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Popular Trigonometry >

sin(x+23)cos(x-37)>(sqrt(3))/2

Solution

sin (x + 2 3^{\circ}) cos (x - 3 7^{\circ}) > \frac{3}{2}

Solution

πn \leq x < 12 7^{\circ} + πn or 15 7^{\circ} + πn < x \leq π + πn

Interval Notation

[πn, 12 7^{\circ} + πn) \cup (15 7^{\circ} + πn, π + πn]

Decimal

πn \leq x < 2.21656 \dots + πn or 2.74016 \dots + πn < x \leq 3.14159 \dots + πn

Solution steps

sin (x + 2 3^{\circ}) cos (x - 3 7^{\circ}) > \frac{3}{2}

Periodicity of

sin (x + 2 3^{\circ}) cos (x - 3 7^{\circ}) : π

sin (x + 2 3^{\circ}) cos (x - 3 7^{\circ})

is composed of the following functions and periods:

sin (x + 2 3^{\circ})

with periodicity of

2 π

The compound periodicity is:

= π

To find the zeroes, set the inequality to zero

sin (x + 2 3^{\circ}) cos (x - 3 7^{\circ}) = 0

Solve

sin (x + 2 3^{\circ}) cos (x - 3 7^{\circ}) = 0

for

0 \leq x < π

sin (x + 2 3^{\circ}) cos (x - 3 7^{\circ}) = 0

Solving each part separately

sin (x + 2 3^{\circ}) = 0 : x = 15 7^{\circ}

sin (x + 2 3^{\circ}) = 0, 0 \leq x < π

General solutions for

sin (x + 2 3^{\circ}) = 0

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x + 2 3^{\circ} = 0 + 36 0^{\circ} n, x + 2 3^{\circ} = 18 0^{\circ} + 36 0^{\circ} n

x + 2 3^{\circ} = 0 + 36 0^{\circ} n, x + 2 3^{\circ} = 18 0^{\circ} + 36 0^{\circ} n

Solve

x + 2 3^{\circ} = 0 + 36 0^{\circ} n : x = 36 0^{\circ} n - 2 3^{\circ}

x + 2 3^{\circ} = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

x + 2 3^{\circ} = 36 0^{\circ} n

Move

2 3^{\circ}

to the right side

x + 2 3^{\circ} = 36 0^{\circ} n

Subtract

2 3^{\circ}

from both sides

x + 2 3^{\circ} - 2 3^{\circ} = 36 0^{\circ} n - 2 3^{\circ}

Simplify

x = 36 0^{\circ} n - 2 3^{\circ}

x = 36 0^{\circ} n - 2 3^{\circ}

Solve

x + 2 3^{\circ} = 18 0^{\circ} + 36 0^{\circ} n : x = 18 0^{\circ} + 36 0^{\circ} n - 2 3^{\circ}

x + 2 3^{\circ} = 18 0^{\circ} + 36 0^{\circ} n

Move

2 3^{\circ}

to the right side

x + 2 3^{\circ} = 18 0^{\circ} + 36 0^{\circ} n

Subtract

2 3^{\circ}

from both sides

x + 2 3^{\circ} - 2 3^{\circ} = 18 0^{\circ} + 36 0^{\circ} n - 2 3^{\circ}

Simplify

x = 18 0^{\circ} + 36 0^{\circ} n - 2 3^{\circ}

x = 18 0^{\circ} + 36 0^{\circ} n - 2 3^{\circ}

x = 36 0^{\circ} n - 2 3^{\circ}, x = 18 0^{\circ} + 36 0^{\circ} n - 2 3^{\circ}

Solutions for the range

0 \leq x < 18 0^{\circ}

x = 15 7^{\circ}

cos (x - 3 7^{\circ}) = 0 : x = 12 7^{\circ}

cos (x - 3 7^{\circ}) = 0, 0 \leq x < π

General solutions for

cos (x - 3 7^{\circ}) = 0

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x - 3 7^{\circ} = 9 0^{\circ} + 36 0^{\circ} n, x - 3 7^{\circ} = 27 0^{\circ} + 36 0^{\circ} n

x - 3 7^{\circ} = 9 0^{\circ} + 36 0^{\circ} n, x - 3 7^{\circ} = 27 0^{\circ} + 36 0^{\circ} n

Solve

x - 3 7^{\circ} = 9 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 12 7^{\circ}

x - 3 7^{\circ} = 9 0^{\circ} + 36 0^{\circ} n

Move

3 7^{\circ}

to the right side

x - 3 7^{\circ} = 9 0^{\circ} + 36 0^{\circ} n

Add

3 7^{\circ}

to both sides

x - 3 7^{\circ} + 3 7^{\circ} = 9 0^{\circ} + 36 0^{\circ} n + 3 7^{\circ}

Simplify

x - 3 7^{\circ} + 3 7^{\circ} = 9 0^{\circ} + 36 0^{\circ} n + 3 7^{\circ}

Simplify

x - 3 7^{\circ} + 3 7^{\circ} : x

x - 3 7^{\circ} + 3 7^{\circ}

Add similar elements:

