What is the value of tan(arcsin(-1/2)+arccos(-(sqrt(3))/2)) ?

The value of tan(arcsin(-1/2)+arccos(-(sqrt(3))/2)) is -sqrt(3)

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tan(arcsin(-1/2)+arccos(-(sqrt(3))/2))

Solution

tan (arcsin (- \frac{1}{2}) + arccos (- \frac{3}{2}))

Solution

- 3

Decimal Notation

- 1.73205 \dots

Solution steps

tan (arcsin (- \frac{1}{2}) + arccos (- \frac{3}{2}))

Use the following property:

arcsin (- x) = - arcsin (x)

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

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

Rewrite using trig identities:

\frac{tan ( arccos ( - \frac{3}{2} ) ) - tan ( arcsin ( \frac{1}{2} ) )}{1 + tan ( arccos ( - \frac{3}{2} ) ) tan ( arcsin ( \frac{1}{2} ) )}

tan (arccos (- \frac{3}{2}) - arcsin (\frac{1}{2}))

Use the Angle Difference identity:

tan (s - t) = \frac{tan ( s ) - tan ( t )}{1 + tan ( s ) tan ( t )}

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

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

Rewrite using trig identities:

tan (arccos (- \frac{3}{2})) = - \frac{3}{3}

tan (arccos (- \frac{3}{2}))

Rewrite using trig identities:

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

Use the following identity:

tan (arccos (x)) = \frac{1 - x ^{2}}{x}

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

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

Simplify

= - \frac{3}{3}

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

tan (arcsin (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{3}{3}

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

Simplify

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

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

Combine the fractions

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

Apply rule

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

= \frac{- 3 - 3}{3}

Add similar elements:

- 3 - 3 = - 23

= \frac{- 2 3}{3}

Apply the fraction rule:

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

= - \frac{2 3}{3}

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

Remove parentheses:

(- a) = - a

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

Apply the fraction rule:

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

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

Apply the fraction rule:

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

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

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

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

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

Multiply fractions:

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

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

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

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

Multiply the numbers:

3 \cdot 3 = 9

= \frac{3}{9}

Cancel the common factor:

3

= \frac{1}{3}

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

Join

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

1 - \frac{1}{3}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

1 \cdot 3 - 1 = 2

1 \cdot 3 - 1

Multiply the numbers:

1 \cdot 3 = 3

= 3 - 1

Subtract the numbers:

3 - 1 = 2

= 2

= \frac{2}{3}

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

Multiply

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

3 \cdot \frac{2}{3}

Multiply fractions:

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

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

Cancel the common factor:

3

= 2

= - \frac{2 3}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 3

= - 3

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Frequently Asked Questions (FAQ)

What is the value of tan(arcsin(-1/2)+arccos(-(sqrt(3))/2)) ?
The value of tan(arcsin(-1/2)+arccos(-(sqrt(3))/2)) is -sqrt(3)