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

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

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

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

- 2

Solution steps

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

Rewrite using trig identities:

2 sin (\frac{π}{2}) cos (π)

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

Simplify:

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

\frac{\frac{3 π}{2} - \frac{π}{2}}{2}

Combine the fractions

\frac{3 π}{2} - \frac{π}{2} : π

Apply rule

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

= \frac{3 π - π}{2}

Add similar elements:

3 π - π = 2 π

= \frac{2 π}{2}

Divide the numbers:

\frac{2}{2} = 1

= π

= \frac{π}{2}

Simplify:

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

\frac{\frac{3 π}{2} + \frac{π}{2}}{2}

Combine the fractions

\frac{3 π}{2} + \frac{π}{2} : 2 π

Apply rule

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

= \frac{3 π + π}{2}

Add similar elements:

3 π + π = 4 π

= \frac{4 π}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 π

= \frac{2 π}{2}

Divide the numbers:

\frac{2}{2} = 1

= π

= 2 sin (\frac{π}{2}) cos (π)

= 2 sin (\frac{π}{2}) cos (π)

Use the following trivial identity:

sin (\frac{π}{2}) = 1

sin (\frac{π}{2})

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}

= 1

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

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}

= (- 1)

= 2 \cdot 1 \cdot (- 1)

Simplify

= - 2

8 (cos (15 0^{\circ}) + i sin (15 0^{\circ}))

e^{1} sin (\frac{π}{2})

sin (- \frac{7}{4} π)

4 \cdot cos (4 0^{\circ})

sinh (- π)

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