Answer:
- 3π/4, 7π/4
- π/6, 5π/6, 7π/6, 11π/6
Step-by-step explanation:
You want the exact solutions on the interval [0, 2π) for the equations ...
- cot(x) +3 = 2
- csc(x)² -10 = -6
Approach
It is helpful to write each equation in the form ...
(trig function) = constant
Then the various solutions will be ...
angle = (inverse trig function)(constant)
along with all other angles in the interval that have the same trig function value.
1. Cot
cot(x) +3 = 2
cot(x) = -1 . . . . . . . subtract 3
x = arccot(-1) = -π/4
The cot function is periodic with period π, so we can add π and 2π to this value to see solutions in the interval of interest:
x = 3π/4, 7π/4
2. Csc
csc(x)² = 4 . . . . . add 10
csc(x) = ±2 . . . . . square root
sin(x) = ±1/2 . . . . relate to function values we know
x = ±π/6
The sine function is symmetrical about x = π/2 and periodic with period 2π, so there are additional solutions:
x = π/6, 5π/6, 7π/6, 11π/6
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Additional comment
A graphing calculator can help you identify and/or check solutions to these equations. It conveniently finds x-intercepts, so we have written the equations in the form f(x) = 0, graphing f(x).
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