What is the value of cos(247.5) ?

The value of cos(247.5) is -(sqrt(2-\sqrt{2)})/2

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cos(247.5)

Solution

cos (247. 5^{\circ})

Solution

- \frac{2 - 2}{2}

Decimal Notation

- 0.38268 \dots

Solution steps

cos (247. 5^{\circ})

Rewrite using trig identities:

- \frac{1 + cos ( 13 5 ^{\circ} )}{2}

cos (247. 5^{\circ})

Write

cos (247. 5^{\circ})

cos (\frac{49 5 ^{\circ}}{2})

= cos (\frac{49 5 ^{\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 ( 49 5 ^{\circ} )}{2}

cos (49 5^{\circ}) = cos (13 5^{\circ})

cos (49 5^{\circ})

Rewrite

49 5^{\circ}

36 0^{\circ} + 13 5^{\circ}

= cos (36 0^{\circ} + 13 5^{\circ})

Apply the periodicity of

cos

cos (x + 36 0^{\circ}) = cos (x)

cos (36 0^{\circ} + 13 5^{\circ}) = cos (13 5^{\circ})

= cos (13 5^{\circ})

= - \frac{1 + cos ( 13 5 ^{\circ} )}{2}

= - \frac{1 + cos ( 13 5 ^{\circ} )}{2}

Use the following trivial identity:

cos (13 5^{\circ}) = - \frac{2}{2}

cos (13 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}

Simplify

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

\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{2 - 2}{2}

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Frequently Asked Questions (FAQ)

What is the value of cos(247.5) ?
The value of cos(247.5) is -(sqrt(2-\sqrt{2)})/2