What is the value of (sin(30))/(sin(22.5)) ?

The value of (sin(30))/(sin(22.5)) is ((2+sqrt(2))sqrt(2-\sqrt{2)})/2

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Popular >

(sin(30))/(sin(22.5))

Solution

\frac{sin ( 3 0 ^{\circ} )}{sin ( 22. 5 ^{\circ} )}

Solution

\frac{( 2 + 2 ) 2 - 2}{2}

Decimal Notation

1.30656 \dots

Solution steps

\frac{sin ( 3 0 ^{\circ} )}{sin ( 22. 5 ^{\circ} )}

Use the following trivial identity:

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 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{1}{2}

Rewrite using trig identities:

sin (22. 5^{\circ}) = \frac{2 - 2}{2}

sin (22. 5^{\circ})

Rewrite using trig identities:

\frac{1 - cos ( 4 5 ^{\circ} )}{2}

sin (22. 5^{\circ})

Write

sin (22. 5^{\circ})

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

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

Use the Half Angle identity:

sin (\frac{θ}{2}) = \frac{1 - cos ( θ )}{2}

Use the Double Angle identity

cos (2 θ) = 1 - 2 sin^{2} (θ)

Substitute

θ

with

\frac{θ}{2}

cos (θ) = 1 - 2 sin^{2} (\frac{θ}{2})

Switch sides

2 sin^{2} (\frac{θ}{2}) = 1 - cos (θ)

Divide both sides by

2

sin^{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

sin (\frac{θ}{2}) = \frac{( 1 - cos ( θ ) )}{2}

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

= \frac{1 - cos ( 4 5 ^{\circ} )}{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}

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

Simplify

\frac{1 - \frac{2}{2}}{2} : \frac{2 - 2}{2}

\frac{1 - \frac{2}{2}}{2}

\frac{1 - \frac{2}{2}}{2} = \frac{2 - 2}{4}

\frac{1 - \frac{2}{2}}{2}

Join

1 - \frac{2}{2} : \frac{2 - 2}{2}

1 - \frac{2}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 - 2}{2}

= \frac{\frac{2 - 2}{2}}{2}

Apply the fraction rule:

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

= \frac{2 - 2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 - 2}{4}

= \frac{2 - 2}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2 - 2}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2 - 2}{2}

= \frac{2 - 2}{2}

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

Simplify

\frac{\frac{1}{2}}{\frac{2 - 2}{2}} : \frac{( 2 + 2 ) 2 - 2}{2}

\frac{\frac{1}{2}}{\frac{2 - 2}{2}}

Divide fractions:

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

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

Refine

= \frac{2}{2 2 - 2}

Cancel the common factor:

2

= \frac{1}{2 - 2}

Rationalize

\frac{1}{2 - 2} : \frac{( 2 + 2 ) 2 - 2}{2}

\frac{1}{2 - 2}

Multiply by the conjugate

\frac{2 - 2}{2 - 2}

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

1 \cdot 2 - 2 = 2 - 2

2 - 2 2 - 2 = 2 - 2

2 - 2 2 - 2

Apply radical rule:

a a = a

2 - 2 2 - 2 = 2 - 2

= 2 - 2

= \frac{2 - 2}{2 - 2}

Multiply by the conjugate

\frac{2 + 2}{2 + 2}

= \frac{2 - 2 ( 2 + 2 )}{( 2 - 2 ) ( 2 + 2 )}

(2 - 2) (2 + 2) = 2

(2 - 2) (2 + 2)

Apply Difference of Two Squares Formula:

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

a = 2, b = 2

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

Simplify

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

2^{2} - (2)^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

(2)^{2} = 2

(2)^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

= 2^{\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

= 2

= 4 - 2

Subtract the numbers:

4 - 2 = 2

= 2

= 2

= \frac{( 2 + 2 ) 2 - 2}{2}

= \frac{( 2 + 2 ) 2 - 2}{2}

= \frac{( 2 + 2 ) 2 - 2}{2}

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Frequently Asked Questions (FAQ)

What is the value of (sin(30))/(sin(22.5)) ?
The value of (sin(30))/(sin(22.5)) is ((2+sqrt(2))sqrt(2-\sqrt{2)})/2