What is the value of (sin(45)+csc^2(60))/(cos(30)-tan^2(45)) ?

The value of (sin(45)+csc^2(60))/(cos(30)-tan^2(45)) is -((3sqrt(2)+8)(sqrt(3)+2))/3

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(sin(45)+csc^2(60))/(cos(30)-tan^2(45))

Solution

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

Solution

- \frac{( 3 2 + 8 ) ( 3 + 2 )}{3}

Decimal Notation

- 15.23005 \dots

Solution steps

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

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:

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

csc (6 0^{\circ})

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

= \frac{2 3}{3}

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:

tan (4 5^{\circ}) = 1

tan (4 5^{\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}

= 1

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

Simplify

\frac{\frac{2}{2} + ( \frac{2 3}{3} ) ^{2}}{\frac{3}{2} - 1 ^{2}} : - \frac{( 3 2 + 8 ) ( 3 + 2 )}{3}

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

Apply rule

1^{a} = 1

1^{2} = 1

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

Join

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

\frac{3}{2} - 1

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{3 - 2}{2}

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

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

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

Apply exponent rule:

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

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

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(23)^{2} = 2^{2} (3)^{2}

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

(3)^{2} : 3

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

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

Cancel the common factor:

3

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

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

2^{2} = 4

= \frac{\frac{2}{2} + \frac{4}{3}}{\frac{3 - 2}{2}}

Apply the fraction rule:

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

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

Join

\frac{2}{2} + \frac{4}{3} : \frac{3 2 + 8}{6}

\frac{2}{2} + \frac{4}{3}

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

2

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{2}{2} :

multiply the denominator and numerator by

3

\frac{2}{2} = \frac{2 \cdot 3}{2 \cdot 3} = \frac{2 \cdot 3}{6}

For

\frac{4}{3} :

multiply the denominator and numerator by

2

\frac{4}{3} = \frac{4 \cdot 2}{3 \cdot 2} = \frac{8}{6}

= \frac{2 \cdot 3}{6} + \frac{8}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{2 \cdot 3 + 8}{6}

= \frac{2 \cdot \frac{3 2 + 8}{6}}{3 - 2}

Multiply

\frac{2 \cdot 3 + 8}{6} \cdot 2 : \frac{3 2 + 8}{3}

\frac{2 \cdot 3 + 8}{6} \cdot 2

Multiply fractions:

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

= \frac{( 2 \cdot 3 + 8 ) \cdot 2}{6}

Cancel the common factor:

2

= \frac{3 2 + 8}{3}

= \frac{\frac{3 2 + 8}{3}}{3 - 2}

Apply the fraction rule:

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

= \frac{3 2 + 8}{3 ( 3 - 2 )}

Rationalize

\frac{3 2 + 8}{3 ( 3 - 2 )} : - \frac{( 2 + 3 ) ( 3 2 + 8 )}{3}

\frac{3 2 + 8}{3 ( 3 - 2 )}

Multiply by the conjugate

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

= \frac{( 3 2 + 8 ) ( 3 + 2 )}{3 ( 3 - 2 ) ( 3 + 2 )}

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

3 (3 - 2) (3 + 2)

Expand

(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} = 4

2^{2}

2^{2} = 4

= 4

= 3 - 4

Subtract the numbers:

3 - 4 = - 1

= - 1

= - 1

= 3 (- 1)

Expand

3 (- 1) : - 3

3 (- 1)

Distribute parentheses

= 3 (- 1)

Apply minus-plus rules

+ (- a) = - a

= - 3 \cdot 1

Multiply the numbers:

3 \cdot 1 = 3

= - 3

= - 3

= \frac{( 3 2 + 8 ) ( 3 + 2 )}{- 3}

Apply the fraction rule:

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

= - \frac{( 3 2 + 8 ) ( 3 + 2 )}{3}

= - \frac{( 3 2 + 8 ) ( 2 + 3 )}{3}

= - \frac{( 3 2 + 8 ) ( 3 + 2 )}{3}

Popular Examples

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sin (16.2 6^{\circ})

0.2cos(30)

0.2 cos (3 0^{\circ})

arctan(0.37)

arctan (0.37)

arctan(-9/8)

arctan (- \frac{9}{8})

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cos^{2} (22 5^{\circ})

Frequently Asked Questions (FAQ)

What is the value of (sin(45)+csc^2(60))/(cos(30)-tan^2(45)) ?
The value of (sin(45)+csc^2(60))/(cos(30)-tan^2(45)) is -((3sqrt(2)+8)(sqrt(3)+2))/3