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Popular Trigonometry >

0.96(cos(x))^2<= 0.83

Solution

0.96 (cos (x))^{2} \leq 0.83

Solution

arccos (\frac{498}{24}) + 2 πn \leq x \leq arccos (- \frac{498}{24}) + 2 πn or - arccos (- \frac{498}{24}) + 2 π + 2 πn \leq x \leq 2 π - arccos (\frac{498}{24}) + 2 πn

Interval Notation

[arccos (\frac{498}{24}) + 2 πn, arccos (- \frac{498}{24}) + 2 πn] \cup [- arccos (- \frac{498}{24}) + 2 π + 2 πn, 2 π - arccos (\frac{498}{24}) + 2 πn]

Decimal

0.37684 \dots + 2 πn \leq x \leq 2.76474 \dots + 2 πn or 3.51843 \dots + 2 πn \leq x \leq 5.90633 \dots + 2 πn

Solution steps

0.96 (cos (x))^{2} \leq 0.83

Multiply both sides by

100

0.96 cos^{2} (x) \leq 0.83

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

0.96 cos^{2} (x) \cdot 100 \leq 0.83 \cdot 100

Refine

96 cos^{2} (x) \leq 83

96 cos^{2} (x) \leq 83

Divide both sides by

96

96 cos^{2} (x) \leq 83

Divide both sides by

96

\frac{96 cos ^{2} ( x )}{96} \leq \frac{83}{96}

Simplify

cos^{2} (x) \leq \frac{83}{96}

cos^{2} (x) \leq \frac{83}{96}

For

u^{n} \leq a

, if

n

is even then

- \frac{83}{96} \leq cos (x) \leq \frac{83}{96}

\frac{83}{96} = \frac{83}{4 6}

\frac{83}{96}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{83}{96}

96 = 46

96

Prime factorization of

96 : 2^{5} \cdot 3

96

96

divides by

296 = 48 \cdot 2

= 2 \cdot 48

48

divides by

248 = 24 \cdot 2

= 2 \cdot 2 \cdot 24

24

divides by

224 = 12 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3

= 2^{5} \cdot 3

= 2^{5} \cdot 3

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{4} \cdot 2 \cdot 3

Apply radical rule:

= 2^{4} 2 \cdot 3

Apply radical rule:

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 2 \cdot 3

Refine

= 46

= \frac{83}{4 6}

- \frac{83}{4 6} \leq cos (x) \leq \frac{83}{4 6}

a \leq u \leq b

then

a \leq u and u \leq b

- \frac{83}{4 6} \leq cos (x) and cos (x) \leq \frac{83}{4 6}

- \frac{83}{4 6} \leq cos (x) : - arccos (- \frac{498}{24}) + 2 πn \leq x \leq arccos (- \frac{498}{24}) + 2 πn

- \frac{83}{4 6} \leq cos (x)

Switch sides

cos (x) \geq - \frac{83}{4 6}

Simplify

- \frac{83}{4 6} : - \frac{498}{24}

- \frac{83}{4 6}

Multiply by the conjugate

\frac{6}{6}

= - \frac{83 6}{4 6 6}

836 = 498

836

Apply radical rule:

a b = a \cdot b

836 = 83 \cdot 6

= 83 \cdot 6

Multiply the numbers:

83 \cdot 6 = 498

= 498

466 = 24

466

Apply radical rule:

a a = a

66 = 6

= 4 \cdot 6

Multiply the numbers:

4 \cdot 6 = 24

= 24

= - \frac{498}{24}

cos (x) \geq - \frac{498}{24}

For

cos (x) \geq a

, if

- 1 < a < 1

then

- arccos (a) + 2 πn \leq x \leq arccos (a) + 2 πn

- arccos (- \frac{498}{24}) + 2 πn \leq x \leq arccos (- \frac{498}{24}) + 2 πn

cos (x) \leq \frac{83}{4 6} : arccos (\frac{498}{24}) + 2 πn \leq x \leq 2 π - arccos (\frac{498}{24}) + 2 πn

cos (x) \leq \frac{83}{4 6}

Simplify

\frac{83}{4 6} : \frac{498}{24}

\frac{83}{4 6}

Multiply by the conjugate

\frac{6}{6}

= \frac{83 6}{4 6 6}

836 = 498

836

Apply radical rule:

a b = a \cdot b

836 = 83 \cdot 6

= 83 \cdot 6

Multiply the numbers:

83 \cdot 6 = 498

= 498

466 = 24

466

Apply radical rule:

a a = a

66 = 6

= 4 \cdot 6

Multiply the numbers:

4 \cdot 6 = 24

= 24

= \frac{498}{24}

cos (x) \leq \frac{498}{24}

For

cos (x) \leq a

, if

- 1 < a < 1

then

arccos (a) + 2 πn \leq x \leq 2 π - arccos (a) + 2 πn

arccos (\frac{498}{24}) + 2 πn \leq x \leq 2 π - arccos (\frac{498}{24}) + 2 πn

Combine the intervals

- arccos (- \frac{498}{24}) + 2 πn \leq x \leq arccos (- \frac{498}{24}) + 2 πn and arccos (\frac{498}{24}) + 2 πn \leq x \leq 2 π - arccos (\frac{498}{24}) + 2 πn

Merge Overlapping Intervals

arccos (\frac{498}{24}) + 2 πn \leq x \leq arccos (- \frac{498}{24}) + 2 πn or - arccos (- \frac{498}{24}) + 2 π + 2 πn \leq x \leq 2 π - arccos (\frac{498}{24}) + 2 πn

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