-459.6) separately.
- sqrt(567.2) ≈ -23.80
- sqrt(-459.6) = √(-1) * sqrt(459.6)
Now, the square root of -1 is represented by the imaginary unit "i." Therefore, sqrt(-459.6) becomes (√459.6)* i, where i is the imaginary unit.
Next, combine the results obtained for sqrt(567.2) and sqrt(-459.6):
-23.80 * (√459.6) * i
Finally, evaluate the arctan of -23.80 * (√459.6) * i. Since this involves complex numbers, the resulting angle will also be a complex number.
In summary, it seems there was an error in your teacher's explanation. You handled the radical correctly by not directly taking the square root of a negative number. However, make sure to further simplify the expression involving the square root of a negative number by representing it as the product of a positive number and the imaginary unit "i."