Question

1)Find all exact solutions on the interval 0 ≤ x < 2π. (Enter your answers as a comma-separated list.)

cot(x) + 3 = 2

2) Find all exact solutions on the interval 0 ≤ x < 2π. (Enter your answers as a comma-separated list.)

csc2(x) − 10 = −6

Answer 1

Answer:

1. 3π/4, 7π/4
2. 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 ...

1. cot(x) +3 = 2
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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