What is the value of (sin(150)+sin(30))/(cos(120)+cos(240)) ?

The value of (sin(150)+sin(30))/(cos(120)+cos(240)) is -1

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(sin(150)+sin(30))/(cos(120)+cos(240))

Solution

\frac{sin ( 15 0 ^{\circ} ) + sin ( 3 0 ^{\circ} )}{cos ( 12 0 ^{\circ} ) + cos ( 24 0 ^{\circ} )}

Solution

- 1

Solution steps

\frac{sin ( 15 0 ^{\circ} ) + sin ( 3 0 ^{\circ} )}{cos ( 12 0 ^{\circ} ) + cos ( 24 0 ^{\circ} )}

Rewrite using trig identities:

cos (24 0^{\circ}) = 2 cos^{2} (12 0^{\circ}) - 1

cos (24 0^{\circ})

Write

cos (24 0^{\circ})

cos (2 \cdot 12 0^{\circ})

= cos (2 \cdot 12 0^{\circ})

Use the Double Angle identity:

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

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

= \frac{sin ( 15 0 ^{\circ} ) + sin ( 3 0 ^{\circ} )}{cos ( 12 0 ^{\circ} ) + 2 cos ^{2} ( 12 0 ^{\circ} ) - 1}

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:

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:

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}

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

Simplify

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

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

Combine the fractions

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

Apply rule

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

= \frac{1 + 1}{2}

Add the numbers:

1 + 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

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

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

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

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

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

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

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

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

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

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

Add similar elements:

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

= \frac{1}{- 1}

Apply the fraction rule:

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

= - \frac{1}{1}

Apply rule

\frac{a}{1} = a

= - 1

= - 1

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Frequently Asked Questions (FAQ)

What is the value of (sin(150)+sin(30))/(cos(120)+cos(240)) ?
The value of (sin(150)+sin(30))/(cos(120)+cos(240)) is -1