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

The value of cos(45)(sin(30)+tan(60)) is (sqrt(2)(1+2sqrt(3)))/4

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

Solution

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

Solution

\frac{2 ( 1 + 2 3 )}{4}

Decimal Notation

1.57829 \dots

Solution steps

cos (4 5^{\circ}) (sin (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:

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 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{1}{2}

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{2}{2} (\frac{1}{2} + 3)

Simplify

\frac{2}{2} (\frac{1}{2} + 3) : \frac{2 ( 1 + 2 3 )}{4}

\frac{2}{2} (\frac{1}{2} + 3)

Multiply fractions:

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

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

Apply radical rule:

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

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

Join

\frac{1}{2} + 3 : \frac{1 + 2 3}{2}

\frac{1}{2} + 3

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

Apply the fraction rule:

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

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

Rationalize

\frac{1 + 2 3}{2 2} : \frac{2 ( 1 + 2 3 )}{4}

\frac{1 + 2 3}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{( 1 + 3 \cdot 2 ) 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 ( 1 + 2 3 )}{4}

= \frac{2 ( 1 + 2 3 )}{4}

= \frac{2 ( 1 + 2 3 )}{4}

Popular Examples

cos(165)-cos(15)

cos (16 5^{\circ}) - cos (1 5^{\circ})

2sin(30)+3cos(60)-3tan(45)

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

cos(arcsin(4/5)+arccos(3/5))

cos (arcsin (\frac{4}{5}) + arccos (\frac{3}{5}))

90sin(60)

90 sin (6 0^{\circ})

cos(2)(90)

cos (2) (90)

Frequently Asked Questions (FAQ)

What is the value of cos(45)(sin(30)+tan(60)) ?
The value of cos(45)(sin(30)+tan(60)) is (sqrt(2)(1+2sqrt(3)))/4