Why is the answer to the exact value of arctan(tan(3pi/4))= -pi/4
Since arctan(anything) has multiple answers, the mathematical convention is to use the answer closest to zero
arctan(tan(3π/4))= 3π/4 would be correct, but -π/4 is closer to zero
(let's look at degrees, most people can relate with degrees better than with radians)
3π/4 radians = 135° an angle in quadrant II
so we would have arctan(tan(135°))= -45°
we know that tan 135° = -1
but so is tan-45°, tan 315°, tan -225° or an infinite number of other angles
If we take arctan(-1) our calculator has been programmed to state the closest angle to zero, and we would get -45
test it with other trig inverses such as " arcsin(sin 210°) " , and you should get -30°
I hope this makes sense.
The answer to the given expression, arctan(tan(3π/4)), is equal to -π/4.
To understand why the answer is -π/4, we need to consider the properties and range of the arctan (inverse tangent) function, as well as the properties of the tangent function.
The tangent function (tan) is defined as the ratio of the sine of an angle to the cosine of that angle. It is periodic with a period of π, which means that for any angle θ, tan(θ + π) = tan(θ).
In the given expression, we have tan(3π/4). First, let's find the reference angle, which is the acute angle whose tangent is equal to the given value. In this case, tan(3π/4) = -1.
In the first quadrant (0 to π/2), the tangent function is positive. In the second quadrant (π/2 to π), the tangent function is negative. Since 3π/4 is in the second quadrant, we know that tan(3π/4) is negative.
Now, let's determine the equivalent angle in the first quadrant. We can use the property mentioned earlier to subtract multiples of π from the given angle until we obtain an angle in the first quadrant with the same tangent value.
3π/4 = π/4 + π/2
Subtracting π/2, we get π/4. This angle, π/4, is in the first quadrant, and its tangent is equal to -1 (the same as 3π/4).
Since the range of the arctan function is (-π/2, π/2), we can conclude that arctan(tan(3π/4)) = arctan(-1) = -π/4.
Note: It is essential to express angles within the appropriate range when using inverse trigonometric functions to ensure unique and accurate solutions.