Question

How do you solve `Tan 3x = -1`?

Answer 1

The answer is 135°; any other angle can be found by remembering the periodicity of the tan-function, that is to say, 180°.

Explanation:

By applying the mathematical definition of tangent:

`tan(x)=sin(x)/cos(x)`

By using our demanded value:

`sin(x)=-cos(x)`

By applying the trigonometric property: `cos(x)=sqrt(1 - sin(x)^2)`

`sin(x)=sqrt(2)/2`

By using a calculator, the easy way, or interpolation, the hard way, we get:

`arcsin(sqrt(2)/2)= 45°`

Remembering `x=3*x`

we get the "basal" value: 135°. and apply:

`x= 135 +- n*180`

where n is integer, you can get any other solution.

PS: there is a nice thing to do, a trick I used to apply to my private students, I will try to make a diagram, you can find the basic angles such as the one just presented herein, easily to remember.