What is the value of (sin(150)+cos(300))/(tan(225)-sin(300)) ?

The value of (sin(150)+cos(300))/(tan(225)-sin(300)) is 2(2-sqrt(3))

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(sin(150)+cos(300))/(tan(225)-sin(300))

Solution

\frac{sin ( 15 0 ^{\circ} ) + cos ( 30 0 ^{\circ} )}{tan ( 22 5 ^{\circ} ) - sin ( 30 0 ^{\circ} )}

Solution

2 (2 - 3)

Decimal Notation

0.53589 \dots

Solution steps

\frac{sin ( 15 0 ^{\circ} ) + cos ( 30 0 ^{\circ} )}{tan ( 22 5 ^{\circ} ) - sin ( 30 0 ^{\circ} )}

tan (22 5^{\circ}) = tan (4 5^{\circ})

tan (22 5^{\circ})

Rewrite

22 5^{\circ}

18 0^{\circ} + 4 5^{\circ}

= tan (18 0^{\circ} + 4 5^{\circ})

Apply the periodicity of

tan

tan (x + 18 0^{\circ}) = tan (x)

tan (18 0^{\circ} + 4 5^{\circ}) = tan (4 5^{\circ})

= tan (4 5^{\circ})

= \frac{sin ( 15 0 ^{\circ} ) + cos ( 30 0 ^{\circ} )}{tan ( 4 5 ^{\circ} ) - sin ( 30 0 ^{\circ} )}

Rewrite using trig identities:

cos (30 0^{\circ}) = 1 - 2 sin^{2} (15 0^{\circ})

cos (30 0^{\circ})

Write

cos (30 0^{\circ})

cos (2 \cdot 15 0^{\circ})

= cos (2 \cdot 15 0^{\circ})

Use the Double Angle identity:

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

= 1 - 2 sin^{2} (15 0^{\circ})

= \frac{sin ( 15 0 ^{\circ} ) + 1 - 2 sin ^{2} ( 15 0 ^{\circ} )}{tan ( 4 5 ^{\circ} ) - sin ( 30 0 ^{\circ} )}

Use the following trivial identity:

sin (15 0^{\circ}) = \frac{1}{2}

sin (15 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 (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

Rewrite using trig identities:

sin (30 0^{\circ}) = - \frac{3}{2}

sin (30 0^{\circ})

Rewrite using trig identities:

sin (18 0^{\circ}) cos (12 0^{\circ}) + cos (18 0^{\circ}) sin (12 0^{\circ})

sin (30 0^{\circ})

Write

sin (30 0^{\circ})

sin (18 0^{\circ} + 12 0^{\circ})

= sin (18 0^{\circ} + 12 0^{\circ})

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (18 0^{\circ}) cos (12 0^{\circ}) + cos (18 0^{\circ}) sin (12 0^{\circ})

= sin (18 0^{\circ}) cos (12 0^{\circ}) + cos (18 0^{\circ}) sin (12 0^{\circ})

Use the following trivial identity:

sin (18 0^{\circ}) = 0

sin (18 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}

= 0

Use the following trivial identity:

cos (12 0^{\circ}) = - \frac{1}{2}

cos (12 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{1}{2}

Use the following trivial identity:

cos (18 0^{\circ}) = (- 1)

cos (18 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}

= (- 1)

Use the following trivial identity:

sin (12 0^{\circ}) = \frac{3}{2}

sin (12 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}

= 0 \cdot (- \frac{1}{2}) + (- 1) \frac{3}{2}

Simplify

= - \frac{3}{2}

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

Simplify

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

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

Apply rule

- (- a) = a

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

Join

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

1 + \frac{3}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 + 3}{2}

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

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

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

Cancel the common factor:

2

= \frac{1}{2}

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

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

\frac{1}{2} + 1 - \frac{1}{2}

Group like terms

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

Add similar elements:

\frac{1}{2} - \frac{1}{2} = 0

= 1

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

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{2}{2 + 3}

Rationalize

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

\frac{2}{2 + 3}

Multiply by the conjugate

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

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

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

(2 + 3) (2 - 3)

Apply Difference of Two Squares Formula:

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

a = 2, b = 3

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

Simplify

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

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

2^{2} = 4

2^{2}

2^{2} = 4

= 4

(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

= 4 - 3

Subtract the numbers:

4 - 3 = 1

= 1

= 1

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

Apply rule

\frac{a}{1} = a

= 2 (2 - 3)

= 2 (2 - 3)

= 2 (2 - 3)

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Frequently Asked Questions (FAQ)

What is the value of (sin(150)+cos(300))/(tan(225)-sin(300)) ?
The value of (sin(150)+cos(300))/(tan(225)-sin(300)) is 2(2-sqrt(3))