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Popular >

30<= sqrt(980)cos(θ)3.19

Solution

30 \leq 980 cos (θ) 3.19

Solution

- arccos (\frac{300 5}{2233}) + 2 πn \leq θ \leq arccos (\frac{300 5}{2233}) + 2 πn

Interval Notation

[- arccos (\frac{300 5}{2233}) + 2 πn, arccos (\frac{300 5}{2233}) + 2 πn]

Decimal Notation

- 1.26567 \dots + 2 πn \leq θ \leq 1.26567 \dots + 2 πn

Solution steps

30 \leq 980 cos (θ) \cdot 3.19

Switch sides

980 cos (θ) \cdot 3.19 \geq 30

Multiply both sides by

100

980 cos (θ) \cdot 3.19 \geq 30

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

980 cos (θ) \cdot 3.19 \cdot 100 \geq 30 \cdot 100

Refine

44665 cos (θ) \geq 3000

44665 cos (θ) \geq 3000

Divide both sides by

44665

44665 cos (θ) \geq 3000

Divide both sides by

44665

\frac{4466 5 cos ( θ )}{4466 5} \geq \frac{3000}{4466 5}

Simplify

\frac{4466 5 cos ( θ )}{4466 5} \geq \frac{3000}{4466 5}

Simplify

\frac{4466 5 cos ( θ )}{4466 5} : cos (θ)

\frac{4466 5 cos ( θ )}{4466 5}

Divide the numbers:

\frac{4466}{4466} = 1

= \frac{5 cos ( θ )}{5}

Cancel the common factor:

5

= cos (θ)

Simplify

\frac{3000}{4466 5} : \frac{300 5}{2233}

\frac{3000}{4466 5}

Cancel the common factor:

2

= \frac{1500}{2233 5}

Factor

1500 : 5^{3} \cdot 2^{2} \cdot 3

Factor

1500 = 5^{3} \cdot 2^{2} \cdot 3

= \frac{5 ^{3} \cdot 2 ^{2} \cdot 3}{2233 5}

Cancel

\frac{2 ^{2} \cdot 3 \cdot 5 ^{3}}{2233 5} : \frac{2 ^{2} \cdot 3 \cdot 5 ^{\frac{5}{2}}}{2233}

\frac{2 ^{2} \cdot 3 \cdot 5 ^{3}}{2233 5}

Apply radical rule:

5 = 5^{\frac{1}{2}}

= \frac{5 ^{3} \cdot 2 ^{2} \cdot 3}{2233 \cdot 5 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{5 ^{3}}{5 ^{\frac{1}{2}}} = 5^{3 - \frac{1}{2}}

= \frac{2 ^{2} \cdot 3 \cdot 5 ^{- \frac{1}{2} + 3}}{2233}

Subtract the numbers:

3 - \frac{1}{2} = \frac{5}{2}

= \frac{2 ^{2} \cdot 3 \cdot 5 ^{\frac{5}{2}}}{2233}

= \frac{2 ^{2} \cdot 3 \cdot 5 ^{\frac{5}{2}}}{2233}

5^{\frac{5}{2}} = 5^{2} 5

5^{\frac{5}{2}}

5^{\frac{5}{2}} = 5^{2 + \frac{1}{2}}

= 5^{2 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 5^{2} \cdot 5^{\frac{1}{2}}

Refine

= 5^{2} 5

= \frac{2 ^{2} \cdot 3 \cdot 5 ^{2} 5}{2233}

2^{2} \cdot 3 \cdot 5^{2} 5 = 3005

2^{2} \cdot 3 \cdot 5^{2} 5

2^{2} = 4

= 5^{2} \cdot 4 \cdot 35

5^{2} = 25

= 4 \cdot 3 \cdot 255

Multiply the numbers:

4 \cdot 3 \cdot 25 = 300

= 3005

= \frac{300 5}{2233}

cos (θ) \geq \frac{300 5}{2233}

cos (θ) \geq \frac{300 5}{2233}

cos (θ) \geq \frac{300 5}{2233}

For

cos (x) \geq a

, if

- 1 < a < 1

then

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

- arccos (\frac{300 5}{2233}) + 2 πn \leq θ \leq arccos (\frac{300 5}{2233}) + 2 πn

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