Question

How do you evaluate `tan^-1 (-sqrt3)` without a calculator?

Answer 1

`x = (2pi)/3 + kpi`

Explanation:

`tan x = - sqrt3`. Find arc x.
Trig table and unit circle give:
`x = (2pi)/3 + kpi`

Answer 2

`arctan(-sqrt{3}) = 120^circ + 180^circ k quad`integer `k`

The principal value is `-60^circ`

Explanation:

Students are only expected to know two triangles, 30/60/90, and 45/45/90, and be able to figure them out in all four quadrants.

It's a goofy way to structure an entire course, but once you understand it problems become easier.

Rule of thumb, `sqrt{3}` means 30/60/90 and `sqrt{2}` means 45/45/90.

We know 30 and 60 degrees have sine and cosine `1/2` and `sqrt{3}/2.`

Tangent is slope, sine over cosine, so we have a sine of `-sqrt{3}/2` and a cosine of `1/2` or a sine of `sqrt{3}/2` and a cosine of `-1/2`. So fourth or second quadrant. When the sine is bigger, that's `60^circ,` so we're looking at `-60^circ` and `120^circ.`

The principal value is in the fourth quadrant, `-60^circ`. I prefer to treat `arctan` as the multivalued inverse,

`arctan(-sqrt{3}) = 120^circ + 180^circ k quad`integer `k`