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Popular >

(23)/(cos(63))

Solution

\frac{23}{cos ( 6 3 ^{\circ} )}

Solution

\frac{23 2 ( 10 - 2 5 + 4 ) ( 3 - 5 ) - 2 5 - 5 + 4}{4}

Decimal Notation

50.66185 \dots

Solution steps

\frac{23}{cos ( 6 3 ^{\circ} )}

Rewrite using trig identities:

cos (6 3^{\circ}) = \frac{2 4 - 2 5 - 5}{4}

cos (6 3^{\circ})

Rewrite using trig identities:

\frac{1 + cos ( 12 6 ^{\circ} )}{2}

cos (6 3^{\circ})

Write

cos (6 3^{\circ})

cos (\frac{12 6 ^{\circ}}{2})

= cos (\frac{12 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 ( 12 6 ^{\circ} )}{2}

= \frac{1 + cos ( 12 6 ^{\circ} )}{2}

Rewrite using trig identities:

cos (12 6^{\circ}) = - \frac{2 5 - 5}{4}

cos (12 6^{\circ})

Rewrite using trig identities:

- sin (3 6^{\circ})

cos (12 6^{\circ})

Use the following identity:

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

= sin (9 0^{\circ} - 12 6^{\circ})

Simplify

= sin (- 3 6^{\circ})

Use the following property:

sin (- x) = - sin (x)

sin (- 3 6^{\circ}) = - sin (3 6^{\circ})

= - sin (3 6^{\circ})

= - sin (3 6^{\circ})

Rewrite using trig identities:

sin (3 6^{\circ}) = \frac{2 5 - 5}{4}

sin (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}

Square both sides

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

Use the following identity:

sin^{2} (x) = 1 - cos^{2} (x)

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

Substitute

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

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

Refine

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

Take the square root of both sides

sin (3 6^{\circ}) = \pm \frac{5 - 5}{8}

sin (3 6^{\circ})

cannot be negative

sin (3 6^{\circ}) = \frac{5 - 5}{8}

Refine

sin (3 6^{\circ}) = \frac{\frac{5 - 5}{2}}{2}

= \frac{\frac{5 - 5}{2}}{2}

Simplify

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

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

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

Simplify

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

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

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

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

Join

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

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

Convert element to fraction:

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

= \frac{1 \cdot 4}{4} - \frac{2 5 - 5}{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 - 2 5 - 5}{4}

Multiply the numbers:

1 \cdot 4 = 4

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

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

Apply the fraction rule:

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

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

Multiply the numbers:

4 \cdot 2 = 8

= \frac{4 - 2 5 - 5}{8}

= \frac{4 - 2 5 - 5}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

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

Rationalize

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

\frac{4 - 2 5 - 5}{2 2}

Multiply by the conjugate

\frac{2}{2}

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

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

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

= \frac{23}{\frac{2 4 - 2 5 - 5}{4}}

Simplify

\frac{23}{\frac{2 4 - 2 5 - 5}{4}} : \frac{23 2 ( 10 - 2 5 + 4 ) ( 3 - 5 ) - 2 5 - 5 + 4}{4}

\frac{23}{\frac{2 4 - 2 5 - 5}{4}}

Apply the fraction rule:

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

= \frac{23 \cdot 4}{2 4 - 2 5 - 5}

Multiply the numbers:

23 \cdot 4 = 92

= \frac{92}{2 - 2 5 - 5 + 4}

Factor

92 : 2^{2} \cdot 23

Factor

92 = 2^{2} \cdot 23

= \frac{2 ^{2} \cdot 23}{2 - 2 5 - 5 + 4}

Cancel

\frac{2 ^{2} \cdot 23}{2 4 - 2 5 - 5} : \frac{23 \cdot 2 ^{\frac{3}{2}}}{4 - 2 5 - 5}

\frac{2 ^{2} \cdot 23}{2 4 - 2 5 - 5}

Apply radical rule:

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

= \frac{2 ^{2} \cdot 23}{2 ^{\frac{1}{2}} - 2 5 - 5 + 4}

Apply exponent rule:

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

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

= \frac{23 \cdot 2 ^{- \frac{1}{2} + 2}}{4 - 2 5 - 5}

Subtract the numbers:

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

= \frac{23 \cdot 2 ^{\frac{3}{2}}}{4 - 2 5 - 5}

= \frac{23 \cdot 2 ^{\frac{3}{2}}}{4 - 2 5 - 5}

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{23 \cdot 2 2}{4 - 2 5 - 5}

Multiply the numbers:

