Is cos(45)=(sin(2*45))/(2*sin(45)) ?

The answer to whether cos(45)=(sin(2*45))/(2*sin(45)) is True

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prove cos(45)=(sin(245))/(2sin(45))

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

cos (4 5^{\circ})

Simplify

= \frac{2}{2}

Manipulating right side

\frac{sin ( 2 c d o t 4 5 ^{\circ} )}{2 sin ( 4 5 ^{\circ} )}

Simplify

\frac{sin ( 2 c d o t 4 5 ^{\circ} )}{2 sin ( 4 5 ^{\circ} )} : \frac{2}{2}

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

sin (2 \cdot 4 5^{\circ}) = 1

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

Simplify:

2 \cdot 4 5^{\circ} = 9 0^{\circ}

2 \cdot 4 5^{\circ}

Multiply fractions:

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

= 9 0^{\circ}

Cancel the common factor:

2

= 9 0^{\circ}

= sin (9 0^{\circ})

Use the following trivial identity:

sin (9 0^{\circ}) = 1

sin (9 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}

= 1

= 1

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

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

2 sin (4 5^{\circ})

Simplify

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

sin (4 5^{\circ})

Use the following trivial identity:

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

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}

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

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

Simplify

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

Multiply

2 \cdot \frac{2}{2} : 2

2 \cdot \frac{2}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 2

= \frac{1}{2}

Rationalize

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

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

= \frac{2}{2}

= \frac{2}{2}

= \frac{2}{2}

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is cos(45)=(sin(2*45))/(2*sin(45)) ?
The answer to whether cos(45)=(sin(2*45))/(2*sin(45)) is True

prove cos(45)=(sin(2*45))/(2*sin(45))

Solution

Solution

Popular Examples

Frequently Asked Questions (FAQ)

Is cos(45)=(sin(2*45))/(2*sin(45)) ?

prove cos(45)=(sin(245))/(2sin(45))

Is cos(45)=(sin(245))/(2sin(45)) ?