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°.