What is the value of 2/(2+cos(45)) ?

The value of 2/(2+cos(45)) is (2(4-sqrt(2)))/7

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2/(2+cos(45))

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

\frac{2 ( 4 - 2 )}{7}

Decimal Notation

0.73879 \dots

Solution steps

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

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

Simplify

\frac{2}{2 + \frac{2}{2}} : \frac{2 ( 4 - 2 )}{7}

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

Join

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

2 + \frac{2}{2}

Convert element to fraction:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 + 2}{2}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4}{4 + 2}

Rationalize

\frac{4}{4 + 2} : \frac{2 ( 4 - 2 )}{7}

\frac{4}{4 + 2}

Multiply by the conjugate

\frac{4 - 2}{4 - 2}

= \frac{4 ( 4 - 2 )}{( 4 + 2 ) ( 4 - 2 )}

(4 + 2) (4 - 2) = 14

(4 + 2) (4 - 2)

Apply Difference of Two Squares Formula:

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

a = 4, b = 2

= 4^{2} - (2)^{2}

Simplify

4^{2} - (2)^{2} : 14

4^{2} - (2)^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

(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

= 16 - 2

Subtract the numbers:

16 - 2 = 14

= 14

= 14

= \frac{4 ( 4 - 2 )}{14}

Cancel the common factor:

2

= \frac{2 ( 4 - 2 )}{7}

= \frac{2 ( 4 - 2 )}{7}

= \frac{2 ( 4 - 2 )}{7}

arccos (\frac{6}{3 14})

cos (- (π))

tan^{2} (3 4^{\circ})

sec^{2} (1.2)

- 4 cos (- π)

What is the value of 2/(2+cos(45)) ?
The value of 2/(2+cos(45)) is (2(4-sqrt(2)))/7