Find the exact value of arctan(-sqrt3)?

Question

I am supposed to find an exact value in radians, but I can't get an exact value. Can someone please show me how to work through this problem? Thank you very much.

Answer 1

Basically, you just have to know what angle yields -√3 when you take the tangent of it, keeping in mind that tangent is opposite over adjacent, and also sine over cosine.

You may decide to start looking to certain families of angles. Perhaps you would start at π/4. At these angles, sine and cosine values are each equal to √(2)/2, so dividing sine by cosine will yield 1, while certainly isn't -√3.

So, let's move on to the π/3 family, which is another common angle. First, let's take the sine and cosine of π/3:

sin(π/3) = √(3)/2
cos(π/3) = 1/2

So, let's use the fact that:

tan(π/3) = sin(π/3) / cos(π/3)
==> plug in known values
tan(π/3) = [√(3)/2] / (1/2)
==> cancel 2's in denominators
tan(π/3) = [√(3)/2] / (1/2) = √(3)

OK, so we've got the right value, but it's positive instead of negative. Therefore, we must have that either sine or cosine is negative. Since we should know that arctangent is only defined in Quardrants I and IV, we know we would be in quadrant IV, where cosine is positive and sine is negative. Therefore, because the range of arctangent is −π/2 ≤ y ≤ π/2, we see that our only acceptable value is -π/3.

So, our answer is −π/3.

Answer 2

here is the foremost to remembering: Sin(00°) = ?0 / 2 — Cos(00°) = ?4 / 2 — Tan(00°) = ?0 / ?4 Sin(30°) = ?a million / 2 — Cos(30°) = ?3 / 2 — Tan(30°) = ?a million / ?3 Sin(40 5°) = ?2 / 2 — Cos(40 5°) = ?2 / 2 — Tan(40 5°) = ?2 / ?2 Sin(60°) = ?3 / 2 — Cos(60°) = ?a million / 2 — Tan(60°) = ?3 / ?a million Sin(ninety°) = ?4 / 2 — Cos(ninety°) = ?0 / 2 — Tan(ninety°) = ?4 / ?0 So, there that's at 30°: ?a million/?3 = a million/?3 = ?3/3

Answer 3

arctan (-sqrt(3)) = 120 or 300 degrees
In radians this would be 2*pi/3 or 5*pi/6. Both of thes are exact answers even though pi is irrational. Please note that the square root of 3 is also irrational.

Answer 4

You have to know the values of tangent at the standard angles. For this problem, you have to find the angle where tangent is -sqrt(3). You should know that the tangent of pi/3 is sqrt(3) and that tangent is an odd function. THis means that the tangent of -pi/3 is -sqrt(3). This, in turn, means that the arctangent of -sqrt(3) is -pi/3.

Answer 5

arctan(√3) = π/3

The first value of arctan(-√3) = π - π/3 = 2π/3.....2nd quadrant

The next one will be 2π - π/3 = 5π/3 .....4th quadrant and so on.