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

The value of (cos(45)*cos(30)+sin(45)*sin(60))/(tan(30)*tan(60)) is sqrt(3/2)

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Popular >

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

Solution

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

Solution

\frac{3}{2}

Decimal Notation

1.22474 \dots

Solution steps

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

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

sin (4 5^{\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{2}{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:

tan (3 0^{\circ}) = \frac{3}{3}

tan (3 0^{\circ})

tan (x)

periodicity table with

18 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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= \frac{3}{3}

Use the following trivial identity:

tan (6 0^{\circ}) = 3

tan (6 0^{\circ})

tan (x)

periodicity table with

18 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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= 3

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

Simplify

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

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

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

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

Factor out common term

\frac{3}{2}

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

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

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

Apply rule

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

= \frac{2 + 2}{2}

Factor

2 + 2 : 22

2 + 2

Factor out common term

2

= 2 (1 + 1)

Refine

= 22

= \frac{2 2}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2

= 2 \frac{3}{2}

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

Multiply

\frac{3}{3} 3 : 1

\frac{3}{3} 3

Multiply fractions:

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

= \frac{3 3}{3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

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

Multiply

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

2 \frac{3}{2}

Multiply fractions:

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

= \frac{3 2}{2}

Apply radical rule:

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

= \frac{3 \cdot 2 ^{\frac{1}{2}}}{2}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{3}{2}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{3}{2}

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

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

\frac{3}{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{2}

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

Apply the fraction rule:

\frac{a}{1} = a

= \frac{3}{2}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{3}{2}

= \frac{3}{2}

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Frequently Asked Questions (FAQ)

What is the value of (cos(45)*cos(30)+sin(45)*sin(60))/(tan(30)*tan(60)) ?
The value of (cos(45)*cos(30)+sin(45)*sin(60))/(tan(30)*tan(60)) is sqrt(3/2)