23 \cdot 2 = 46

= \frac{46 2}{4 - 2 5 - 5}

Rationalize

\frac{46 2}{4 - 2 5 - 5} : \frac{23 2 ( 3 - 5 ) ( 10 - 2 5 + 4 ) - 2 5 - 5 + 4}{4}

\frac{46 2}{4 - 2 5 - 5}

Multiply by the conjugate

\frac{4 - 2 5 - 5}{4 - 2 5 - 5}

= \frac{46 2 4 - 2 5 - 5}{4 - 2 5 - 5 4 - 2 5 - 5}

4 - 2 5 - 5 4 - 2 5 - 5 = 4 - 10 - 25

4 - 2 5 - 5 4 - 2 5 - 5

Apply radical rule:

a a = a

- 2 5 - 5 + 4 - 2 5 - 5 + 4 = 4 - 2 5 - 5

= 4 - 2 5 - 5

2 5 - 5 = 10 - 25

2 5 - 5

Apply radical rule:

a b = a \cdot b

2 5 - 5 = 2 (5 - 5)

= 2 (5 - 5)

Expand

2 (5 - 5) : 10 - 25

2 (5 - 5)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 5, c = 5

= 2 \cdot 5 - 25

Multiply the numbers:

2 \cdot 5 = 10

= 10 - 25

= 10 - 25

= 4 - 10 - 25

= \frac{46 2 4 - 2 5 - 5}{4 - 10 - 2 5}

Multiply by the conjugate

\frac{4 + 10 - 2 5}{4 + 10 - 2 5}

= \frac{46 2 4 - 2 5 - 5 ( 4 + 10 - 2 5 )}{( 4 - 10 - 2 5 ) ( 4 + 10 - 2 5 )}

(4 - 10 - 25) (4 + 10 - 25) = 6 + 25

(4 - 10 - 25) (4 + 10 - 25)

Apply Difference of Two Squares Formula:

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

a = 4, b = 10 - 25

= 4^{2} - (10 - 25)^{2}

Simplify

4^{2} - (10 - 25)^{2} : 6 + 25

4^{2} - (10 - 25)^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

(10 - 25)^{2} = 10 - 25

(10 - 25)^{2}

Apply radical rule:

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

= ((10 - 25)^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (10 - 25)^{\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

= 10 - 25

= 16 - (10 - 25)

- (10 - 25) : - 10 + 25

- (10 - 25)

Distribute parentheses

= - (10) - (- 25)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 10 + 25

= 16 - 10 + 25

Subtract the numbers:

16 - 10 = 6

= 6 + 25

= 6 + 25

= \frac{46 2 ( 4 + 10 - 2 5 ) 4 - 2 5 - 5}{6 + 2 5}

Factor

6 + 25 : 2 (3 + 5)

6 + 25

Rewrite as

= 2 \cdot 3 + 25

Factor out common term

2

= 2 (3 + 5)

= \frac{46 2 ( 4 + 10 - 2 5 ) 4 - 2 5 - 5}{2 ( 3 + 5 )}

Divide the numbers:

\frac{46}{2} = 23

= \frac{23 2 ( 10 - 2 5 + 4 ) - 2 5 - 5 + 4}{( 3 + 5 )}

Remove parentheses:

(a) = a

= \frac{23 2 ( 10 - 2 5 + 4 ) - 2 5 - 5 + 4}{3 + 5}

Multiply by the conjugate

\frac{3 - 5}{3 - 5}

= \frac{23 2 ( 10 - 2 5 + 4 ) - 2 5 - 5 + 4 ( 3 - 5 )}{( 3 + 5 ) ( 3 - 5 )}

(3 + 5) (3 - 5) = 4

(3 + 5) (3 - 5)

Apply Difference of Two Squares Formula:

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

a = 3, b = 5

= 3^{2} - (5)^{2}

Simplify

3^{2} - (5)^{2} : 4

3^{2} - (5)^{2}

3^{2} = 9

3^{2}

3^{2} = 9

= 9

(5)^{2} = 5

(5)^{2}

Apply radical rule:

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

= (5^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 5^{\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

= 5

= 9 - 5

Subtract the numbers:

9 - 5 = 4

= 4

= 4

= \frac{23 2 ( 10 - 2 5 + 4 ) ( 3 - 5 ) - 2 5 - 5 + 4}{4}

= \frac{23 2 ( 10 - 2 5 + 4 ) ( 3 - 5 ) - 2 5 - 5 + 4}{4}

= \frac{23 2 ( 10 - 2 5 + 4 ) ( 3 - 5 ) - 2 5 - 5 + 4}{4}

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