What is the general solution for (13)/(sin(108))= 9/(sin(x)) ?

The general solution for (13)/(sin(108))= 9/(sin(x)) is x=0.71872…+360n,x=180-0.71872…+360n

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(13)/(sin(108))= 9/(sin(x))

Solution

\frac{13}{sin ( 10 8 ^{\circ} )} = \frac{9}{sin ( x )}

Solution

x = 0.71872 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.71872 \dots + 36 0^{\circ} n

Radians

x = 0.71872 \dots + 2 πn, x = π - 0.71872 \dots + 2 πn

Solution steps

\frac{13}{sin ( 10 8 ^{\circ} )} = \frac{9}{sin ( x )}

sin (10 8^{\circ}) = \frac{2 5 + 5}{4}

sin (10 8^{\circ})

Rewrite using trig identities:

cos (1 8^{\circ})

sin (10 8^{\circ})

Use the following identity:

sin (x) = cos (9 0^{\circ} - x)

= cos (9 0^{\circ} - 10 8^{\circ})

Simplify:

9 0^{\circ} - 10 8^{\circ} = - 1 8^{\circ}

9 0^{\circ} - 10 8^{\circ}

Least Common Multiplier of

2, 5 : 10

2, 5

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

5 : 5

5

5

is a prime number, therefore no factorization is possible

= 5

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

2

5

= 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 10

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

10

For

9 0^{\circ} :

multiply the denominator and numerator by

5

9 0^{\circ} = \frac{18 0 ^{\circ} 5}{2 \cdot 5} = 9 0^{\circ}

For

10 8^{\circ} :

multiply the denominator and numerator by

2

10 8^{\circ} = \frac{54 0 ^{\circ} 2}{5 \cdot 2} = 10 8^{\circ}

= 9 0^{\circ} - 10 8^{\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} 5 - 108 0 ^{\circ}}{10}

Add similar elements:

90 0^{\circ} - 108 0^{\circ} = - 18 0^{\circ}

= \frac{- 18 0 ^{\circ}}{10}

Apply the fraction rule:

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

= - 1 8^{\circ}

= cos (- 1 8^{\circ})

Use the following property:

cos (- x) = cos (x)

cos (- 1 8^{\circ}) = cos (1 8^{\circ})

= cos (1 8^{\circ})

= cos (1 8^{\circ})

Rewrite using trig identities:

\frac{1 + cos ( 3 6 ^{\circ} )}{2}

cos (1 8^{\circ})

Write

cos (1 8^{\circ})

cos (\frac{3 6 ^{\circ}}{2})

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

Use the Half Angle identity:

cos (\frac{θ}{2}) = \frac{1 + cos ( θ )}{2}

Use the Double Angle identity

cos (2 θ) = 2 cos^{2} (θ) - 1

Substitute

θ

with

\frac{θ}{2}

cos (θ) = 2 cos^{2} (\frac{θ}{2}) - 1

Switch sides

2 cos^{2} (\frac{θ}{2}) = 1 + cos (θ)

Divide both sides by

2

cos^{2} (\frac{θ}{2}) = \frac{( 1 + cos ( θ ) )}{2}

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

range [0, 9 0^{\circ}] [9 0^{\circ}, 18 0^{\circ}] [18 0^{\circ}, 27 0^{\circ}] [27 0^{\circ}, 36 0^{\circ}] quadrant I II III I V sin positive positive negative negative cos positive negative negative positive

cos (\frac{θ}{2}) = \frac{( 1 + cos ( θ ) )}{2}

= \frac{1 + cos ( 3 6 ^{\circ} )}{2}

= \frac{1 + cos ( 3 6 ^{\circ} )}{2}

Rewrite using trig identities:

cos (3 6^{\circ}) = \frac{5 + 1}{4}

cos (3 6^{\circ})

Show that:

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

Use the following product to sum identity:

2 sin (x) cos (y) = sin (x + y) - sin (x - y)

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = sin (5 4^{\circ}) - sin (1 8^{\circ})

Show that:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Use the following identity:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

Divide both sides by

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

Substitute

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

\frac{1}{2} = sin (5 4^{\circ}) - sin (1 8^{\circ})

sin (5 4^{\circ}) = cos (9 0^{\circ} - 5 4^{\circ})

\frac{1}{2} = cos (9 0^{\circ} - 5 4^{\circ}) - sin (1 8^{\circ})

\frac{1}{2} = cos (3 6^{\circ}) - sin (1 8^{\circ})

Show that:

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

Use the factorization rule:

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

a = cos (3 6^{\circ}) + sin (1 8^{\circ})

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = ((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) ((cos (3 6^{\circ}) + sin (1 8^{\circ})) - (cos (3 6^{\circ}) - sin (1 8^{\circ})))

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 2 (2 cos (3 6^{\circ}) sin (1 8^{\circ}))

Show that:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Use the following identity:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

Divide both sides by

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

Substitute

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 1

Substitute

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (\frac{1}{2})^{2} = 1

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} = 1

Add

\frac{1}{4}

to both sides

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} + \frac{1}{4} = 1 + \frac{1}{4}

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} = \frac{5}{4}

Take the square root of both sides

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \pm \frac{5}{4}

cos (3 6^{\circ})

cannot be negative

sin (1 8^{\circ})

cannot be negative

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

Add the following equations

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{2}

((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) = (\frac{5}{2} + \frac{1}{2})

