What is the value of sin(2pi)-sin((3pi)/2) ?

The value of sin(2pi)-sin((3pi)/2) is 1

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sin(2pi)-sin((3pi)/2)

Solution

sin (2 π) - sin (\frac{3 π}{2})

Solution

1

Solution steps

sin (2 π) - sin (\frac{3 π}{2})

sin (2 π) = sin (0)

sin (2 π)

Rewrite

2 π

2 π + 0

= sin (2 π + 0)

Apply the periodicity of

sin

sin (x + 2 π) = sin (x)

sin (2 π + 0) = sin (0)

= sin (0)

= sin (0) - sin (\frac{3 π}{2})

Rewrite using trig identities:

- 2 sin (\frac{3 π}{4}) cos (\frac{3 π}{4})

sin (0) - sin (\frac{3 π}{2})

Use the Sum to Product identity:

sin (s) - sin (t) = 2 sin (\frac{s - t}{2}) cos (\frac{s + t}{2})

= 2 sin (\frac{0 - \frac{3 π}{2}}{2}) cos (\frac{0 + \frac{3 π}{2}}{2})

Simplify:

\frac{0 - \frac{3 π}{2}}{2} = - \frac{3 π}{4}

\frac{0 - \frac{3 π}{2}}{2}

0 - \frac{3 π}{2} = - \frac{3 π}{2}

= \frac{- \frac{3 π}{2}}{2}

Apply the fraction rule:

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

= - \frac{\frac{3 π}{2}}{2}

Apply the fraction rule:

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

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

= - \frac{3 π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= - \frac{3 π}{4}

Simplify:

\frac{0 + \frac{3 π}{2}}{2} = \frac{3 π}{4}

\frac{0 + \frac{3 π}{2}}{2}

0 + \frac{3 π}{2} = \frac{3 π}{2}

= \frac{\frac{3 π}{2}}{2}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 π}{4}

= 2 sin (- \frac{3 π}{4}) cos (\frac{3 π}{4})

Use the following property:

sin (- x) = - sin (x)

sin (- \frac{3 π}{4}) = - sin (\frac{3 π}{4})

= 2 (- sin (\frac{3 π}{4})) cos (\frac{3 π}{4})

Simplify

= - 2 sin (\frac{3 π}{4}) cos (\frac{3 π}{4})

= - 2 sin (\frac{3 π}{4}) cos (\frac{3 π}{4})

Use the following trivial identity:

sin (\frac{3 π}{4}) = \frac{2}{2}

sin (\frac{3 π}{4})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

Use the following trivial identity:

cos (\frac{3 π}{4}) = - \frac{2}{2}

cos (\frac{3 π}{4})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{2}{2}

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

Simplify

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

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

Apply rule

- (- a) = a

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

Multiply fractions:

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

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

Cancel the common factor:

2

= \frac{2 2}{2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

= 1

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Frequently Asked Questions (FAQ)

What is the value of sin(2pi)-sin((3pi)/2) ?
The value of sin(2pi)-sin((3pi)/2) is 1