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Popular Trigonometry >

6.5<= 5+3sin(30t)

Solution

6.5 \leq 5 + 3 sin (30 t)

Solution

\frac{π}{180} + \frac{π}{15} n \leq t \leq \frac{π}{36} + \frac{π}{15} n

Interval Notation

[\frac{π}{180} + \frac{π}{15} n, \frac{π}{36} + \frac{π}{15} n]

Decimal

0.01745 \dots + \frac{π}{15} n \leq t \leq 0.08726 \dots + \frac{π}{15} n

Solution steps

6.5 \leq 5 + 3 sin (30 t)

Switch sides

5 + 3 sin (30 t) \geq 6.5

Move

5

to the right side

5 + 3 sin (30 t) \geq 6.5

Subtract

5

from both sides

5 + 3 sin (30 t) - 5 \geq 6.5 - 5

Simplify

3 sin (30 t) \geq 1.5

3 sin (30 t) \geq 1.5

Divide both sides by

3

3 sin (30 t) \geq 1.5

Divide both sides by

3

\frac{3 sin ( 30 t )}{3} \geq \frac{1.5}{3}

Simplify

sin (30 t) \geq 0.5

sin (30 t) \geq 0.5

For

sin (x) \geq a

, if

- 1 < a < 1

then

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

arcsin (0.5) + 2 πn \leq 30 t \leq π - arcsin (0.5) + 2 πn

a \leq u \leq b

then

a \leq u and u \leq b

arcsin (0.5) + 2 πn \leq 30 t and 30 t \leq π - arcsin (0.5) + 2 πn

arcsin (0.5) + 2 πn \leq 30 t : t \geq \frac{π}{180} + \frac{πn}{15}

arcsin (0.5) + 2 πn \leq 30 t

Switch sides

30 t \geq arcsin (0.5) + 2 πn

Simplify

arcsin (0.5) + 2 πn : \frac{π}{6} + 2 πn

arcsin (0.5) + 2 πn

arcsin (0.5) = \frac{π}{6}

arcsin (0.5)

= arcsin (\frac{1}{2})

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

arcsin (\frac{1}{2})

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{6}

= \frac{π}{6}

= \frac{π}{6} + 2 πn

30 t \geq \frac{π}{6} + 2 πn

Divide both sides by

30

30 t \geq \frac{π}{6} + 2 πn

Divide both sides by

30

\frac{30 t}{30} \geq \frac{\frac{π}{6}}{30} + \frac{2 πn}{30}

Simplify

\frac{30 t}{30} \geq \frac{\frac{π}{6}}{30} + \frac{2 πn}{30}

Simplify

\frac{30 t}{30} : t

\frac{30 t}{30}

Divide the numbers:

\frac{30}{30} = 1

= t

Simplify

\frac{\frac{π}{6}}{30} + \frac{2 πn}{30} : \frac{π}{180} + \frac{πn}{15}

\frac{\frac{π}{6}}{30} + \frac{2 πn}{30}

\frac{\frac{π}{6}}{30} = \frac{π}{180}

\frac{\frac{π}{6}}{30}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π}{6 \cdot 30}

Multiply the numbers:

6 \cdot 30 = 180

= \frac{π}{180}

\frac{2 πn}{30} = \frac{πn}{15}

\frac{2 πn}{30}

Cancel the common factor:

2

= \frac{πn}{15}

= \frac{π}{180} + \frac{πn}{15}

t \geq \frac{π}{180} + \frac{πn}{15}

t \geq \frac{π}{180} + \frac{πn}{15}

t \geq \frac{π}{180} + \frac{πn}{15}

30 t \leq π - arcsin (0.5) + 2 πn : t \leq \frac{π}{36} + \frac{π}{15} n

30 t \leq π - arcsin (0.5) + 2 πn

Simplify

π - arcsin (0.5) + 2 πn : π - \frac{π}{6} + 2 πn

π - arcsin (0.5) + 2 πn

arcsin (0.5) = \frac{π}{6}

arcsin (0.5)

= arcsin (\frac{1}{2})

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

arcsin (\frac{1}{2})

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{6}

= \frac{π}{6}

= π - \frac{π}{6} + 2 πn

30 t \leq π - \frac{π}{6} + 2 πn

Divide both sides by

30

30 t \leq π - \frac{π}{6} + 2 πn

Divide both sides by

30

\frac{30 t}{30} \leq \frac{π}{30} - \frac{\frac{π}{6}}{30} + \frac{2 πn}{30}

Simplify

\frac{30 t}{30} \leq \frac{π}{30} - \frac{\frac{π}{6}}{30} + \frac{2 πn}{30}

Simplify

\frac{30 t}{30} : t

\frac{30 t}{30}

Divide the numbers:

\frac{30}{30} = 1

= t

Simplify

\frac{π}{30} - \frac{\frac{π}{6}}{30} + \frac{2 πn}{30} : \frac{π}{30} - \frac{π}{180} + \frac{πn}{15}

\frac{π}{30} - \frac{\frac{π}{6}}{30} + \frac{2 πn}{30}

\frac{\frac{π}{6}}{30} = \frac{π}{180}

\frac{\frac{π}{6}}{30}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π}{6 \cdot 30}

Multiply the numbers:

6 \cdot 30 = 180

= \frac{π}{180}

\frac{2 πn}{30} = \frac{πn}{15}

\frac{2 πn}{30}

Cancel the common factor:

2

= \frac{πn}{15}

= \frac{π}{30} - \frac{π}{180} + \frac{πn}{15}

t \leq \frac{π}{30} - \frac{π}{180} + \frac{πn}{15}

t \leq \frac{π}{30} - \frac{π}{180} + \frac{πn}{15}

Simplify

\frac{π}{30} - \frac{π}{180} : \frac{π}{36}

\frac{π}{30} - \frac{π}{180}

Least Common Multiplier of

30, 180 : 180

30, 180

Least Common Multiplier (LCM)

Prime factorization of

30 : 2 \cdot 3 \cdot 5

30

30

divides by

230 = 15 \cdot 2

= 2 \cdot 15

15

divides by

315 = 5 \cdot 3

= 2 \cdot 3 \cdot 5

2, 3, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 5

Prime factorization of

180 : 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

180

180

divides by

2180 = 90 \cdot 2

= 2 \cdot 90

90

divides by

290 = 45 \cdot 2

= 2 \cdot 2 \cdot 45

45

divides by

345 = 15 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 15

15

divides by

315 = 5 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

2, 3, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

Multiply each factor the greatest number of times it occurs in either

30

180

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

Multiply the numbers:

2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 = 180

= 180

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

180

For

\frac{π}{30} :

multiply the denominator and numerator by

6

\frac{π}{30} = \frac{π 6}{30 \cdot 6} = \frac{π 6}{180}

= \frac{π 6}{180} - \frac{π}{180}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π 6 - π}{180}

Add similar elements:

6 π - π = 5 π

= \frac{5 π}{180}

Cancel the common factor:

5

= \frac{π}{36}

t \leq \frac{π}{36} + \frac{π}{15} n

t \leq \frac{π}{36} + \frac{π}{15} n

Combine the intervals

t \geq \frac{π}{180} + \frac{πn}{15} and t \leq \frac{π}{36} + \frac{π}{15} n

Merge Overlapping Intervals

\frac{π}{180} + \frac{π}{15} n \leq t \leq \frac{π}{36} + \frac{π}{15} n

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