What is the value of (cos(30))/(sin(60)+cos(45)) ?

The value of (cos(30))/(sin(60)+cos(45)) is 3-sqrt(6)

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(cos(30))/(sin(60)+cos(45))

Solution

\frac{cos ( 3 0 ^{\circ} )}{sin ( 6 0 ^{\circ} ) + cos ( 4 5 ^{\circ} )}

Solution

3 - 6

Decimal Notation

0.55051 \dots

Solution steps

\frac{cos ( 3 0 ^{\circ} )}{sin ( 6 0 ^{\circ} ) + cos ( 4 5 ^{\circ} )}

Use the following trivial identity:

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

cos (3 0^{\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{3}{2}

Use the following trivial identity:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 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{3}{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{\frac{3}{2}}{\frac{3}{2} + \frac{2}{2}}

Simplify

\frac{\frac{3}{2}}{\frac{3}{2} + \frac{2}{2}} : 3 - 6

\frac{\frac{3}{2}}{\frac{3}{2} + \frac{2}{2}}

Combine the fractions

\frac{3}{2} + \frac{2}{2} : \frac{3 + 2}{2}

Apply rule

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

= \frac{3 + 2}{2}

= \frac{\frac{3}{2}}{\frac{3 + 2}{2}}

Divide fractions:

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

= \frac{3 \cdot 2}{2 ( 3 + 2 )}

Cancel the common factor:

2

= \frac{3}{3 + 2}

Rationalize

\frac{3}{3 + 2} : 3 - 6

\frac{3}{3 + 2}

Multiply by the conjugate

\frac{3 - 2}{3 - 2}

= \frac{3 ( 3 - 2 )}{( 3 + 2 ) ( 3 - 2 )}

3 (3 - 2) = 3 - 6

3 (3 - 2)

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = 3, c = 2

= 33 - 32

Simplify

33 - 32 : 3 - 6

33 - 32

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

32 = 6

32

Apply radical rule:

a b = a \cdot b

32 = 3 \cdot 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

= 3 - 6

= 3 - 6

(3 + 2) (3 - 2) = 1

(3 + 2) (3 - 2)

Apply Difference of Two Squares Formula:

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

a = 3, b = 2

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

Simplify

(3)^{2} - (2)^{2} : 1

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

(3)^{2} = 3

(3)^{2}

Apply radical rule:

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

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 3^{\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

= 3

(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

= 3 - 2

Subtract the numbers:

3 - 2 = 1

= 1

= 1

= \frac{3 - 6}{1}

Apply rule

\frac{a}{1} = a

= 3 - 6

= 3 - 6

= 3 - 6

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Frequently Asked Questions (FAQ)

What is the value of (cos(30))/(sin(60)+cos(45)) ?
The value of (cos(30))/(sin(60)+cos(45)) is 3-sqrt(6)