What is the value of ((sin(150)))/((1+cos(150))) ?

The value of ((sin(150)))/((1+cos(150))) is 2+sqrt(3)

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((sin(150)))/((1+cos(150)))

Solution

\frac{( sin ( 15 0 ^{\circ} ) )}{( 1 + cos ( 15 0 ^{\circ} ) )}

Solution

2 + 3

Decimal Notation

3.73205 \dots

Solution steps

\frac{( sin ( 15 0 ^{\circ} ) )}{( 1 + cos ( 15 0 ^{\circ} ) )}

= \frac{sin ( 15 0 ^{\circ} )}{1 + cos ( 15 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:

cos (15 0^{\circ}) = - \frac{3}{2}

cos (15 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}

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

Simplify

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

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

Apply the fraction rule:

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

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

Multiply

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 2 - 3

= \frac{1}{2 - 3}

Rationalize

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

\frac{1}{2 - 3}

Multiply by the conjugate

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

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

1 \cdot (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 + 3}{1}

Apply rule

\frac{a}{1} = a

= 2 + 3

= 2 + 3

= 2 + 3

Popular Examples

((tan(70)+tan(50)))/((1-tan(70)tan(50)))

\frac{( tan ( 7 0 ^{\circ} ) + tan ( 5 0 ^{\circ} ) )}{( 1 - tan ( 7 0 ^{\circ} ) tan ( 5 0 ^{\circ} ) )}

ln(2+tan((3pi)/(16)))

ln (2 + tan (\frac{3 π}{16}))

arcsin(4718)

arcsin (4718)

(1)(1-10/100+1/(2pi)(sin(2pi)(10/100)))

(1) (1 - \frac{10}{100} + \frac{1}{2 π} (sin (2 π) (\frac{10}{100})))

cos(pi)+i*sin(pi)

cos (π) + i \cdot sin (π)

Frequently Asked Questions (FAQ)

What is the value of ((sin(150)))/((1+cos(150))) ?
The value of ((sin(150)))/((1+cos(150))) is 2+sqrt(3)