What is the general solution for sin(105)sin(x)=(sqrt(3)+2)/4 ?

The general solution for sin(105)sin(x)=(sqrt(3)+2)/4 is No Solution for x\in\mathbb{R}

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sin(105)sin(x)=(sqrt(3)+2)/4

Solution

sin (10 5^{\circ}) sin (x^{\circ}) = \frac{3 + 2}{4}

Solution

No Solution for x \in R

Solution steps

sin (10 5^{\circ}) sin (x^{\circ}) = \frac{3 + 2}{4}

sin (10 5^{\circ}) = \frac{2 ( 3 + 1 )}{4}

sin (10 5^{\circ})

Rewrite using trig identities:

sin (6 0^{\circ}) cos (4 5^{\circ}) + cos (6 0^{\circ}) sin (4 5^{\circ})

sin (10 5^{\circ})

Write

sin (10 5^{\circ})

sin (6 0^{\circ} + 4 5^{\circ})

= sin (6 0^{\circ} + 4 5^{\circ})

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (6 0^{\circ}) cos (4 5^{\circ}) + cos (6 0^{\circ}) sin (4 5^{\circ})

= sin (6 0^{\circ}) cos (4 5^{\circ}) + cos (6 0^{\circ}) sin (4 5^{\circ})

Use the following trivial identity:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 0^{\circ})

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}

= \frac{3}{2}

Use the following trivial identity:

cos (4 5^{\circ}) = \frac{2}{2}

cos (4 5^{\circ})

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}

= \frac{2}{2}

Use the following trivial identity:

cos (6 0^{\circ}) = \frac{1}{2}

cos (6 0^{\circ})

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}

= \frac{1}{2}

Use the following trivial identity:

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 5^{\circ})

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}

= \frac{2}{2}

= \frac{3}{2} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{2}{2}

Simplify

\frac{3}{2} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{2}{2} : \frac{2 ( 3 + 1 )}{4}

\frac{3}{2} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{2}{2}

Factor out common term

\frac{2}{2}

= \frac{2}{2} (\frac{3}{2} + \frac{1}{2})

\frac{3}{2} + \frac{1}{2} = \frac{3 + 1}{2}

\frac{3}{2} + \frac{1}{2}

Apply rule

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

= \frac{3 + 1}{2}

= \frac{2}{2} \cdot \frac{1 + 3}{2}

Multiply fractions:

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

= \frac{( 3 + 1 ) 2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 ( 1 + 3 )}{4}

= \frac{2 ( 3 + 1 )}{4}

\frac{2 ( 3 + 1 )}{4} sin (x^{\circ}) = \frac{3 + 2}{4}

Multiply both sides by

4

\frac{2 ( 3 + 1 )}{4} sin (x^{\circ}) = \frac{3 + 2}{4}

Multiply both sides by

4

4 \cdot \frac{2 ( 3 + 1 )}{4} sin (x^{\circ}) = \frac{4 ( 3 + 2 )}{4}

Simplify

2 (3 + 1) sin (x^{\circ}) = 3 + 2

2 (3 + 1) sin (x^{\circ}) = 3 + 2

Divide both sides by

2 (3 + 1)

2 (3 + 1) sin (x^{\circ}) = 3 + 2

Divide both sides by

2 (3 + 1)

\frac{2 ( 3 + 1 ) sin ( x ^{\circ} )}{2 ( 3 + 1 )} = \frac{3}{2 ( 3 + 1 )} + \frac{2}{2 ( 3 + 1 )}

Simplify

\frac{2 ( 3 + 1 ) sin ( x ^{\circ} )}{2 ( 3 + 1 )} = \frac{3}{2 ( 3 + 1 )} + \frac{2}{2 ( 3 + 1 )}

Simplify

\frac{2 ( 3 + 1 ) sin ( x ^{\circ} )}{2 ( 3 + 1 )} : sin (x^{\circ})

\frac{2 ( 3 + 1 ) sin ( x ^{\circ} )}{2 ( 3 + 1 )}

Cancel the common factor:

2

= \frac{( 1 + 3 ) sin ( x ^{\circ} )}{3 + 1}

Cancel the common factor:

3 + 1

= sin (x^{\circ})

