What are the exact values for which tan t = -(sqrt3)

t = - 1.73205

120 degrees and 300 degrees

To find the exact values of t for which tan t = -√3, we can use the inverse function of tangent, which is called arctan or atan. The arctan function yields the angle whose tangent is a given value.

First, we can take the arctan of both sides of the equation:

arctan(tan t) = arctan(-√3)

Since arctan and tan are inverse functions, they cancel out on the left side, leaving us with:

t = arctan(-√3)

Now, we need to evaluate the arctan function. It is important to note that the arctan function has a range of -π/2 to π/2 (or -90° to 90°).

In this case, since we know that tan t = -√3, it implies that t is in the 2nd and 4th quadrants since tangent is negative in those quadrants.

For the principal value, we can use the calculator to find arctan(-√3). By evaluating this in degrees, we get approximately:

t ≈ -60°

However, since t appears in the 2nd and 4th quadrants, we can add or subtract multiples of 180° to the principal value to find other possible values.

Thus, t can also be:

t ≈ -60° + 180° = 120°

or

t ≈ -60° - 180° = -240°

So the exact values for which tan t = -√3 are -60°, 120°, and -240°.