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

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

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sin(pi/3-arcsin(-1/2))

Solution

sin (\frac{π}{3} - arcsin (- \frac{1}{2}))

Solution

1

Solution steps

sin (\frac{π}{3} - arcsin (- \frac{1}{2}))

Rewrite using trig identities:

sin (\frac{π}{3}) cos (arcsin (\frac{1}{2})) + cos (\frac{π}{3}) sin (arcsin (\frac{1}{2}))

sin (\frac{π}{3} - arcsin (- \frac{1}{2}))

Use the Angle Difference identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= sin (\frac{π}{3}) cos (arcsin (- \frac{1}{2})) - cos (\frac{π}{3}) sin (arcsin (- \frac{1}{2}))

Use the following property:

arcsin (- x) = - arcsin (x)

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

= sin (\frac{π}{3}) cos (arcsin (- \frac{1}{2})) - cos (\frac{π}{3}) sin (- arcsin (\frac{1}{2}))

Use the following property:

sin (- x) = - sin (x)

sin (- arcsin (\frac{1}{2})) = - sin (arcsin (\frac{1}{2}))

= sin (\frac{π}{3}) cos (arcsin (- \frac{1}{2})) - cos (\frac{π}{3}) (- sin (arcsin (\frac{1}{2})))

Use the following property:

arcsin (- x) = - arcsin (x)

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

= sin (\frac{π}{3}) cos (- arcsin (\frac{1}{2})) - cos (\frac{π}{3}) (- sin (arcsin (\frac{1}{2})))

Use the following property:

cos (- x) = cos (x)

cos (- arcsin (\frac{1}{2})) = cos (arcsin (\frac{1}{2}))

= sin (\frac{π}{3}) cos (arcsin (\frac{1}{2})) - cos (\frac{π}{3}) (- sin (arcsin (\frac{1}{2})))

Simplify

= sin (\frac{π}{3}) cos (arcsin (\frac{1}{2})) + cos (\frac{π}{3}) sin (arcsin (\frac{1}{2}))

= sin (\frac{π}{3}) cos (arcsin (\frac{1}{2})) + cos (\frac{π}{3}) sin (arcsin (\frac{1}{2}))

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{2}

sin (\frac{π}{3})

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{3}{2}

Rewrite using trig identities:

cos (arcsin (\frac{1}{2})) = \frac{3}{2}

cos (arcsin (\frac{1}{2}))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{3}{2}

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

cos (\frac{π}{3})

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{1}{2}

Rewrite using trig identities:

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

Use the following identity:

sin (arcsin (x)) = x

= \frac{1}{2}

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

Simplify

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

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

\frac{3}{2} \cdot \frac{3}{2} = \frac{3}{4}

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

Multiply fractions:

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

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

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3}{4}

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= \frac{3}{4} + \frac{1}{4}

Apply rule

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

= \frac{3 + 1}{4}

Add the numbers:

3 + 1 = 4

= \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= 1

= 1

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Frequently Asked Questions (FAQ)

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