What is the value of (sin(30)-cos(30))(sin(30)+cos(30)) ?

The value of (sin(30)-cos(30))(sin(30)+cos(30)) is -1/2

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(sin(30)-cos(30))(sin(30)+cos(30))

Solution

(sin (3 0^{\circ}) - cos (3 0^{\circ})) (sin (3 0^{\circ}) + cos (3 0^{\circ}))

Solution

- \frac{1}{2}

Decimal Notation

- 0.5

Solution steps

(sin (3 0^{\circ}) - cos (3 0^{\circ})) (sin (3 0^{\circ}) + cos (3 0^{\circ}))

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 (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}

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

Simplify

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

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

Simplify

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

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

Apply rule

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

= \frac{1 - 3}{2}

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

Combine the fractions

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

Apply rule

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

= \frac{1 + 3}{2}

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

Remove parentheses:

(a) = a

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

Multiply fractions:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{( 1 - 3 ) ( 1 + 3 )}{4}

Expand

(1 - 3) (1 + 3) : - 2

(1 - 3) (1 + 3)

Apply Difference of Two Squares Formula:

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

a = 1, b = 3

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

Simplify

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

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

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - (3)^{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

= 1 - 3

Subtract the numbers:

1 - 3 = - 2

= - 2

= - 2

= \frac{- 2}{4}

Apply the fraction rule:

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

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

= - \frac{1}{2}

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Frequently Asked Questions (FAQ)

What is the value of (sin(30)-cos(30))(sin(30)+cos(30)) ?
The value of (sin(30)-cos(30))(sin(30)+cos(30)) is -1/2