- 3 7^{\circ} + 3 7^{\circ} = 0

= x

Simplify

9 0^{\circ} + 36 0^{\circ} n + 3 7^{\circ} : 36 0^{\circ} n + 12 7^{\circ}

9 0^{\circ} + 36 0^{\circ} n + 3 7^{\circ}

Group like terms

= 36 0^{\circ} n + 9 0^{\circ} + 3 7^{\circ}

Least Common Multiplier of

2, 180 : 180

2, 180

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

2

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

9 0^{\circ} :

multiply the denominator and numerator by

90

9 0^{\circ} = \frac{18 0 ^{\circ} 90}{2 \cdot 90} = 9 0^{\circ}

= 9 0^{\circ} + 3 7^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{18 0 ^{\circ} 90 + 666 0 ^{\circ}}{180}

Add similar elements:

1620 0^{\circ} + 666 0^{\circ} = 2286 0^{\circ}

= 36 0^{\circ} n + 12 7^{\circ}

x = 36 0^{\circ} n + 12 7^{\circ}

x = 36 0^{\circ} n + 12 7^{\circ}

x = 36 0^{\circ} n + 12 7^{\circ}

Solve

x - 3 7^{\circ} = 27 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 30 7^{\circ}

x - 3 7^{\circ} = 27 0^{\circ} + 36 0^{\circ} n

Move

3 7^{\circ}

to the right side

x - 3 7^{\circ} = 27 0^{\circ} + 36 0^{\circ} n

Add

3 7^{\circ}

to both sides

x - 3 7^{\circ} + 3 7^{\circ} = 27 0^{\circ} + 36 0^{\circ} n + 3 7^{\circ}

Simplify

x - 3 7^{\circ} + 3 7^{\circ} = 27 0^{\circ} + 36 0^{\circ} n + 3 7^{\circ}

Simplify

x - 3 7^{\circ} + 3 7^{\circ} : x

x - 3 7^{\circ} + 3 7^{\circ}

Add similar elements:

- 3 7^{\circ} + 3 7^{\circ} = 0

= x

Simplify

27 0^{\circ} + 36 0^{\circ} n + 3 7^{\circ} : 36 0^{\circ} n + 30 7^{\circ}

27 0^{\circ} + 36 0^{\circ} n + 3 7^{\circ}

Group like terms

= 36 0^{\circ} n + 27 0^{\circ} + 3 7^{\circ}

Least Common Multiplier of

2, 180 : 180

2, 180

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

2

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

27 0^{\circ} :

multiply the denominator and numerator by

90

27 0^{\circ} = \frac{54 0 ^{\circ} 90}{2 \cdot 90} = 27 0^{\circ}

= 27 0^{\circ} + 3 7^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{4860 0 ^{\circ} + 666 0 ^{\circ}}{180}

Add similar elements:

4860 0^{\circ} + 666 0^{\circ} = 5526 0^{\circ}

= 36 0^{\circ} n + 30 7^{\circ}

x = 36 0^{\circ} n + 30 7^{\circ}

x = 36 0^{\circ} n + 30 7^{\circ}

x = 36 0^{\circ} n + 30 7^{\circ}

x = 36 0^{\circ} n + 12 7^{\circ}, x = 36 0^{\circ} n + 30 7^{\circ}

Solutions for the range

0 \leq x < 18 0^{\circ}

x = 12 7^{\circ}

Combine all the solutions

12 7^{\circ} or 15 7^{\circ}

The intervals between the zeros

0 < x < 12 7^{\circ}, 12 7^{\circ} < x < 15 7^{\circ}, 15 7^{\circ} < x < π

Summarize in a table:

sin (x + 2 3^{\circ}) cos (x - 3 7^{\circ}) sin (x + 2 3^{\circ}) cos (x - 3 7^{\circ}) x = 0 + + + 0 < x < 12 7^{\circ} + + + x = 12 7^{\circ} + 00 12 7^{\circ} < x < 15 7^{\circ} + - - x = 15 7^{\circ} 0 - 0 15 7^{\circ} < x < π - - + x = π - - +

Identify the intervals that satisfy the required condition:

> 0

x = 0 or 0 < x < 12 7^{\circ} or 15 7^{\circ} < x < π or x = π

Merge Overlapping Intervals

0 \leq x < 12 7^{\circ} or 15 7^{\circ} < x < π or x = π

The union of two intervals is the set of numbers which are in either interval

x = 0

0 < x < 12 7^{\circ}

0 \leq x < 12 7^{\circ}

The union of two intervals is the set of numbers which are in either interval

0 \leq x < 12 7^{\circ}

15 7^{\circ} < x < π

0 \leq x < 12 7^{\circ} or 15 7^{\circ} < x < π

The union of two intervals is the set of numbers which are in either interval

0 \leq x < 12 7^{\circ} or 15 7^{\circ} < x < π

x = π

0 \leq x < 12 7^{\circ} or 15 7^{\circ} < x \leq π

0 \leq x < 12 7^{\circ} or 15 7^{\circ} < x \leq π

Apply the periodicity of

sin (x + 2 3^{\circ}) cos (x - 3 7^{\circ})

πn \leq x < 12 7^{\circ} + πn or 15 7^{\circ} + πn < x \leq π + πn

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