What is the value of sin^2(240) ?

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sin^2(240)

Solution

sin^{2} (24 0^{\circ})

Solution

\frac{3}{4}

Decimal Notation

0.75

Solution steps

sin^{2} (24 0^{\circ})

Rewrite using trig identities:

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

sin^{2} (24 0^{\circ})

Use the following identity:

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

Use the Double Angle identity

cos (2 θ) = 1 - 2 sin^{2} (θ)

Switch sides

2 sin^{2} (θ) - 1 = - cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 - cos (2 θ)

Divide both sides by

2

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

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

Simplify

= \frac{1 - cos ( 48 0 ^{\circ} )}{2}

cos (48 0^{\circ}) = cos (12 0^{\circ})

cos (48 0^{\circ})

Rewrite

48 0^{\circ}

36 0^{\circ} + 12 0^{\circ}

= cos (36 0^{\circ} + 12 0^{\circ})

Apply the periodicity of

cos

cos (x + 36 0^{\circ}) = cos (x)

cos (36 0^{\circ} + 12 0^{\circ}) = cos (12 0^{\circ})

= cos (12 0^{\circ})

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

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

Simplify

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

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

Apply rule

- (- a) = a

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

Join

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

1 + \frac{1}{2}

Convert element to fraction:

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

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

1 \cdot 2 + 1 = 3

1 \cdot 2 + 1

Multiply the numbers:

1 \cdot 2 = 2

= 2 + 1

Add the numbers:

2 + 1 = 3

= 3

= \frac{3}{2}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3}{4}

= \frac{3}{4}

Popular Examples

arctan(0/10)

arctan (\frac{0}{10})

1/(cot(0))

\frac{1}{cot ( 0 )}

1/2 sin(1)

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

6cos(480)

6 cos (48 0^{\circ})

2-sin(pi)

2 - sin (π)

Frequently Asked Questions (FAQ)

What is the value of sin^2(240) ?
The value of sin^2(240) is 3/4