What is the value of sin(arccos(2/3)-arctan(1/2)) ?

The value of sin(arccos(2/3)-arctan(1/2)) is 2/3-(2sqrt(5))/(15)

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sin(arccos(2/3)-arctan(1/2))

Solution

sin (arccos (\frac{2}{3}) - arctan (\frac{1}{2}))

Solution

\frac{2}{3} - \frac{2 5}{15}

Decimal Notation

0.36852 \dots

Solution steps

sin (arccos (\frac{2}{3}) - arctan (\frac{1}{2}))

Rewrite using trig identities:

sin (arccos (\frac{2}{3})) cos (arctan (\frac{1}{2})) - cos (arccos (\frac{2}{3})) sin (arctan (\frac{1}{2}))

sin (arccos (\frac{2}{3}) - arctan (\frac{1}{2}))

Use the Angle Difference identity:

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

= sin (arccos (\frac{2}{3})) cos (arctan (\frac{1}{2})) - cos (arccos (\frac{2}{3})) sin (arctan (\frac{1}{2}))

= sin (arccos (\frac{2}{3})) cos (arctan (\frac{1}{2})) - cos (arccos (\frac{2}{3})) sin (arctan (\frac{1}{2}))

Rewrite using trig identities:

sin (arccos (\frac{2}{3})) = \frac{5}{3}

sin (arccos (\frac{2}{3}))

Rewrite using trig identities:

sin (arccos (\frac{2}{3})) = 1 - (\frac{2}{3})^{2}

Use the following identity:

sin (arccos (x)) = 1 - x^{2}

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

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

Simplify

= \frac{5}{3}

Rewrite using trig identities:

cos (arctan (\frac{1}{2})) = \frac{2 5}{5}

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

Rewrite using trig identities:

cos (arctan (\frac{1}{2})) = \frac{1 + ( \frac{1}{2} ) ^{2}}{1 + ( \frac{1}{2} ) ^{2}}

Use the following identity:

cos (arctan (x)) = \frac{1 + x ^{2}}{1 + x ^{2}}

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

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

Simplify

= \frac{2 5}{5}

Rewrite using trig identities:

cos (arccos (\frac{2}{3})) = \frac{2}{3}

Use the following identity:

cos (arccos (x)) = x

= \frac{2}{3}

Rewrite using trig identities:

sin (arctan (\frac{1}{2})) = \frac{5}{5}

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

Rewrite using trig identities:

sin (arctan (\frac{1}{2})) = \frac{( \frac{1}{2} ) 1 + ( \frac{1}{2} ) ^{2}}{1 + ( \frac{1}{2} ) ^{2}}

Use the following identity:

sin (arctan (x)) = \frac{x 1 + x ^{2}}{1 + x ^{2}}

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

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

Simplify

= \frac{5}{5}

= \frac{5}{3} \cdot \frac{2 5}{5} - \frac{2}{3} \cdot \frac{5}{5}

Simplify

\frac{5}{3} \cdot \frac{2 5}{5} - \frac{2}{3} \cdot \frac{5}{5} : \frac{2}{3} - \frac{2 5}{15}

\frac{5}{3} \cdot \frac{2 5}{5} - \frac{2}{3} \cdot \frac{5}{5}

\frac{5}{3} \cdot \frac{2 5}{5} = \frac{2}{3}

\frac{5}{3} \cdot \frac{2 5}{5}

Multiply fractions:

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

= \frac{5 \cdot 2 5}{3 \cdot 5}

5 \cdot 25 = 10

5 \cdot 25

Apply radical rule:

a a = a

55 = 5

= 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 10

= \frac{10}{3 \cdot 5}

Multiply the numbers:

3 \cdot 5 = 15

= \frac{10}{15}

Cancel the common factor:

5

= \frac{2}{3}

\frac{2}{3} \cdot \frac{5}{5} = \frac{2 5}{15}

\frac{2}{3} \cdot \frac{5}{5}

Multiply fractions:

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

= \frac{2 5}{3 \cdot 5}

Multiply the numbers:

3 \cdot 5 = 15

= \frac{2 5}{15}

= \frac{2}{3} - \frac{2 5}{15}

= \frac{2}{3} - \frac{2 5}{15}

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Frequently Asked Questions (FAQ)

What is the value of sin(arccos(2/3)-arctan(1/2)) ?
The value of sin(arccos(2/3)-arctan(1/2)) is 2/3-(2sqrt(5))/(15)