Find two standard position primary angles in radians by solving for the unknown. "Primary angles" are those angles which exist between 0 and 2pi. As usual, use exact values in your calculations.
cotx + 1 = 0
To find the two standard position primary angles in radians, we can solve the equation cot(x) + 1 = 0.
Step 1: Subtract 1 from both sides to isolate cot(x) term:
cot(x) = -1
Step 2: Take the reciprocal of both sides to isolate x:
1/cot(x) = -1
tan(x) = -1
Step 3: Use the inverse tangent function to solve for x:
x = arctan(-1)
Now, let's find the value of arctan(-1) using exact values.
The value of arctan(-1) is an angle whose tangent is -1. This occurs in the third and fourth quadrants.
In the third quadrant (negative x and negative y), the tangent is negative. So, we have:
x = arctan(-1) = π + arctan(1)
In the fourth quadrant (positive x and negative y), the tangent is also negative. So, we have:
x = arctan(-1) = 2π + arctan(1)
Therefore, the two standard position primary angles in radians for the given equation cot(x) + 1 = 0 are:
x = π + arctan(1)
x = 2π + arctan(1)
Note: arctan(1) can be simplified further using exact values, but since the question requests the use of exact values in calculations, we will leave it as it is.