Simplify

\frac{3}{2 ( 3 + 1 )} + \frac{2}{2 ( 3 + 1 )} : \frac{2 + 6}{4}

\frac{3}{2 ( 3 + 1 )} + \frac{2}{2 ( 3 + 1 )}

Apply rule

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

= \frac{3 + 2}{2 ( 1 + 3 )}

Multiply by the conjugate

\frac{2}{2}

= \frac{( 3 + 2 ) 2}{2 ( 1 + 3 ) 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2 ( 3 + 2 )}{( 1 + 3 ) \cdot 2}

Multiply by the conjugate

\frac{1 - 3}{1 - 3}

= \frac{2 ( 3 + 2 ) ( 1 - 3 )}{( 1 + 3 ) \cdot 2 ( 1 - 3 )}

2 (3 + 2) (1 - 3) = - 2 - 6

2 (3 + 2) (1 - 3)

Expand

(3 + 2) (1 - 3) : - 1 - 3

(3 + 2) (1 - 3)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 3, b = 2, c = 1, d = - 3

= 3 \cdot 1 + 3 (- 3) + 2 \cdot 1 + 2 (- 3)

Apply minus-plus rules

+ (- a) = - a

= 1 \cdot 3 - 33 + 2 \cdot 1 - 23

Simplify

1 \cdot 3 - 33 + 2 \cdot 1 - 23 : - 1 - 3

1 \cdot 3 - 33 + 2 \cdot 1 - 23

Add similar elements:

1 \cdot 3 - 23 = - 3

= - 3 - 33 + 2 \cdot 1

Apply radical rule:

a a = a

33 = 3

= - 3 - 3 + 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= - 3 - 3 + 2

Add/Subtract the numbers:

- 3 + 2 = - 1

= - 1 - 3

= - 1 - 3

= 2 (- 1 - 3)

Expand

2 (- 1 - 3) : - 2 - 6

2 (- 1 - 3)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = - 1, c = 3

= 2 (- 1) - 23

Apply minus-plus rules

+ (- a) = - a

= - 1 \cdot 2 - 23

Simplify

- 1 \cdot 2 - 23 : - 2 - 6

- 1 \cdot 2 - 23

1 \cdot 2 = 2

1 \cdot 2

Multiply:

1 \cdot 2 = 2

= 2

23 = 6

23

Apply radical rule:

a b = a \cdot b

23 = 2 \cdot 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= - 2 - 6

= - 2 - 6

= - 2 - 6

(1 + 3) \cdot 2 (1 - 3) = - 4

(1 + 3) \cdot 2 (1 - 3)

= 2 (1 + 3) (1 - 3)

Expand

(1 + 3) (1 - 3) : - 2

(1 + 3) (1 - 3)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 1, b = 3

= 1^{2} - (3)^{2}

Simplify

1^{2} - (3)^{2} : - 2

1^{2} - (3)^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - (3)^{2}

(3)^{2} = 3

(3)^{2}

Apply radical rule:

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

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 3

= 1 - 3

Subtract the numbers:

1 - 3 = - 2

= - 2

= - 2

= 2 (- 2)

Expand

2 (- 2) : - 4

2 (- 2)

Distribute parentheses

= 2 (- 2)

Apply minus-plus rules

+ (- a) = - a

= - 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= - 4

= - 4

= \frac{- 2 - 6}{- 4}

Apply the fraction rule:

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

- 2 - 6 = - (2 + 6)

= \frac{2 + 6}{4}

sin (x^{\circ}) = \frac{2 + 6}{4}

sin (x^{\circ}) = \frac{2 + 6}{4}

sin (x^{\circ}) = \frac{2 + 6}{4}

Apply trig inverse properties

sin (x^{\circ}) = \frac{2 + 6}{4}

General solutions for

sin (x^{\circ}) = \frac{2 + 6}{4}

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

x^{\circ} = arcsin (\frac{2 + 6}{4}) + 36 0^{\circ} n, x^{\circ} = 18 0^{\circ} - arcsin (\frac{2 + 6}{4}) + 36 0^{\circ} n

x^{\circ} = arcsin (\frac{2 + 6}{4}) + 36 0^{\circ} n, x^{\circ} = 18 0^{\circ} - arcsin (\frac{2 + 6}{4}) + 36 0^{\circ} n

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for sin(105)sin(x)=(sqrt(3)+2)/4 ?
The general solution for sin(105)sin(x)=(sqrt(3)+2)/4 is No Solution for x\in\mathbb{R}