Solve csc(4x) − 4 = 0 for the four smallest positive solutions

Thank you!

  x ∈ {0.063170, 0.722228, 1.633966, 2.293024}

Step-by-step explanation:

It is probably convenient to replace csc(4x) with its equivalent, 1/sin(4x). Then add 4 and multiply by sin(4x)/4.

  1/sin(4x) -4 = 0

  1/sin(4x) = 4

  1/4 = sin(4x)

Using the inverse sine function, we find ...

  4x = arcsin(1/4) +2kπ . . . . in radians

and the supplementary solution, ...

  4x = π -arcsin(1/4) +2kπ

Dividing these by 4 gives the possible values of x.

  x = arcsin(1/4)/4 +kπ/2

  x = π/4 -arcsin(1/4) +kπ/2

The four smallest positive solutions will be found where k = 0 or 1.

  x ∈ {0.063170, 0.722228, 1.633966, 2.293024}

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I like a graphing calculator for solving these quickly and easily.

The null and alternative hypotheses:

H0: attitudes between first survey and second survey have not changed.

H1:attitudes between first survey and second survey have changed.

Let's find the observed and expected value using the following:

Observed value = sample size * observed proportion

Expected value = sample size * expected proportion.

Optimistic:

expected= 6%*380=22.8

observed=1%*380=3.8  

(fo-fe)²/fe = 15.83333

Slightly Optimistic:

expected: 29%*380=110.2);

Observed: 30%*380= 114

(fo-fe)²/fe  = 0.131034

Slightly Pessimistic:

expected= 24%*380= 91.2

Observed = 28%*380=106.4  

(fo-fe)²/fe =  2.533333

Pessimistic:

Expected = 41%*380 =155.8

Observed = 41%*380 =155.8

(fo-fe)²/fe = 0

Total (fo-fe)²/fe =

15.83333 + 0.131034 + 2.533333 + 0 = 18.4977

Degrees of freedom, df = 4 - 1 = 3

For chi square test statistic:

Σ(fo-fe)²/fe = 15.83333 + 0.131034 + 2.533333 + 0 =  18.498

Calculate pvalue.

Calculating for pvalue at, df = 3, chi-square score = 18.4977, α=0.05

Pvalue = 0.000347

Decision:

Since pvalue, 0.000347 is less than significance level, 0.05, reject null hypothesis H0.

Conclusion:

There is enough evidence to conclude that there is a change in attitudes between  first survey and the second survey at a level of significance, 0.05.