Use a reference triangle in an appropriate quadrant to find the angle.
-$\cot ^ { - 1 } ( - 1 )$
A) $- \frac { \pi } { 4 }$
B) $\frac { \pi } { 4 }$
C) $- \frac { 3 \pi } { 4 }$
D) $\frac { 3 \pi } { 4 }$

Question

Quizplus · Accepted Answer

We want to find the angle whose cotangent is -1. Recall that the cotangent is the adjacent over the opposite. Thus, we can draw a reference triangle in the third quadrant where the adjacent is -1 and the opposite is 1. 
[asy]
unitsize(0.5cm);
draw((0,-5)--(0,5),black+1.5bp,EndArrow(5));
draw((-5,0)--(5,0),black+1.5bp,EndArrow(5));
label("$x$",(6,0),fontsize(10));
label("$y$",(0,6),fontsize(10));
draw((0,0)--(-4,-4)--(-4,0));
label("$1$",(0,-2),E);
label("$-1$",(-2,-4),N);
label("$\theta$",(0.3,0.3),NE);
[/asy]
Now we can use the inverse cotangent function to find the angle theta. Since we are in the third quadrant, the angle theta will be between pi and 3pi/2. 
$$\cot^{-1}(-1) = \theta = \pi + \theta_0 = \pi + \frac{\pi}{4} = \frac{5\pi}{4}$$
However, we need to find the angle in the appropriate quadrant, which is between -pi/2 and pi/2. Since the cotangent function has a period of pi, we can subtract pi from our answer to get an equivalent angle between -pi/2 and pi/2. 
$$\frac{5\pi}{4} - \pi = \frac{\pi}{4}$$
Therefore, the answer is A.

Use a Reference Triangle in an Appropriate Quadrant to Find cot⁡−1(−1)\cot ^ { - 1 } ( - 1 )cot−1(−1)

Use a Reference Triangle in an Appropriate Quadrant to Find $\cot ^ { - 1 } ( - 1 )$