Refine

cos (3 6^{\circ}) = \frac{5 + 1}{4}

= \frac{5 + 1}{4}

= \frac{1 + \frac{5 + 1}{4}}{2}

Simplify

\frac{1 + \frac{5 + 1}{4}}{2} : \frac{2 5 + 5}{4}

\frac{1 + \frac{5 + 1}{4}}{2}

\frac{1 + \frac{5 + 1}{4}}{2} = \frac{5 + 5}{8}

\frac{1 + \frac{5 + 1}{4}}{2}

Join

1 + \frac{5 + 1}{4} : \frac{5 + 5}{4}

1 + \frac{5 + 1}{4}

Convert element to fraction:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} + \frac{5 + 1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 4 + 5 + 1}{4}

1 \cdot 4 + 5 + 1 = 5 + 5

1 \cdot 4 + 5 + 1

Multiply the numbers:

1 \cdot 4 = 4

= 4 + 5 + 1

Add the numbers:

4 + 1 = 5

= 5 + 5

= \frac{5 + 5}{4}

= \frac{\frac{5 + 5}{4}}{2}

Apply the fraction rule:

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

= \frac{5 + 5}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{5 + 5}{8}

= \frac{5 + 5}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{5 + 5}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

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

Rationalize

\frac{5 + 5}{2 2} : \frac{2 5 + 5}{4}

\frac{5 + 5}{2 2}

Multiply by the conjugate

\frac{2}{2}

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

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 5 + 5}{4}

= \frac{2 5 + 5}{4}

= \frac{2 5 + 5}{4}

\frac{13}{\frac{2 5 + 5}{4}} = \frac{9}{sin ( x )}

Cross multiply

\frac{13}{\frac{2 5 + 5}{4}} = \frac{9}{sin ( x )}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

13 sin (x) = \frac{2 5 + 5}{4} \cdot 9

Simplify

\frac{2 5 + 5}{4} \cdot 9 : \frac{9 2 5 + 5}{4}

\frac{2 5 + 5}{4} \cdot 9

Multiply fractions:

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

= \frac{2 5 + 5 \cdot 9}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{9 2 5 + 5}{2 ^{2}}

Cancel

\frac{2 5 + 5 \cdot 9}{2 ^{2}} : \frac{9 5 + 5}{2 ^{\frac{3}{2}}}

\frac{2 5 + 5 \cdot 9}{2 ^{2}}

Apply radical rule:

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

= \frac{9 \cdot 2 ^{\frac{1}{2}} 5 + 5}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{9 5 + 5}{2 ^{2 - \frac{1}{2}}}

Subtract the numbers:

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

= \frac{9 5 + 5}{2 ^{\frac{3}{2}}}

= \frac{9 5 + 5}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

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

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

Apply exponent rule:

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

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

Refine

= 22

= \frac{9 5 + 5}{2 2}

Rationalize

\frac{9 5 + 5}{2 2} : \frac{9 2 5 + 5}{4}

\frac{9 5 + 5}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{9 5 + 5 2}{2 2 2}

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{9 2 5 + 5}{4}

= \frac{9 2 5 + 5}{4}

13 sin (x) = \frac{9 2 5 + 5}{4}

13 sin (x) = \frac{9 2 5 + 5}{4}

Divide both sides by

13

13 sin (x) = \frac{9 2 5 + 5}{4}

Divide both sides by

13

\frac{13 sin ( x )}{13} = \frac{\frac{9 2 5 + 5}{4}}{13}

Simplify

\frac{13 sin ( x )}{13} = \frac{\frac{9 2 5 + 5}{4}}{13}

Simplify

\frac{13 sin ( x )}{13} : sin (x)

\frac{13 sin ( x )}{13}

Divide the numbers:

\frac{13}{13} = 1

= sin (x)

Simplify

\frac{\frac{9 2 5 + 5}{4}}{13} : \frac{9 2 5 + 5}{52}

\frac{\frac{9 2 5 + 5}{4}}{13}

Apply the fraction rule:

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

= \frac{9 2 5 + 5}{4 \cdot 13}

Multiply the numbers:

4 \cdot 13 = 52

= \frac{9 2 5 + 5}{52}

sin (x) = \frac{9 2 5 + 5}{52}

sin (x) = \frac{9 2 5 + 5}{52}

sin (x) = \frac{9 2 5 + 5}{52}

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Take the denominator(s) of

\frac{9}{sin ( x )}

and compare to zero

sin (x) = 0

The following points are undefined

sin (x) = 0

Combine undefined points with solutions:

sin (x) = \frac{9 2 5 + 5}{52}

Apply trig inverse properties

sin (x) = \frac{9 2 5 + 5}{52}

General solutions for

sin (x) = \frac{9 2 5 + 5}{52}

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

x = arcsin (\frac{9 2 5 + 5}{52}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{9 2 5 + 5}{52}) + 36 0^{\circ} n

x = arcsin (\frac{9 2 5 + 5}{52}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{9 2 5 + 5}{52}) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.71872 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.71872 \dots + 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for (13)/(sin(108))= 9/(sin(x)) ?
The general solution for (13)/(sin(108))= 9/(sin(x)) is x=0.71872…+360n,x=180-0.71872…+360n